System.Random.MWC.Distributions:truncatedExp from mwc-random-0.13.3.2

Average Error: 25.4 → 9.1

Time: 7.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -3197164.26069925865:\\ \;\;\;\;x - \frac{2 \cdot \left(2 \cdot \log \left(\sqrt[3]{\sqrt[3]{1 + y \cdot \mathsf{expm1}\left(z\right)}}\right) + \log \left(\sqrt[3]{\sqrt[3]{1 + y \cdot \mathsf{expm1}\left(z\right)}}\right)\right) + \log \left(\sqrt[3]{1 + y \cdot \mathsf{expm1}\left(z\right)}\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \left(1 \cdot \frac{z \cdot y}{t} + \frac{\log 1}{t}\right)\\ \end{array}\]

C

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 <= -3197164.2606992587)) {
		VAR = ((double) (x - ((double) (((double) (((double) (2.0 * ((double) (((double) (2.0 * ((double) log(((double) cbrt(((double) cbrt(((double) (1.0 + ((double) (y * ((double) expm1(z)))))))))))))) + ((double) log(((double) cbrt(((double) cbrt(((double) (1.0 + ((double) (y * ((double) expm1(z)))))))))))))))) + ((double) log(((double) cbrt(((double) (1.0 + ((double) (y * ((double) expm1(z)))))))))))) / t))));
	} else {
		VAR = ((double) (x - ((double) (((double) (1.0 * ((double) (((double) (z * y)) / t)))) + ((double) (((double) log(1.0)) / t))))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	25.4
Target	15.8
Herbie	9.1

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

Derivation

1. Split input into 2 regimes

2. if z < -3197164.2606992587

   1. Initial program 11.2

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

   3. Applied sub-neg 11.2

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

   4. Applied associate-+l+ 11.2

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

   5. Simplified 11.2

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

   7. Applied add-cube-cbrt 11.2

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

   8. Applied log-prod 11.2

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

   9. Simplified 11.2

      \[\leadsto x - \frac{\color{blue}{2 \cdot \log \left(\sqrt[3]{1 + y \cdot \mathsf{expm1}\left(z\right)}\right)} + \log \left(\sqrt[3]{1 + y \cdot \mathsf{expm1}\left(z\right)}\right)}{t}\]
   10. Using strategy rm

   11. Applied add-cube-cbrt 11.3

       \[\leadsto x - \frac{2 \cdot \log \color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{1 + y \cdot \mathsf{expm1}\left(z\right)}} \cdot \sqrt[3]{\sqrt[3]{1 + y \cdot \mathsf{expm1}\left(z\right)}}\right) \cdot \sqrt[3]{\sqrt[3]{1 + y \cdot \mathsf{expm1}\left(z\right)}}\right)} + \log \left(\sqrt[3]{1 + y \cdot \mathsf{expm1}\left(z\right)}\right)}{t}\]

   12. Applied log-prod 11.3

       \[\leadsto x - \frac{2 \cdot \color{blue}{\left(\log \left(\sqrt[3]{\sqrt[3]{1 + y \cdot \mathsf{expm1}\left(z\right)}} \cdot \sqrt[3]{\sqrt[3]{1 + y \cdot \mathsf{expm1}\left(z\right)}}\right) + \log \left(\sqrt[3]{\sqrt[3]{1 + y \cdot \mathsf{expm1}\left(z\right)}}\right)\right)} + \log \left(\sqrt[3]{1 + y \cdot \mathsf{expm1}\left(z\right)}\right)}{t}\]

   13. Simplified 11.3

       \[\leadsto x - \frac{2 \cdot \left(\color{blue}{2 \cdot \log \left(\sqrt[3]{\sqrt[3]{1 + y \cdot \mathsf{expm1}\left(z\right)}}\right)} + \log \left(\sqrt[3]{\sqrt[3]{1 + y \cdot \mathsf{expm1}\left(z\right)}}\right)\right) + \log \left(\sqrt[3]{1 + y \cdot \mathsf{expm1}\left(z\right)}\right)}{t}\]

   if -3197164.2606992587 < z

   1. Initial program 31.1

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

   3. Applied sub-neg 31.1

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

   4. Applied associate-+l+ 16.4

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

   5. Simplified 11.4

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

   6. Taylor expanded around 0 8.2

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

4. Final simplification 9.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3197164.26069925865:\\ \;\;\;\;x - \frac{2 \cdot \left(2 \cdot \log \left(\sqrt[3]{\sqrt[3]{1 + y \cdot \mathsf{expm1}\left(z\right)}}\right) + \log \left(\sqrt[3]{\sqrt[3]{1 + y \cdot \mathsf{expm1}\left(z\right)}}\right)\right) + \log \left(\sqrt[3]{1 + y \cdot \mathsf{expm1}\left(z\right)}\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \left(1 \cdot \frac{z \cdot y}{t} + \frac{\log 1}{t}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020122 +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)))