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

Average Error: 25.1 → 9.2

Time: 7.2s

Precision: 64

• could not determine a ground truth for program body

  1. x = 5.2835940265373795e-264
  2. y = 1.6930423734106654e+204
  3. z = 2.641081399364923e+158
  4. t = 2.955234747016825e-260

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -5.35935564879756704 \cdot 10^{-216}:\\ \;\;\;\;x - \frac{\log \left(1 + \left(y \cdot \left(\sqrt[3]{\mathsf{expm1}\left(z\right)} \cdot \sqrt[3]{\mathsf{expm1}\left(z\right)}\right)\right) \cdot \sqrt[3]{\mathsf{expm1}\left(z\right)}\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \mathsf{fma}\left(\frac{z \cdot y}{t}, 1, \mathsf{fma}\left(0.5, \frac{{z}^{2} \cdot y}{t}, \frac{\log 1}{t}\right)\right)\\ \end{array}\]

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 ((z <= -5.359355648797567e-216)) {
		temp = (x - (log((1.0 + ((y * (cbrt(expm1(z)) * cbrt(expm1(z)))) * cbrt(expm1(z))))) / 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	25.1
Target	16.1
Herbie	9.2

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

Derivation

1. Split input into 2 regimes

2. if z < -5.359355648797567e-216

   1. Initial program 19.8

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

   3. Applied sub-neg 19.8

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

   4. Applied associate-+l+ 14.4

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

   5. Simplified 11.6

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

   7. Applied add-cube-cbrt 11.6

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

   8. Applied associate-*r* 11.6

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

   if -5.359355648797567e-216 < z

   1. Initial program 31.3

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

   2. Taylor expanded around 0 6.3

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

   3. Simplified 6.3

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

4. Final simplification 9.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -5.35935564879756704 \cdot 10^{-216}:\\ \;\;\;\;x - \frac{\log \left(1 + \left(y \cdot \left(\sqrt[3]{\mathsf{expm1}\left(z\right)} \cdot \sqrt[3]{\mathsf{expm1}\left(z\right)}\right)\right) \cdot \sqrt[3]{\mathsf{expm1}\left(z\right)}\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \mathsf{fma}\left(\frac{z \cdot y}{t}, 1, \mathsf{fma}\left(0.5, \frac{{z}^{2} \cdot y}{t}, \frac{\log 1}{t}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020058 +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)))