Average Error: 24.4 → 7.8
Time: 15.1s
Precision: binary64
\[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(1 - y\right) + y \cdot e^{z} \leq 0:\\ \;\;\;\;x - \frac{\log \left(1 + y \cdot z\right)}{t}\\ \mathbf{elif}\;\left(1 - y\right) + y \cdot e^{z} \leq 1:\\ \;\;\;\;x - \left(\left(\frac{e^{z} \cdot {y}^{2}}{t} + \frac{y \cdot e^{z}}{t}\right) - \left(\frac{y}{t} + \left(0.5 \cdot \frac{{y}^{2} \cdot {\left(e^{z}\right)}^{2}}{t} + 0.5 \cdot \frac{{y}^{2}}{t}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(\left(1 - y\right) + \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(e^{z} \cdot \sqrt[3]{y}\right)\right)}{t}\\ \end{array}\]
x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}
↓
\begin{array}{l}
\mathbf{if}\;\left(1 - y\right) + y \cdot e^{z} \leq 0:\\
\;\;\;\;x - \frac{\log \left(1 + y \cdot z\right)}{t}\\

\mathbf{elif}\;\left(1 - y\right) + y \cdot e^{z} \leq 1:\\
\;\;\;\;x - \left(\left(\frac{e^{z} \cdot {y}^{2}}{t} + \frac{y \cdot e^{z}}{t}\right) - \left(\frac{y}{t} + \left(0.5 \cdot \frac{{y}^{2} \cdot {\left(e^{z}\right)}^{2}}{t} + 0.5 \cdot \frac{{y}^{2}}{t}\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;x - \frac{\log \left(\left(1 - y\right) + \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(e^{z} \cdot \sqrt[3]{y}\right)\right)}{t}\\

\end{array}
(FPCore (x y z t)
 :precision binary64
 (- x (/ (log (+ (- 1.0 y) (* y (exp z)))) t)))
↓
(FPCore (x y z t)
 :precision binary64
 (if (<= (+ (- 1.0 y) (* y (exp z))) 0.0)
   (- x (/ (log (+ 1.0 (* y z))) t))
   (if (<= (+ (- 1.0 y) (* y (exp z))) 1.0)
     (-
      x
      (-
       (+ (/ (* (exp z) (pow y 2.0)) t) (/ (* y (exp z)) t))
       (+
        (/ y t)
        (+
         (* 0.5 (/ (* (pow y 2.0) (pow (exp z) 2.0)) t))
         (* 0.5 (/ (pow y 2.0) t))))))
     (-
      x
      (/
       (log (+ (- 1.0 y) (* (* (cbrt y) (cbrt y)) (* (exp z) (cbrt y)))))
       t)))))
double code(double x, double y, double z, double t) {
	return x - (log((1.0 - y) + (y * exp(z))) / t);
}
↓
double code(double x, double y, double z, double t) {
	double tmp;
	if (((1.0 - y) + (y * exp(z))) <= 0.0) {
		tmp = x - (log(1.0 + (y * z)) / t);
	} else if (((1.0 - y) + (y * exp(z))) <= 1.0) {
		tmp = x - ((((exp(z) * pow(y, 2.0)) / t) + ((y * exp(z)) / t)) - ((y / t) + ((0.5 * ((pow(y, 2.0) * pow(exp(z), 2.0)) / t)) + (0.5 * (pow(y, 2.0) / t)))));
	} else {
		tmp = x - (log((1.0 - y) + ((cbrt(y) * cbrt(y)) * (exp(z) * cbrt(y)))) / t);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original24.4
Target16.1
Herbie7.8
\[\begin{array}{l} \mathbf{if}\;z < -2.8874623088207947 \cdot 10^{+119}:\\ \;\;\;\;\left(x - \frac{\frac{-0.5}{y \cdot t}}{z \cdot z}\right) - \frac{-0.5}{y \cdot t} \cdot \frac{\frac{2}{z}}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + z \cdot y\right)}{t}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (+.f64 (-.f64 1 y) (*.f64 y (exp.f64 z))) < 0.0

    1. Initial program 64.0

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\]

    2. Taylor expanded around 0 14.5

      \[\leadsto x - \frac{\log \color{blue}{\left(z \cdot y + 1\right)}}{t}\]

    3. Simplified14.5

      \[\leadsto x - \frac{\log \color{blue}{\left(1 + z \cdot y\right)}}{t}\]

    if 0.0 < (+.f64 (-.f64 1 y) (*.f64 y (exp.f64 z))) < 1

    1. Initial program 12.0

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\]

    2. Taylor expanded around 0 6.0

      \[\leadsto x - \color{blue}{\left(\left(\frac{e^{z} \cdot {y}^{2}}{t} + \frac{e^{z} \cdot y}{t}\right) - \left(\frac{y}{t} + \left(0.5 \cdot \frac{{\left(e^{z}\right)}^{2} \cdot {y}^{2}}{t} + 0.5 \cdot \frac{{y}^{2}}{t}\right)\right)\right)}\]

    if 1 < (+.f64 (-.f64 1 y) (*.f64 y (exp.f64 z)))

    1. Initial program 2.6

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\]
    2. Using strategy rm

    3. Applied add-cube-cbrt_binary642.7

      \[\leadsto x - \frac{\log \left(\left(1 - y\right) + \color{blue}{\left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}\right)} \cdot e^{z}\right)}{t}\]

    4. Applied associate-*l*_binary642.7

      \[\leadsto x - \frac{\log \left(\left(1 - y\right) + \color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(\sqrt[3]{y} \cdot e^{z}\right)}\right)}{t}\]

    5. Simplified2.7

      \[\leadsto x - \frac{\log \left(\left(1 - y\right) + \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \color{blue}{\left(e^{z} \cdot \sqrt[3]{y}\right)}\right)}{t}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification7.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - y\right) + y \cdot e^{z} \leq 0:\\ \;\;\;\;x - \frac{\log \left(1 + y \cdot z\right)}{t}\\ \mathbf{elif}\;\left(1 - y\right) + y \cdot e^{z} \leq 1:\\ \;\;\;\;x - \left(\left(\frac{e^{z} \cdot {y}^{2}}{t} + \frac{y \cdot e^{z}}{t}\right) - \left(\frac{y}{t} + \left(0.5 \cdot \frac{{y}^{2} \cdot {\left(e^{z}\right)}^{2}}{t} + 0.5 \cdot \frac{{y}^{2}}{t}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(\left(1 - y\right) + \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(e^{z} \cdot \sqrt[3]{y}\right)\right)}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2021174 
(FPCore (x y z t)
  :name "System.Random.MWC.Distributions:truncatedExp from mwc-random-0.13.3.2"
  :precision binary64

  :herbie-target
  (if (< z -2.8874623088207947e+119) (- (- x (/ (/ (- 0.5) (* y t)) (* z z))) (* (/ (- 0.5) (* y t)) (/ (/ 2.0 z) (* z z)))) (- x (/ (log (+ 1.0 (* z y))) t)))

  (- x (/ (log (+ (- 1.0 y) (* y (exp z)))) t)))