Simplify SFORB analytical solution, whitespace

I do not know why the formulae for b1 and b2 on page 64 of FOCUS
kinetics (2014) were not simplified. Clearly, the term

  k12 * k21 - (k12 + k1output) * k21)

can be simplified to

  - k1output * k21

The test for equivalence of DFOP and SFORB fits verifies that the change
is OK. I also removed trailing whitespaces, substituted tab characters
by two whitespaces and removed indenting of text in paragraphs
describing parameters in roxygen comments to unify formatting.

4 files changed, 88 insertions, 88 deletions

diff --git a/R/parent_solutions.R b/R/parent_solutions.R
index 04226b73..f11cbff9 100644
--- a/R/parent_solutions.R
+++ b/R/parent_solutions.R
@@ -7,7 +7,7 @@
 #' @param parent_0 Starting value for the response variable at time zero.
 #' @param k Kinetic rate constant.
 #' @return The value of the response variable at time \code{t}.
-#' @references 
+#' @references
 #' FOCUS (2006) \dQuote{Guidance Document on Estimating Persistence
 #'   and Degradation Kinetics from Environmental Fate Studies on Pesticides in
 #'   EU Registration} Report of the FOCUS Work Group on Degradation Kinetics,
@@ -25,33 +25,33 @@
 #' @export
 SFO.solution <- function(t, parent_0, k)
 {
-	parent = parent_0 * exp(-k * t)
+  parent = parent_0 * exp(-k * t)
 }
 
 #' First-Order Multi-Compartment kinetics
-#' 
+#'
 #' Function describing exponential decline from a defined starting value, with
 #' a decreasing rate constant.
-#' 
+#'
 #' The form given here differs slightly from the original reference by
 #' Gustafson and Holden (1990). The parameter \code{beta} corresponds to 1/beta
 #' in the original equation.
-#' 
+#'
 #' @family parent solutions
-#' @inherit SFO.solution 
+#' @inherit SFO.solution
 #' @param alpha Shape parameter determined by coefficient of variation of rate
-#'   constant values.
+#' constant values.
 #' @param beta Location parameter.
 #' @note The solution of the FOMC kinetic model reduces to the
-#'   \code{\link{SFO.solution}} for large values of \code{alpha} and
-#'   \code{beta} with \eqn{k = \frac{\beta}{\alpha}}{k = beta/alpha}.
-#' @references 
+#' \code{\link{SFO.solution}} for large values of \code{alpha} and \code{beta}
+#' with \eqn{k = \frac{\beta}{\alpha}}{k = beta/alpha}.
+#' @references
 #' FOCUS (2006) \dQuote{Guidance Document on Estimating Persistence
 #'   and Degradation Kinetics from Environmental Fate Studies on Pesticides in
 #'   EU Registration} Report of the FOCUS Work Group on Degradation Kinetics,
 #'   EC Document Reference Sanco/10058/2005 version 2.0, 434 pp,
 #'   \url{http://esdac.jrc.ec.europa.eu/projects/degradation-kinetics}
-#' 
+#'
 #' FOCUS (2014) \dQuote{Generic guidance for Estimating Persistence
 #'   and Degradation Kinetics from Environmental Fate Studies on Pesticides in
 #'   EU Registration} Report of the FOCUS Work Group on Degradation Kinetics,
@@ -62,119 +62,119 @@ SFO.solution <- function(t, parent_0, k)
 #'   A new model based on spatial variability. \emph{Environmental Science and
 #'   Technology} \bold{24}, 1032-1038
 #' @examples
-#' 
+#'
 #'   plot(function(x) FOMC.solution(x, 100, 10, 2), 0, 2, ylim = c(0, 100))
-#' 
+#'
 #' @export
 FOMC.solution <- function(t, parent_0, alpha, beta)
 {
-	parent = parent_0 / (t/beta + 1)^alpha
+  parent = parent_0 / (t/beta + 1)^alpha
 }
 
 #' Indeterminate order rate equation kinetics
-#' 
+#'
 #' Function describing exponential decline from a defined starting value, with
 #' a concentration dependent rate constant.
-#' 
+#'
 #' @family parent solutions
-#' @inherit SFO.solution 
+#' @inherit SFO.solution
 #' @param k__iore Rate constant. Note that this depends on the concentration
-#'   units used.
+#' units used.
 #' @param N Exponent describing the nonlinearity of the rate equation
 #' @note The solution of the IORE kinetic model reduces to the
-#'   \code{\link{SFO.solution}} if N = 1.  The parameters of the IORE model can
-#'   be transformed to equivalent parameters of the FOMC mode - see the NAFTA
-#'   guidance for details.
+#' \code{\link{SFO.solution}} if N = 1.  The parameters of the IORE model can
+#' be transformed to equivalent parameters of the FOMC mode - see the NAFTA
+#' guidance for details.
 #' @references NAFTA Technical Working Group on Pesticides (not dated) Guidance
-#'   for Evaluating and Calculating Degradation Kinetics in Environmental Media
+#' for Evaluating and Calculating Degradation Kinetics in Environmental Media
 #' @examples
-#' 
+#'
 #'   plot(function(x) IORE.solution(x, 100, 0.2, 1.3), 0, 2, ylim = c(0, 100))
 #'   \dontrun{
 #'     fit.fomc <- mkinfit("FOMC", FOCUS_2006_C, quiet = TRUE)
 #'     fit.iore <- mkinfit("IORE", FOCUS_2006_C, quiet = TRUE)
 #'     fit.iore.deS <- mkinfit("IORE", FOCUS_2006_C, solution_type = "deSolve", quiet = TRUE)
-#' 
-#'     print(data.frame(fit.fomc$par, fit.iore$par, fit.iore.deS$par, 
+#'
+#'     print(data.frame(fit.fomc$par, fit.iore$par, fit.iore.deS$par,
 #'                      row.names = paste("model par", 1:4)))
-#'     print(rbind(fomc = endpoints(fit.fomc)$distimes, iore = endpoints(fit.iore)$distimes, 
+#'     print(rbind(fomc = endpoints(fit.fomc)$distimes, iore = endpoints(fit.iore)$distimes,
 #'                 iore.deS = endpoints(fit.iore)$distimes))
 #'   }
-#' 
+#'
 #' @export
 IORE.solution <- function(t, parent_0, k__iore, N)
 {
-	parent = (parent_0^(1 - N) - (1 - N) * k__iore * t)^(1/(1 - N))
+  parent = (parent_0^(1 - N) - (1 - N) * k__iore * t)^(1/(1 - N))
 }
 
 #' Double First-Order in Parallel kinetics
-#' 
+#'
 #' Function describing decline from a defined starting value using the sum of
 #' two exponential decline functions.
-#' 
+#'
 #' @family parent solutions
-#' @inherit SFO.solution 
+#' @inherit SFO.solution
 #' @param t Time.
 #' @param k1 First kinetic constant.
 #' @param k2 Second kinetic constant.
 #' @param g Fraction of the starting value declining according to the first
-#'   kinetic constant.
+#' kinetic constant.
 #' @examples
-#' 
+#'
 #'   plot(function(x) DFOP.solution(x, 100, 5, 0.5, 0.3), 0, 4, ylim = c(0,100))
-#' 
+#'
 #' @export
 DFOP.solution <- function(t, parent_0, k1, k2, g)
 {
-	parent = g * parent_0 * exp(-k1 * t) +
-		 (1 - g) * parent_0 * exp(-k2 * t)
+  parent = g * parent_0 * exp(-k1 * t) +
+     (1 - g) * parent_0 * exp(-k2 * t)
 }
 
 #' Hockey-Stick kinetics
-#' 
+#'
 #' Function describing two exponential decline functions with a break point
 #' between them.
-#' 
+#'
 #' @family parent solutions
-#' @inherit DFOP.solution 
+#' @inherit DFOP.solution
 #' @param tb Break point. Before this time, exponential decline according to
-#'   \code{k1} is calculated, after this time, exponential decline proceeds
-#'   according to \code{k2}.
+#' \code{k1} is calculated, after this time, exponential decline proceeds
+#' according to \code{k2}.
 #' @examples
-#' 
+#'
 #'   plot(function(x) HS.solution(x, 100, 2, 0.3, 0.5), 0, 2, ylim=c(0,100))
-#' 
+#'
 #' @export
 HS.solution <- function(t, parent_0, k1, k2, tb)
 {
-	parent = ifelse(t <= tb, 
-		parent_0 * exp(-k1 * t),
-		parent_0 * exp(-k1 * tb) * exp(-k2 * (t - tb)))
+  parent = ifelse(t <= tb,
+    parent_0 * exp(-k1 * t),
+    parent_0 * exp(-k1 * tb) * exp(-k2 * (t - tb)))
 }
 
 #' Single First-Order Reversible Binding kinetics
-#' 
+#'
 #' Function describing the solution of the differential equations describing
 #' the kinetic model with first-order terms for a two-way transfer from a free
 #' to a bound fraction, and a first-order degradation term for the free
 #' fraction.  The initial condition is a defined amount in the free fraction
 #' and no substance in the bound fraction.
-#' 
+#'
 #' @family parent solutions
-#' @inherit SFO.solution 
+#' @inherit SFO.solution
 #' @param k_12 Kinetic constant describing transfer from free to bound.
 #' @param k_21 Kinetic constant describing transfer from bound to free.
 #' @param k_1output Kinetic constant describing degradation of the free
-#'   fraction.
+#' fraction.
 #' @return The value of the response variable, which is the sum of free and
-#'   bound fractions at time \code{t}.
+#' bound fractions at time \code{t}.
 #' @examples
-#' 
+#'
 #'   \dontrun{plot(function(x) SFORB.solution(x, 100, 0.5, 2, 3), 0, 2)}
-#' 
+#'
 #' @export
 SFORB.solution = function(t, parent_0, k_12, k_21, k_1output) {
-  sqrt_exp = sqrt(1/4 * (k_12 + k_21 + k_1output)^2 + k_12 * k_21 - (k_12 + k_1output) * k_21)
+  sqrt_exp = sqrt(1/4 * (k_12 + k_21 + k_1output)^2 - k_1output * k_21)
   b1 = 0.5 * (k_12 + k_21 + k_1output) + sqrt_exp
   b2 = 0.5 * (k_12 + k_21 + k_1output) - sqrt_exp
 
@@ -184,19 +184,19 @@ SFORB.solution = function(t, parent_0, k_12, k_21, k_1output) {
 }
 
 #' Logistic kinetics
-#' 
+#'
 #' Function describing exponential decline from a defined starting value, with
 #' an increasing rate constant, supposedly caused by microbial growth
-#' 
+#'
 #' @family parent solutions
-#' @inherit SFO.solution 
+#' @inherit SFO.solution
 #' @param kmax Maximum rate constant.
 #' @param k0 Minimum rate constant effective at time zero.
 #' @param r Growth rate of the increase in the rate constant.
 #' @note The solution of the logistic model reduces to the
-#'   \code{\link{SFO.solution}} if \code{k0} is equal to \code{kmax}.
+#' \code{\link{SFO.solution}} if \code{k0} is equal to \code{kmax}.
 #' @examples
-#' 
+#'
 #'   # Reproduce the plot on page 57 of FOCUS (2014)
 #'   plot(function(x) logistic.solution(x, 100, 0.08, 0.0001, 0.2),
 #'        from = 0, to = 100, ylim = c(0, 100),
@@ -213,10 +213,10 @@ SFORB.solution = function(t, parent_0, k_12, k_21, k_1output) {
 #'          legend = paste0("k0 = ", c(0.0001, 0.0001, 0.0001, 0.001, 0.08),
 #'                          ", r = ", c(0.2, 0.4, 0.8, 0.2, 0.2)),
 #'          lty = 1:5, col = 1:5)
-#' 
+#'
 #'   # Fit with synthetic data
 #'   logistic <- mkinmod(parent = mkinsub("logistic"))
-#' 
+#'
 #'   sampling_times = c(0, 1, 3, 7, 14, 28, 60, 90, 120)
 #'   parms_logistic <- c(kmax = 0.08, k0 = 0.0001, r = 0.2)
 #'   d_logistic <- mkinpredict(logistic,
@@ -225,14 +225,14 @@ SFORB.solution = function(t, parent_0, k_12, k_21, k_1output) {
 #'   d_2_1 <- add_err(d_logistic,
 #'     sdfunc = function(x) sigma_twocomp(x, 0.5, 0.07),
 #'     n = 1, reps = 2, digits = 5, LOD = 0.1, seed = 123456)[[1]]
-#' 
+#'
 #'   m <- mkinfit("logistic", d_2_1, quiet = TRUE)
 #'   plot_sep(m)
 #'   summary(m)$bpar
 #'   endpoints(m)$distimes
-#' 
+#'
 #' @export
 logistic.solution <- function(t, parent_0, kmax, k0, r)
 {
-	parent = parent_0 * (kmax / (kmax - k0 + k0 * exp (r * t))) ^(kmax/r)
+  parent = parent_0 * (kmax / (kmax - k0 + k0 * exp (r * t))) ^(kmax/r)
 }
diff --git a/log/test.log b/log/test.log
index 95d50017..52b92245 100644
--- a/log/test.log
+++ b/log/test.log
@@ -1,53 +1,53 @@
 ℹ Testing mkin
 ✔ | F W S  OK | Context
 ✔ |         5 | AIC calculation
-✔ |         5 | Analytical solutions for coupled models [3.4s]
+✔ |         5 | Analytical solutions for coupled models [3.3s]
 ✔ |         5 | Calculation of Akaike weights
 ✔ |         3 | Export dataset for reading into CAKE
 ✔ |        12 | Confidence intervals and p-values [1.0s]
-✔ |     1  12 | Dimethenamid data from 2018 [33.0s]
+✔ |     1  12 | Dimethenamid data from 2018 [31.5s]
 ────────────────────────────────────────────────────────────────────────────────
 Skip (test_dmta.R:98:3): Different backends get consistent results for SFO-SFO3+, dimethenamid data
 Reason: Fitting this ODE model with saemix takes about 15 minutes on my system
 ────────────────────────────────────────────────────────────────────────────────
-✔ |        14 | Error model fitting [5.6s]
+✔ |        14 | Error model fitting [4.9s]
 ✔ |         5 | Time step normalisation
 ✔ |         4 | Calculation of FOCUS chi2 error levels [0.6s]
-✔ |        14 | Results for FOCUS D established in expertise for UBA (Ranke 2014) [0.9s]
+✔ |        14 | Results for FOCUS D established in expertise for UBA (Ranke 2014) [0.8s]
 ✔ |         4 | Test fitting the decline of metabolites from their maximum [0.4s]
 ✔ |         1 | Fitting the logistic model [0.2s]
-✔ |        10 | Batch fitting and diagnosing hierarchical kinetic models [26.9s]
+✔ |        10 | Batch fitting and diagnosing hierarchical kinetic models [23.7s]
 ✔ |     1  12 | Nonlinear mixed-effects models [0.3s]
 ────────────────────────────────────────────────────────────────────────────────
 Skip (test_mixed.R:74:3): saemix results are reproducible for biphasic fits
 Reason: Fitting with saemix takes around 10 minutes when using deSolve
 ────────────────────────────────────────────────────────────────────────────────
 ✔ |         3 | Test dataset classes mkinds and mkindsg
-✔ |        10 | Special cases of mkinfit calls [0.6s]
-✔ |         3 | mkinfit features [0.8s]
-✔ |         8 | mkinmod model generation and printing [0.3s]
-✔ |         3 | Model predictions with mkinpredict [0.4s]
-✔ |         7 | Multistart method for saem.mmkin models [44.7s]
-✔ |        16 | Evaluations according to 2015 NAFTA guidance [2.8s]
-✔ |         9 | Nonlinear mixed-effects models with nlme [9.9s]
-✔ |        16 | Plotting [10.5s]
+✔ |        10 | Special cases of mkinfit calls [0.5s]
+✔ |         3 | mkinfit features [0.7s]
+✔ |         8 | mkinmod model generation and printing [0.2s]
+✔ |         3 | Model predictions with mkinpredict [0.3s]
+✔ |         7 | Multistart method for saem.mmkin models [36.3s]
+✔ |        16 | Evaluations according to 2015 NAFTA guidance [2.3s]
+✔ |         9 | Nonlinear mixed-effects models with nlme [8.9s]
+✔ |        16 | Plotting [10.0s]
 ✔ |         4 | Residuals extracted from mkinfit models
-✔ |     1  37 | saemix parent models [76.1s]
+✔ |     1  37 | saemix parent models [72.0s]
 ────────────────────────────────────────────────────────────────────────────────
 Skip (test_saemix_parent.R:153:3): We can also use mkin solution methods for saem
 Reason: This still takes almost 2.5 minutes although we do not solve ODEs
 ────────────────────────────────────────────────────────────────────────────────
-✔ |         2 | Complex test case from Schaefer et al. (2007) Piacenza paper [1.5s]
+✔ |         2 | Complex test case from Schaefer et al. (2007) Piacenza paper [1.4s]
 ✔ |        11 | Processing of residue series
-✔ |         7 | Fitting the SFORB model [3.9s]
+✔ |         7 | Fitting the SFORB model [3.7s]
 ✔ |         1 | Summaries of old mkinfit objects
 ✔ |         5 | Summary [0.2s]
-✔ |         4 | Results for synthetic data established in expertise for UBA (Ranke 2014) [2.3s]
-✔ |         9 | Hypothesis tests [8.4s]
-✔ |         4 | Calculation of maximum time weighted average concentrations (TWAs) [2.2s]
+✔ |         4 | Results for synthetic data established in expertise for UBA (Ranke 2014) [2.2s]
+✔ |         9 | Hypothesis tests [8.1s]
+✔ |         4 | Calculation of maximum time weighted average concentrations (TWAs) [2.4s]
 
 ══ Results ═════════════════════════════════════════════════════════════════════
-Duration: 237.3 s
+Duration: 216.5 s
 
 ── Skipped tests  ──────────────────────────────────────────────────────────────
 • Fitting this ODE model with saemix takes about 15 minutes on my system (1)
diff --git a/man/FOMC.solution.Rd b/man/FOMC.solution.Rd
index eeee85f9..d645113c 100644
--- a/man/FOMC.solution.Rd
+++ b/man/FOMC.solution.Rd
@@ -30,8 +30,8 @@ in the original equation.
 }
 \note{
 The solution of the FOMC kinetic model reduces to the
-\code{\link{SFO.solution}} for large values of \code{alpha} and
-\code{beta} with \eqn{k = \frac{\beta}{\alpha}}{k = beta/alpha}.
+\code{\link{SFO.solution}} for large values of \code{alpha} and \code{beta}
+with \eqn{k = \frac{\beta}{\alpha}}{k = beta/alpha}.
 }
 \examples{
 
diff --git a/man/IORE.solution.Rd b/man/IORE.solution.Rd
index 1150f023..5d0126a7 100644
--- a/man/IORE.solution.Rd
+++ b/man/IORE.solution.Rd
@@ -37,9 +37,9 @@ guidance for details.
     fit.iore <- mkinfit("IORE", FOCUS_2006_C, quiet = TRUE)
     fit.iore.deS <- mkinfit("IORE", FOCUS_2006_C, solution_type = "deSolve", quiet = TRUE)
 
-    print(data.frame(fit.fomc$par, fit.iore$par, fit.iore.deS$par, 
+    print(data.frame(fit.fomc$par, fit.iore$par, fit.iore.deS$par,
                      row.names = paste("model par", 1:4)))
-    print(rbind(fomc = endpoints(fit.fomc)$distimes, iore = endpoints(fit.iore)$distimes, 
+    print(rbind(fomc = endpoints(fit.fomc)$distimes, iore = endpoints(fit.iore)$distimes,
                 iore.deS = endpoints(fit.iore)$distimes))
   }