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

Average Error: 25.1 → 8.6

Time: 6.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -3.2314583272364048 \cdot 10^{-62}:\\ \;\;\;\;x - \left(\sqrt[3]{\log \left(\mathsf{fma}\left(\mathsf{expm1}\left(z\right), y, 1\right)\right)} \cdot \sqrt[3]{\log \left(\mathsf{fma}\left(\mathsf{expm1}\left(z\right), y, 1\right)\right)}\right) \cdot \frac{\sqrt[3]{\log \left(\mathsf{fma}\left(\mathsf{expm1}\left(z\right), y, 1\right)\right)}}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot z, z, 1 \cdot z\right), y, \log 1\right)}{t}\\ \end{array}\]

C

double code(double x, double y, double z, double t) {
	return (x - (log(((1.0 - y) + (y * exp(z)))) / t));
}

↓

double code(double x, double y, double z, double t) {
	double VAR;
	if ((z <= -3.2314583272364048e-62)) {
		VAR = (x - ((cbrt(log(fma(expm1(z), y, 1.0))) * cbrt(log(fma(expm1(z), y, 1.0)))) * (cbrt(log(fma(expm1(z), y, 1.0))) / t)));
	} else {
		VAR = (x - (fma(fma((0.5 * z), z, (1.0 * z)), y, 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	25.1
Target	16.3
Herbie	8.6

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

Derivation

1. Split input into 2 regimes

2. if z < -3.2314583272364048e-62

   1. Initial program 14.8

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

   3. Applied sub-neg 14.8

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

   4. Applied associate-+l+ 13.5

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

   5. Simplified 12.0

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

   7. Applied add-log-exp 12.0

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

   8. Simplified 12.0

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

   10. Applied *-un-lft-identity 12.0

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

   11. Applied add-cube-cbrt 12.1

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

   12. Applied times-frac 12.1

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

   13. Simplified 12.1

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

   if -3.2314583272364048e-62 < z

   1. Initial program 30.9

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

   3. Applied sub-neg 30.9

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

   4. Applied associate-+l+ 15.3

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

   5. Simplified 10.9

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

   7. Applied add-log-exp 10.9

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

   8. Simplified 10.9

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

   9. Taylor expanded around 0 6.6

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

   10. Simplified 6.6

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

4. Final simplification 8.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.2314583272364048 \cdot 10^{-62}:\\ \;\;\;\;x - \left(\sqrt[3]{\log \left(\mathsf{fma}\left(\mathsf{expm1}\left(z\right), y, 1\right)\right)} \cdot \sqrt[3]{\log \left(\mathsf{fma}\left(\mathsf{expm1}\left(z\right), y, 1\right)\right)}\right) \cdot \frac{\sqrt[3]{\log \left(\mathsf{fma}\left(\mathsf{expm1}\left(z\right), y, 1\right)\right)}}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot z, z, 1 \cdot z\right), y, \log 1\right)}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2020105 +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)))