Average Error: 25.0 → 8.5
Time: 10.3s
Precision: 64
\[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.31107669511558345 \cdot 10^{-82}:\\ \;\;\;\;x - \frac{1}{t} \cdot \log \left(\mathsf{fma}\left(\mathsf{expm1}\left(z\right), y, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{fma}\left(0.5, {z}^{2} \cdot y, \mathsf{fma}\left(1, z \cdot y, \log 1\right)\right)}{t}\\ \end{array}\]
x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.31107669511558345 \cdot 10^{-82}:\\
\;\;\;\;x - \frac{1}{t} \cdot \log \left(\mathsf{fma}\left(\mathsf{expm1}\left(z\right), y, 1\right)\right)\\

\mathbf{else}:\\
\;\;\;\;x - \frac{\mathsf{fma}\left(0.5, {z}^{2} \cdot y, \mathsf{fma}\left(1, z \cdot y, \log 1\right)\right)}{t}\\

\end{array}
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 VAR;
	if ((z <= -1.3110766951155834e-82)) {
		VAR = (x - ((1.0 / t) * log(fma(expm1(z), y, 1.0))));
	} else {
		VAR = (x - (fma(0.5, (pow(z, 2.0) * y), fma(1.0, (z * y), log(1.0))) / t));
	}
	return VAR;
}

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

Original25.0
Target16.5
Herbie8.5
\[\begin{array}{l} \mathbf{if}\;z \lt -2.88746230882079466 \cdot 10^{119}:\\ \;\;\;\;\left(x - \frac{\frac{-0.5}{y \cdot t}}{z \cdot z}\right) - \frac{-0.5}{y \cdot t} \cdot \frac{\frac{2}{z}}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + z \cdot y\right)}{t}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -1.3110766951155834e-82

    1. Initial program 15.7

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

    3. Applied sub-neg15.7

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

    4. Applied associate-+l+13.8

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

    5. Simplified11.9

      \[\leadsto x - \frac{\log \left(1 + \color{blue}{y \cdot \mathsf{expm1}\left(z\right)}\right)}{t}\]
    6. Using strategy rm

    7. Applied clear-num11.9

      \[\leadsto x - \color{blue}{\frac{1}{\frac{t}{\log \left(1 + y \cdot \mathsf{expm1}\left(z\right)\right)}}}\]
    8. Using strategy rm

    9. Applied div-inv11.9

      \[\leadsto x - \frac{1}{\color{blue}{t \cdot \frac{1}{\log \left(1 + y \cdot \mathsf{expm1}\left(z\right)\right)}}}\]

    10. Applied add-cube-cbrt11.9

      \[\leadsto x - \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{t \cdot \frac{1}{\log \left(1 + y \cdot \mathsf{expm1}\left(z\right)\right)}}\]

    11. Applied times-frac11.9

      \[\leadsto x - \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{t} \cdot \frac{\sqrt[3]{1}}{\frac{1}{\log \left(1 + y \cdot \mathsf{expm1}\left(z\right)\right)}}}\]

    12. Simplified11.9

      \[\leadsto x - \color{blue}{\frac{1}{t}} \cdot \frac{\sqrt[3]{1}}{\frac{1}{\log \left(1 + y \cdot \mathsf{expm1}\left(z\right)\right)}}\]

    13. Simplified11.9

      \[\leadsto x - \frac{1}{t} \cdot \color{blue}{\log \left(\mathsf{fma}\left(\mathsf{expm1}\left(z\right), y, 1\right)\right)}\]

    if -1.3110766951155834e-82 < z

    1. Initial program 31.0

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

    2. Taylor expanded around 0 6.3

      \[\leadsto x - \frac{\color{blue}{0.5 \cdot \left({z}^{2} \cdot y\right) + \left(1 \cdot \left(z \cdot y\right) + \log 1\right)}}{t}\]

    3. Simplified6.3

      \[\leadsto x - \frac{\color{blue}{\mathsf{fma}\left(0.5, {z}^{2} \cdot y, \mathsf{fma}\left(1, z \cdot y, \log 1\right)\right)}}{t}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification8.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.31107669511558345 \cdot 10^{-82}:\\ \;\;\;\;x - \frac{1}{t} \cdot \log \left(\mathsf{fma}\left(\mathsf{expm1}\left(z\right), y, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{fma}\left(0.5, {z}^{2} \cdot y, \mathsf{fma}\left(1, z \cdot y, \log 1\right)\right)}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2020103 +o rules:numerics
(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 z) (* z z)))) (- x (/ (log (+ 1 (* z y))) t)))

  (- x (/ (log (+ (- 1 y) (* y (exp z)))) t)))