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

Average Error: 25.4 → 11.9

Time: 8.3s

Precision: 64

• could not determine a ground truth for program body

  1. x = -3.413161699705947e-213
  2. y = 6.010790371481058e-241
  3. z = 1.1215786563862367e+204
  4. t = 3.45929965396414e-189

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -7.8788764985442077 \cdot 10^{-149}:\\ \;\;\;\;x - \frac{1}{t} \cdot \log \left(\mathsf{fma}\left(\mathsf{expm1}\left(z\right), y, 1\right)\right)\\ \mathbf{elif}\;x \le 1.23289972704431142 \cdot 10^{-76}:\\ \;\;\;\;x - \left(1 \cdot \frac{z \cdot y}{t} + \frac{\log 1}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(\sqrt{1 + y \cdot \mathsf{expm1}\left(z\right)}\right) + \log \left(\sqrt{1 + y \cdot \mathsf{expm1}\left(z\right)}\right)}{t}\\ \end{array}\]

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 ((x <= -7.878876498544208e-149)) {
		VAR = ((double) (x - ((double) (((double) (1.0 / t)) * ((double) log(((double) fma(((double) expm1(z)), y, 1.0))))))));
	} else {
		double VAR_1;
		if ((x <= 1.2328997270443114e-76)) {
			VAR_1 = ((double) (x - ((double) (((double) (1.0 * ((double) (((double) (z * y)) / t)))) + ((double) (((double) log(1.0)) / t))))));
		} else {
			VAR_1 = ((double) (x - ((double) (((double) (((double) log(((double) sqrt(((double) (1.0 + ((double) (y * ((double) expm1(z)))))))))) + ((double) log(((double) sqrt(((double) (1.0 + ((double) (y * ((double) expm1(z)))))))))))) / t))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	25.4
Target	16.2
Herbie	11.9

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

Derivation

1. Split input into 3 regimes

2. if x < -7.878876498544208e-149

   1. Initial program 21.8

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

   3. Applied sub-neg 21.8

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

   4. Applied associate-+l+ 8.7

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

   5. Simplified 6.7

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

   7. Applied clear-num 6.7

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

   9. Applied div-inv 6.7

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

   10. Applied add-cube-cbrt 6.7

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

   11. Applied times-frac 6.7

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

   12. Simplified 6.7

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

   13. Simplified 6.7

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

   if -7.878876498544208e-149 < x < 1.2328997270443114e-76

   1. Initial program 34.7

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

   3. Applied sub-neg 34.7

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

   4. Applied associate-+l+ 29.3

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

   5. Simplified 24.1

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

   6. Taylor expanded around 0 24.8

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

   if 1.2328997270443114e-76 < x

   1. Initial program 20.2

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

   3. Applied sub-neg 20.2

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

   4. Applied associate-+l+ 6.5

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

   5. Simplified 4.6

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

   7. Applied add-sqr-sqrt 4.6

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

   8. Applied log-prod 4.6

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

4. Final simplification 11.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -7.8788764985442077 \cdot 10^{-149}:\\ \;\;\;\;x - \frac{1}{t} \cdot \log \left(\mathsf{fma}\left(\mathsf{expm1}\left(z\right), y, 1\right)\right)\\ \mathbf{elif}\;x \le 1.23289972704431142 \cdot 10^{-76}:\\ \;\;\;\;x - \left(1 \cdot \frac{z \cdot y}{t} + \frac{\log 1}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(\sqrt{1 + y \cdot \mathsf{expm1}\left(z\right)}\right) + \log \left(\sqrt{1 + y \cdot \mathsf{expm1}\left(z\right)}\right)}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2020114 +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)))