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

Average Error: 25.0 → 11.2

Time: 9.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -4.99429414839628993:\\ \;\;\;\;x - \frac{\log \left(\left(1 - y\right) + \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(\left(\left(\sqrt[3]{\sqrt[3]{y}} \cdot \left(\sqrt[3]{e^{\log \left(\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right)}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{y}}}\right)\right) \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot e^{z}\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + y \cdot \left(\frac{1}{2} \cdot {z}^{2} + z\right)\right)}{t}\\ \end{array}\]

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 <= -4.99429414839629)) {
		VAR = ((double) (x - ((double) (((double) log(((double) (((double) (1.0 - y)) + ((double) (((double) (((double) cbrt(y)) * ((double) cbrt(y)))) * ((double) (((double) (((double) (((double) cbrt(((double) cbrt(y)))) * ((double) (((double) cbrt(((double) exp(((double) log(((double) cbrt(((double) (((double) cbrt(y)) * ((double) cbrt(y)))))))))))) * ((double) cbrt(((double) cbrt(((double) cbrt(y)))))))))) * ((double) cbrt(((double) cbrt(y)))))) * ((double) exp(z)))))))))) / t))));
	} else {
		VAR = ((double) (x - ((double) (((double) log(((double) (1.0 + ((double) (y * ((double) (((double) (0.5 * ((double) pow(z, 2.0)))) + z)))))))) / t))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	25.0
Target	16.4
Herbie	11.2

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

Derivation

1. Split input into 2 regimes

2. if z < -4.99429414839628993

   1. Initial program 11.3

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

   3. Applied add-cube-cbrt 11.3

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

   4. Applied associate-*l* 11.3

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

   6. Applied add-cube-cbrt 11.3

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

   8. Applied add-cube-cbrt 11.3

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

   9. Applied cbrt-prod 11.3

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

   10. Applied cbrt-prod 11.3

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

   12. Applied add-exp-log 11.3

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

   if -4.99429414839628993 < z

   1. Initial program 30.9

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

   2. Taylor expanded around 0 11.1

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

   3. Simplified 11.1

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

4. Final simplification 11.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -4.99429414839628993:\\ \;\;\;\;x - \frac{\log \left(\left(1 - y\right) + \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(\left(\left(\sqrt[3]{\sqrt[3]{y}} \cdot \left(\sqrt[3]{e^{\log \left(\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right)}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{y}}}\right)\right) \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot e^{z}\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + y \cdot \left(\frac{1}{2} \cdot {z}^{2} + z\right)\right)}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2020147 
(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 (/ (/ (neg 0.5) (* y t)) (* z z))) (* (/ (neg 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)))