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

Average Error: 25.5 → 8.7

Time: 9.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -296098434438560.62:\\ \;\;\;\;x - \frac{\log \left(1 + \left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(\sqrt[3]{y} \cdot e^{z}\right) - y\right)\right)}{t}\\ \mathbf{elif}\;z \le 3.25761963951068856 \cdot 10^{-122}:\\ \;\;\;\;x - \left(1 \cdot \left(y \cdot \frac{z}{t}\right) + \left(\frac{\log 1}{t} + 0.5 \cdot \left(y \cdot \frac{z}{\frac{t}{z}}\right)\right)\right)\\ \mathbf{elif}\;z \le 2.74095623860811759 \cdot 10^{-34}:\\ \;\;\;\;x - \frac{\log \left(1 + y \cdot \left(\frac{1}{6} \cdot {z}^{3} + \left(z + z \cdot \left(z \cdot \frac{1}{2}\right)\right)\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \left(\left(\frac{\log 1}{t} + 0.5 \cdot \left(y \cdot \frac{z}{\frac{t}{z}}\right)\right) + 1 \cdot \left(\sqrt[3]{y \cdot \frac{z}{t}} \cdot \left(\sqrt[3]{y \cdot \frac{z}{t}} \cdot \sqrt[3]{y \cdot \frac{z}{t}}\right)\right)\right)\\ \end{array}\]

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 <= -296098434438560.6)) {
		VAR = ((double) (x - ((double) (((double) log(((double) (1.0 + ((double) (((double) (((double) (((double) cbrt(y)) * ((double) cbrt(y)))) * ((double) (((double) cbrt(y)) * ((double) exp(z)))))) - y)))))) / t))));
	} else {
		double VAR_1;
		if ((z <= 3.2576196395106886e-122)) {
			VAR_1 = ((double) (x - ((double) (((double) (1.0 * ((double) (y * ((double) (z / t)))))) + ((double) (((double) (((double) log(1.0)) / t)) + ((double) (0.5 * ((double) (y * ((double) (z / ((double) (t / z))))))))))))));
		} else {
			double VAR_2;
			if ((z <= 2.7409562386081176e-34)) {
				VAR_2 = ((double) (x - ((double) (((double) log(((double) (1.0 + ((double) (y * ((double) (((double) (0.16666666666666666 * ((double) pow(z, 3.0)))) + ((double) (z + ((double) (z * ((double) (z * 0.5)))))))))))))) / t))));
			} else {
				VAR_2 = ((double) (x - ((double) (((double) (((double) (((double) log(1.0)) / t)) + ((double) (0.5 * ((double) (y * ((double) (z / ((double) (t / z)))))))))) + ((double) (1.0 * ((double) (((double) cbrt(((double) (y * ((double) (z / t)))))) * ((double) (((double) cbrt(((double) (y * ((double) (z / t)))))) * ((double) cbrt(((double) (y * ((double) (z / t))))))))))))))));
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	25.5
Target	16.3
Herbie	8.7

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

Derivation

1. Split input into 4 regimes

2. if z < -296098434438560.62

   1. Initial program 11.8

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

   2. Simplified 11.8

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

   4. Applied add-cube-cbrt 11.8

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

   5. Applied associate-*l* 11.8

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

   6. Simplified 11.8

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

   if -296098434438560.62 < z < 3.25761963951068856e-122

   1. Initial program 30.9

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

   2. Simplified 15.2

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

   3. Taylor expanded around 0 7.0

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

   4. Simplified 6.1

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

   if 3.25761963951068856e-122 < z < 2.74095623860811759e-34

   1. Initial program 32.7

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

   2. Simplified 20.7

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

   3. Taylor expanded around 0 12.7

      \[\leadsto x - \frac{\log \left(1 + \color{blue}{\left(\frac{1}{6} \cdot \left({z}^{3} \cdot y\right) + \left(z \cdot y + \frac{1}{2} \cdot \left({z}^{2} \cdot y\right)\right)\right)}\right)}{t}\]

   4. Simplified 12.7

      \[\leadsto x - \frac{\log \left(1 + \color{blue}{y \cdot \left(\frac{1}{6} \cdot {z}^{3} + \left(z + z \cdot \left(z \cdot \frac{1}{2}\right)\right)\right)}\right)}{t}\]

   if 2.74095623860811759e-34 < z

   1. Initial program 26.5

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

   2. Simplified 21.8

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

   3. Taylor expanded around 0 14.8

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

   4. Simplified 14.8

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

   6. Applied add-cube-cbrt 15.0

      \[\leadsto x - \left(1 \cdot \color{blue}{\left(\left(\sqrt[3]{y \cdot \frac{z}{t}} \cdot \sqrt[3]{y \cdot \frac{z}{t}}\right) \cdot \sqrt[3]{y \cdot \frac{z}{t}}\right)} + \left(\frac{\log 1}{t} + 0.5 \cdot \left(y \cdot \frac{z}{\frac{t}{z}}\right)\right)\right)\]
3. Recombined 4 regimes into one program.

4. Final simplification 8.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -296098434438560.62:\\ \;\;\;\;x - \frac{\log \left(1 + \left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(\sqrt[3]{y} \cdot e^{z}\right) - y\right)\right)}{t}\\ \mathbf{elif}\;z \le 3.25761963951068856 \cdot 10^{-122}:\\ \;\;\;\;x - \left(1 \cdot \left(y \cdot \frac{z}{t}\right) + \left(\frac{\log 1}{t} + 0.5 \cdot \left(y \cdot \frac{z}{\frac{t}{z}}\right)\right)\right)\\ \mathbf{elif}\;z \le 2.74095623860811759 \cdot 10^{-34}:\\ \;\;\;\;x - \frac{\log \left(1 + y \cdot \left(\frac{1}{6} \cdot {z}^{3} + \left(z + z \cdot \left(z \cdot \frac{1}{2}\right)\right)\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \left(\left(\frac{\log 1}{t} + 0.5 \cdot \left(y \cdot \frac{z}{\frac{t}{z}}\right)\right) + 1 \cdot \left(\sqrt[3]{y \cdot \frac{z}{t}} \cdot \left(\sqrt[3]{y \cdot \frac{z}{t}} \cdot \sqrt[3]{y \cdot \frac{z}{t}}\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020185 
(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)))