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

Average Error: 25.0 → 10.7

Time: 9.8s

Precision: 64

• could not determine a ground truth for program body

  1. x = -1.8195025401945306e+260
  2. y = 6.646099188021102e+261
  3. z = 1.1234894225173824e+95
  4. t = -3.904071768339897e-136

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le 3.0282374466509755 \cdot 10^{-40} \lor \neg \left(y \le 2.29997582564267609 \cdot 10^{127}\right):\\ \;\;\;\;x - \left(\frac{1}{2} \cdot \log \left(1 + y \cdot \mathsf{expm1}\left(z\right)\right) + \log \left(\sqrt{1 + y \cdot \mathsf{expm1}\left(z\right)}\right)\right) \cdot \frac{1}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \mathsf{fma}\left(\frac{z \cdot y}{t}, 1, \mathsf{fma}\left(0.5, \frac{{z}^{2} \cdot y}{t}, \frac{\log 1}{t}\right)\right)\\ \end{array}\]

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 temp;
	if (((y <= 3.0282374466509755e-40) || !(y <= 2.299975825642676e+127))) {
		temp = (x - ((((1.0 / 2.0) * log((1.0 + (y * expm1(z))))) + log(sqrt((1.0 + (y * expm1(z)))))) * (1.0 / t)));
	} else {
		temp = (x - fma(((z * y) / t), 1.0, fma(0.5, ((pow(z, 2.0) * y) / t), (log(1.0) / t))));
	}
	return temp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	25.0
Target	16.1
Herbie	10.7

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

Derivation

1. Split input into 2 regimes

2. if y < 3.0282374466509755e-40 or 2.299975825642676e+127 < y

   1. Initial program 22.6

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

   3. Applied sub-neg 22.6

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

   4. Applied associate-+l+ 14.0

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

   5. Simplified 10.8

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

   7. Applied div-inv 10.8

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

   9. Applied add-sqr-sqrt 10.8

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

   10. Applied log-prod 10.8

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

   12. Applied pow1 10.8

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

   13. Applied sqrt-pow1 10.8

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

   14. Applied log-pow 10.8

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

   if 3.0282374466509755e-40 < y < 2.299975825642676e+127

   1. Initial program 42.7

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

   2. Taylor expanded around 0 10.1

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

   3. Simplified 10.1

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

4. Final simplification 10.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le 3.0282374466509755 \cdot 10^{-40} \lor \neg \left(y \le 2.29997582564267609 \cdot 10^{127}\right):\\ \;\;\;\;x - \left(\frac{1}{2} \cdot \log \left(1 + y \cdot \mathsf{expm1}\left(z\right)\right) + \log \left(\sqrt{1 + y \cdot \mathsf{expm1}\left(z\right)}\right)\right) \cdot \frac{1}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \mathsf{fma}\left(\frac{z \cdot y}{t}, 1, \mathsf{fma}\left(0.5, \frac{{z}^{2} \cdot y}{t}, \frac{\log 1}{t}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 +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)))