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

Average Error: 25.1 → 8.4

Time: 7.9s

Precision: 64

• could not determine a ground truth for program body

  1. x = 1.3518618556854088e-159
  2. y = 5.370988944379436e-172
  3. z = 2.0341971232729314e+104
  4. t = -4.662252843282357e+284

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -4.75482599374740609 \cdot 10^{-50}:\\ \;\;\;\;x - \frac{2 \cdot \left(\log \left(\sqrt{\sqrt[3]{1 + y \cdot \mathsf{expm1}\left(z\right)}}\right) + \log \left(\sqrt{\sqrt[3]{1 + y \cdot \mathsf{expm1}\left(z\right)}}\right)\right) + \log \left(\sqrt[3]{1 + y \cdot \mathsf{expm1}\left(z\right)}\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{fma}\left(0.5, {z}^{2} \cdot y, \mathsf{fma}\left(1, z \cdot y, \log 1\right)\right)}{t}\\ \end{array}\]

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 <= -4.754825993747406e-50)) {
		VAR = (x - (((2.0 * (log(sqrt(cbrt((1.0 + (y * expm1(z)))))) + log(sqrt(cbrt((1.0 + (y * expm1(z)))))))) + log(cbrt((1.0 + (y * expm1(z)))))) / t));
	} else {
		VAR = (x - (fma(0.5, (pow(z, 2.0) * y), fma(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	16.4
Herbie	8.4

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

Derivation

1. Split input into 2 regimes

2. if z < -4.754825993747406e-50

   1. Initial program 13.1

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

   3. Applied sub-neg 13.1

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

   4. Applied associate-+l+ 11.9

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

   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-cube-cbrt 10.9

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

   8. Applied log-prod 10.9

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

   9. Simplified 10.9

      \[\leadsto x - \frac{\color{blue}{2 \cdot \log \left(\sqrt[3]{1 + y \cdot \mathsf{expm1}\left(z\right)}\right)} + \log \left(\sqrt[3]{1 + y \cdot \mathsf{expm1}\left(z\right)}\right)}{t}\]
   10. Using strategy rm

   11. Applied add-sqr-sqrt 10.9

       \[\leadsto x - \frac{2 \cdot \log \color{blue}{\left(\sqrt{\sqrt[3]{1 + y \cdot \mathsf{expm1}\left(z\right)}} \cdot \sqrt{\sqrt[3]{1 + y \cdot \mathsf{expm1}\left(z\right)}}\right)} + \log \left(\sqrt[3]{1 + y \cdot \mathsf{expm1}\left(z\right)}\right)}{t}\]

   12. Applied log-prod 10.9

       \[\leadsto x - \frac{2 \cdot \color{blue}{\left(\log \left(\sqrt{\sqrt[3]{1 + y \cdot \mathsf{expm1}\left(z\right)}}\right) + \log \left(\sqrt{\sqrt[3]{1 + y \cdot \mathsf{expm1}\left(z\right)}}\right)\right)} + \log \left(\sqrt[3]{1 + y \cdot \mathsf{expm1}\left(z\right)}\right)}{t}\]

   if -4.754825993747406e-50 < z

   1. Initial program 31.6

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

   2. Taylor expanded around 0 7.1

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

   3. Simplified 7.1

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

4. Final simplification 8.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -4.75482599374740609 \cdot 10^{-50}:\\ \;\;\;\;x - \frac{2 \cdot \left(\log \left(\sqrt{\sqrt[3]{1 + y \cdot \mathsf{expm1}\left(z\right)}}\right) + \log \left(\sqrt{\sqrt[3]{1 + y \cdot \mathsf{expm1}\left(z\right)}}\right)\right) + \log \left(\sqrt[3]{1 + y \cdot \mathsf{expm1}\left(z\right)}\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{fma}\left(0.5, {z}^{2} \cdot y, \mathsf{fma}\left(1, z \cdot y, \log 1\right)\right)}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2020102 +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)))