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

Average Error: 24.7 → 8.6

Time: 8.0s

Precision: 64

• could not determine a ground truth for program body

  1. x = 9.272403512084269e-250
  2. y = 5.757546373319306e-53
  3. z = 1.1261706166891039e+234
  4. t = -8.163887581353229e-84

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -1.2015657265463743 \cdot 10^{-9}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}, \sqrt[3]{1 - y}, y \cdot e^{z}\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \mathsf{fma}\left(\frac{z \cdot y}{t}, 1, \mathsf{fma}\left(0.5, \frac{\sqrt[3]{{\left({z}^{2} \cdot y\right)}^{3}}}{t}, \frac{\log 1}{t}\right)\right)\\ \end{array}\]

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 <= -1.2015657265463743e-09)) {
		VAR = (x - (log(fma((cbrt((1.0 - y)) * cbrt((1.0 - y))), cbrt((1.0 - y)), (y * exp(z)))) / t));
	} else {
		VAR = (x - fma(((z * y) / t), 1.0, fma(0.5, (cbrt(pow((pow(z, 2.0) * y), 3.0)) / 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	24.7
Target	15.7
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 < -1.2015657265463743e-09

   1. Initial program 10.7

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

   3. Applied add-cube-cbrt 10.8

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

   4. Applied fma-def 10.8

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

   if -1.2015657265463743e-09 < z

   1. Initial program 30.7

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

   2. Taylor expanded around 0 7.1

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

   3. Simplified 7.1

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

   5. Applied add-cbrt-cube 17.8

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

   6. Applied add-cbrt-cube 17.8

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

   7. Applied cbrt-unprod 17.8

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

   8. Simplified 7.6

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

4. Final simplification 8.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.2015657265463743 \cdot 10^{-9}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}, \sqrt[3]{1 - y}, y \cdot e^{z}\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \mathsf{fma}\left(\frac{z \cdot y}{t}, 1, \mathsf{fma}\left(0.5, \frac{\sqrt[3]{{\left({z}^{2} \cdot y\right)}^{3}}}{t}, \frac{\log 1}{t}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 +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)))