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

Average Error: 25.1 → 8.0

Time: 7.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -2.7490043504004516 \cdot 10^{-5}:\\ \;\;\;\;x - \frac{\log \left(1 + \left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(\sqrt[3]{y} \cdot e^{z}\right) - y\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \left(1 \cdot \left(y \cdot \frac{z}{t}\right) + \left(\frac{\log 1}{t} + 0.5 \cdot \left(\left(\sqrt[3]{y \cdot \left(\frac{\sqrt[3]{z}}{\sqrt[3]{t}} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{t}}\right)} \cdot \left(\sqrt[3]{y \cdot \left(\frac{\sqrt[3]{z}}{\sqrt[3]{t}} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{t}}\right)} \cdot \sqrt[3]{y \cdot \left(\frac{\sqrt[3]{z}}{\sqrt[3]{t}} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{t}}\right)}\right)\right) \cdot \frac{z}{\frac{\sqrt[3]{t}}{\sqrt[3]{z}}}\right)\right)\right)\\ \end{array}\]

C

double code(double x, double y, double z, double t) {
	return ((double) (x - (((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 <= -2.7490043504004516e-05)) {
		VAR = ((double) (x - (((double) log(((double) (1.0 + ((double) (((double) (((double) (((double) cbrt(y)) * ((double) cbrt(y)))) * ((double) (((double) cbrt(y)) * ((double) exp(z)))))) - y)))))) / t)));
	} else {
		VAR = ((double) (x - ((double) (((double) (1.0 * ((double) (y * (z / t))))) + ((double) ((((double) log(1.0)) / t) + ((double) (0.5 * ((double) (((double) (((double) cbrt(((double) (y * ((double) ((((double) cbrt(z)) / ((double) cbrt(t))) * (((double) cbrt(z)) / ((double) cbrt(t))))))))) * ((double) (((double) cbrt(((double) (y * ((double) ((((double) cbrt(z)) / ((double) cbrt(t))) * (((double) cbrt(z)) / ((double) cbrt(t))))))))) * ((double) cbrt(((double) (y * ((double) ((((double) cbrt(z)) / ((double) cbrt(t))) * (((double) cbrt(z)) / ((double) cbrt(t))))))))))))) * (z / (((double) cbrt(t)) / ((double) cbrt(z))))))))))))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	25.1
Target	16.0
Herbie	8.0

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

Derivation

1. Split input into 2 regimes

2. if z < -2.7490043504004516e-5

   1. Initial program 11.5

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

   2. Simplified 11.5

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

   4. Applied add-cube-cbrt 11.5

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

   5. Applied associate-*l* 11.5

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

   6. Simplified 11.5

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

   if -2.7490043504004516e-5 < z

   1. Initial program 31.2

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

   2. Simplified 15.7

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

   3. Taylor expanded around 0 7.2

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

   4. Simplified 6.5

      \[\leadsto x - \color{blue}{\left(1 \cdot \left(y \cdot \frac{z}{t}\right) + \left(\frac{\log 1}{t} + 0.5 \cdot \left(y \cdot \frac{z}{\frac{t}{z}}\right)\right)\right)}\]
   5. Using strategy rm

   6. Applied add-cube-cbrt 6.5

      \[\leadsto x - \left(1 \cdot \left(y \cdot \frac{z}{t}\right) + \left(\frac{\log 1}{t} + 0.5 \cdot \left(y \cdot \frac{z}{\frac{t}{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}}\right)\right)\right)\]

   7. Applied add-cube-cbrt 6.5

      \[\leadsto x - \left(1 \cdot \left(y \cdot \frac{z}{t}\right) + \left(\frac{\log 1}{t} + 0.5 \cdot \left(y \cdot \frac{z}{\frac{\color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}\right)\right)\right)\]

   8. Applied times-frac 6.5

      \[\leadsto x - \left(1 \cdot \left(y \cdot \frac{z}{t}\right) + \left(\frac{\log 1}{t} + 0.5 \cdot \left(y \cdot \frac{z}{\color{blue}{\frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{\sqrt[3]{t}}{\sqrt[3]{z}}}}\right)\right)\right)\]

   9. Applied *-un-lft-identity 6.5

      \[\leadsto x - \left(1 \cdot \left(y \cdot \frac{z}{t}\right) + \left(\frac{\log 1}{t} + 0.5 \cdot \left(y \cdot \frac{\color{blue}{1 \cdot z}}{\frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{\sqrt[3]{t}}{\sqrt[3]{z}}}\right)\right)\right)\]

   10. Applied times-frac 6.5

       \[\leadsto x - \left(1 \cdot \left(y \cdot \frac{z}{t}\right) + \left(\frac{\log 1}{t} + 0.5 \cdot \left(y \cdot \color{blue}{\left(\frac{1}{\frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}} \cdot \frac{z}{\frac{\sqrt[3]{t}}{\sqrt[3]{z}}}\right)}\right)\right)\right)\]

   11. Applied associate-*r* 6.5

       \[\leadsto x - \left(1 \cdot \left(y \cdot \frac{z}{t}\right) + \left(\frac{\log 1}{t} + 0.5 \cdot \color{blue}{\left(\left(y \cdot \frac{1}{\frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}}\right) \cdot \frac{z}{\frac{\sqrt[3]{t}}{\sqrt[3]{z}}}\right)}\right)\right)\]

   12. Simplified 6.5

       \[\leadsto x - \left(1 \cdot \left(y \cdot \frac{z}{t}\right) + \left(\frac{\log 1}{t} + 0.5 \cdot \left(\color{blue}{\left(y \cdot \left(\frac{\sqrt[3]{z}}{\sqrt[3]{t}} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{t}}\right)\right)} \cdot \frac{z}{\frac{\sqrt[3]{t}}{\sqrt[3]{z}}}\right)\right)\right)\]
   13. Using strategy rm

   14. Applied add-cube-cbrt 6.5

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

4. Final simplification 8.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.7490043504004516 \cdot 10^{-5}:\\ \;\;\;\;x - \frac{\log \left(1 + \left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(\sqrt[3]{y} \cdot e^{z}\right) - y\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \left(1 \cdot \left(y \cdot \frac{z}{t}\right) + \left(\frac{\log 1}{t} + 0.5 \cdot \left(\left(\sqrt[3]{y \cdot \left(\frac{\sqrt[3]{z}}{\sqrt[3]{t}} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{t}}\right)} \cdot \left(\sqrt[3]{y \cdot \left(\frac{\sqrt[3]{z}}{\sqrt[3]{t}} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{t}}\right)} \cdot \sqrt[3]{y \cdot \left(\frac{\sqrt[3]{z}}{\sqrt[3]{t}} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{t}}\right)}\right)\right) \cdot \frac{z}{\frac{\sqrt[3]{t}}{\sqrt[3]{z}}}\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(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)))