Average Error: 25.2 → 8.8
Time: 7.8s
Precision: 64
\[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -7.6731932909166638 \cdot 10^{-25}:\\ \;\;\;\;x - \sqrt{\log \left(1 + y \cdot \mathsf{expm1}\left(z\right)\right)} \cdot \frac{\sqrt{\log \left(1 + y \cdot \mathsf{expm1}\left(z\right)\right)}}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{fma}\left(0.5, {z}^{2} \cdot y, \mathsf{fma}\left(1, z \cdot y, \log 1\right)\right)}{t}\\ \end{array}\]
x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}
↓
\begin{array}{l}
\mathbf{if}\;z \le -7.6731932909166638 \cdot 10^{-25}:\\
\;\;\;\;x - \sqrt{\log \left(1 + y \cdot \mathsf{expm1}\left(z\right)\right)} \cdot \frac{\sqrt{\log \left(1 + y \cdot \mathsf{expm1}\left(z\right)\right)}}{t}\\

\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 ((double) (x - ((double) (((double) log(((double) (((double) (1.0 - y)) + ((double) (y * ((double) exp(z)))))))) / t))));
}
↓
double code(double x, double y, double z, double t) {
	double VAR;
	if ((z <= -7.673193290916664e-25)) {
		VAR = ((double) (x - ((double) (((double) sqrt(((double) log(((double) (1.0 + ((double) (y * ((double) expm1(z)))))))))) * ((double) (((double) sqrt(((double) log(((double) (1.0 + ((double) (y * ((double) expm1(z)))))))))) / t))))));
	} else {
		VAR = ((double) (x - ((double) (((double) fma(0.5, ((double) (((double) pow(z, 2.0)) * y)), ((double) fma(1.0, ((double) (z * y)), ((double) 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

Original25.2
Target16.3
Herbie8.8
\[\begin{array}{l} \mathbf{if}\;z \lt -2.88746230882079466 \cdot 10^{119}:\\ \;\;\;\;\left(x - \frac{\frac{-0.5}{y \cdot t}}{z \cdot z}\right) - \frac{-0.5}{y \cdot t} \cdot \frac{\frac{2}{z}}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + z \cdot y\right)}{t}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -7.673193290916664e-25

    1. Initial program 12.1

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

    3. Applied sub-neg12.1

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

    4. Applied associate-+l+11.9

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

    5. Simplified11.5

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

    7. Applied *-un-lft-identity11.5

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

    8. Applied add-sqr-sqrt12.6

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

    9. Applied times-frac12.6

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

    10. Simplified12.6

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

    if -7.673193290916664e-25 < z

    1. Initial program 31.4

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

    2. Taylor expanded around 0 7.0

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

    3. Simplified7.0

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

  4. Final simplification8.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -7.6731932909166638 \cdot 10^{-25}:\\ \;\;\;\;x - \sqrt{\log \left(1 + y \cdot \mathsf{expm1}\left(z\right)\right)} \cdot \frac{\sqrt{\log \left(1 + y \cdot \mathsf{expm1}\left(z\right)\right)}}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{fma}\left(0.5, {z}^{2} \cdot y, \mathsf{fma}\left(1, z \cdot y, \log 1\right)\right)}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2020120 +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)))