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

Average Error: 25.1 → 8.8

Time: 7.4s

Precision: 64

• could not determine a ground truth for program body

  1. x = 8.677094401051118e+132
  2. y = 1.662100557100748e-24
  3. z = 2.2737189935205076e+82
  4. t = -6.442308509514324e-171

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -7.2452238851260002 \cdot 10^{-14}:\\ \;\;\;\;x - \frac{\log \left(\sqrt{\left(1 - y\right) + y \cdot e^{z}}\right) + \log \left(\sqrt{\left(1 - y\right) + y \cdot e^{z}}\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \mathsf{fma}\left(\left(\sqrt[3]{\frac{z \cdot y}{t}} \cdot \sqrt[3]{\frac{z \cdot y}{t}}\right) \cdot \left(\sqrt[3]{z \cdot y} \cdot \sqrt[3]{\frac{1}{t}}\right), 1, \mathsf{fma}\left(0.5, \frac{{z}^{2} \cdot y}{t}, \frac{\log 1}{t}\right)\right)\\ \end{array}\]

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 VAR;
	if ((z <= -7.245223885126e-14)) {
		VAR = (x - ((log(sqrt(((1.0 - y) + (y * exp(z))))) + log(sqrt(((1.0 - y) + (y * exp(z)))))) / t));
	} else {
		VAR = (x - fma(((cbrt(((z * y) / t)) * cbrt(((z * y) / t))) * (cbrt((z * y)) * cbrt((1.0 / t)))), 1.0, fma(0.5, ((pow(z, 2.0) * y) / t), (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	16.2
Herbie	8.8

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

Derivation

1. Split input into 2 regimes

2. if z < -7.245223885126e-14

   1. Initial program 11.9

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

   3. Applied add-sqr-sqrt 11.9

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

   4. Applied log-prod 11.9

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

   if -7.245223885126e-14 < z

   1. Initial program 31.0

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

   2. Taylor expanded around 0 7.2

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

   3. Simplified 7.2

      \[\leadsto x - \color{blue}{\mathsf{fma}\left(\frac{z \cdot y}{t}, 1, \mathsf{fma}\left(0.5, \frac{{z}^{2} \cdot y}{t}, \frac{\log 1}{t}\right)\right)}\]
   4. Using strategy rm

   5. Applied add-cube-cbrt 7.4

      \[\leadsto x - \mathsf{fma}\left(\color{blue}{\left(\sqrt[3]{\frac{z \cdot y}{t}} \cdot \sqrt[3]{\frac{z \cdot y}{t}}\right) \cdot \sqrt[3]{\frac{z \cdot y}{t}}}, 1, \mathsf{fma}\left(0.5, \frac{{z}^{2} \cdot y}{t}, \frac{\log 1}{t}\right)\right)\]
   6. Using strategy rm

   7. Applied div-inv 7.4

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

   8. Applied cbrt-prod 7.4

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

4. Final simplification 8.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -7.2452238851260002 \cdot 10^{-14}:\\ \;\;\;\;x - \frac{\log \left(\sqrt{\left(1 - y\right) + y \cdot e^{z}}\right) + \log \left(\sqrt{\left(1 - y\right) + y \cdot e^{z}}\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \mathsf{fma}\left(\left(\sqrt[3]{\frac{z \cdot y}{t}} \cdot \sqrt[3]{\frac{z \cdot y}{t}}\right) \cdot \left(\sqrt[3]{z \cdot y} \cdot \sqrt[3]{\frac{1}{t}}\right), 1, \mathsf{fma}\left(0.5, \frac{{z}^{2} \cdot y}{t}, \frac{\log 1}{t}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020075 +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)))