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

Average Error: 25.0 → 10.2

Time: 7.5s

Precision: 64

• could not determine a ground truth for program body

  1. x = 2.176255397046734e-102
  2. y = 6.188380233264385e+102
  3. z = 8.769655940674386e+155
  4. t = 6.60673625524331e-128

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -2.77926232449136807 \cdot 10^{44}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(\sqrt{1 - y}, \sqrt{1 - y}, y \cdot e^{z}\right)\right)}{t}\\ \mathbf{elif}\;z \le -8.89077092915690779 \cdot 10^{-119}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(\frac{1}{2}, {z}^{2} \cdot y, \mathsf{fma}\left(z, y, 1\right)\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \mathsf{fma}\left(\frac{z \cdot y}{t}, 1, \mathsf{fma}\left(0.5, \frac{{z}^{2} \cdot y}{t}, \frac{\log 1}{t}\right)\right)\\ \end{array}\]

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 <= -2.779262324491368e+44)) {
		temp = (x - (log(fma(sqrt((1.0 - y)), sqrt((1.0 - y)), (y * exp(z)))) / t));
	} else {
		double temp_1;
		if ((z <= -8.890770929156908e-119)) {
			temp_1 = (x - (log(fma(0.5, (pow(z, 2.0) * y), fma(z, y, 1.0))) / t));
		} else {
			temp_1 = (x - fma(((z * y) / t), 1.0, fma(0.5, ((pow(z, 2.0) * y) / t), (log(1.0) / t))));
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	25.0
Target	16.6
Herbie	10.2

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

Derivation

1. Split input into 3 regimes

2. if z < -2.779262324491368e+44

   1. Initial program 12.2

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

   3. Applied add-sqr-sqrt 12.2

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

   4. Applied fma-def 12.2

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

   if -2.779262324491368e+44 < z < -8.890770929156908e-119

   1. Initial program 25.1

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

   2. Taylor expanded around 0 18.6

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

   3. Simplified 18.6

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

   if -8.890770929156908e-119 < 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 6.9

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

   3. Simplified 6.9

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

4. Final simplification 10.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.77926232449136807 \cdot 10^{44}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(\sqrt{1 - y}, \sqrt{1 - y}, y \cdot e^{z}\right)\right)}{t}\\ \mathbf{elif}\;z \le -8.89077092915690779 \cdot 10^{-119}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(\frac{1}{2}, {z}^{2} \cdot y, \mathsf{fma}\left(z, y, 1\right)\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \mathsf{fma}\left(\frac{z \cdot y}{t}, 1, \mathsf{fma}\left(0.5, \frac{{z}^{2} \cdot y}{t}, \frac{\log 1}{t}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020066 +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)))