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

Average Error: 24.3 → 7.8

Time: 12.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -1.93917976820741827 \cdot 10^{-4}:\\ \;\;\;\;x - \log \left(\left(1 - y\right) + y \cdot e^{z}\right) \cdot \frac{1}{t}\\ \mathbf{elif}\;z \le 1.4219674450745225 \cdot 10^{-132}:\\ \;\;\;\;x - \left(1 \cdot \left(\frac{z}{t} \cdot y\right) + \frac{\log 1}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + y \cdot \left(\frac{1}{2} \cdot {z}^{2} + z\right)\right)}{t}\\ \end{array}\]

C

double code(double x, double y, double z, double t) {
	return ((double) (x - ((double) (((double) log(((double) (((double) (1.0 - y)) + ((double) (y * ((double) exp(z)))))))) / t))));
}

↓

double code(double x, double y, double z, double t) {
	double VAR;
	if ((z <= -0.00019391797682074183)) {
		VAR = ((double) (x - ((double) (((double) log(((double) (((double) (1.0 - y)) + ((double) (y * ((double) exp(z)))))))) * ((double) (1.0 / t))))));
	} else {
		double VAR_1;
		if ((z <= 1.4219674450745225e-132)) {
			VAR_1 = ((double) (x - ((double) (((double) (1.0 * ((double) (((double) (z / t)) * y)))) + ((double) (((double) log(1.0)) / t))))));
		} else {
			VAR_1 = ((double) (x - ((double) (((double) log(((double) (1.0 + ((double) (y * ((double) (((double) (0.5 * ((double) pow(z, 2.0)))) + z)))))))) / t))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	24.3
Target	15.3
Herbie	7.8

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

Derivation

1. Split input into 3 regimes

2. if z < -1.93917976820741827e-4

   1. Initial program 9.9

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

   3. Applied div-inv 9.9

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

   if -1.93917976820741827e-4 < z < 1.4219674450745225e-132

   1. Initial program 30.8

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

   2. Taylor expanded around 0 6.1

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

   3. Simplified 6.1

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

   4. Taylor expanded around 0 6.1

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

   6. Applied associate-/l* 8.1

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

   8. Applied associate-/r/ 5.4

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

   if 1.4219674450745225e-132 < z

   1. Initial program 30.4

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

   2. Taylor expanded around 0 12.4

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

   3. Simplified 12.4

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

4. Final simplification 7.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.93917976820741827 \cdot 10^{-4}:\\ \;\;\;\;x - \log \left(\left(1 - y\right) + y \cdot e^{z}\right) \cdot \frac{1}{t}\\ \mathbf{elif}\;z \le 1.4219674450745225 \cdot 10^{-132}:\\ \;\;\;\;x - \left(1 \cdot \left(\frac{z}{t} \cdot y\right) + \frac{\log 1}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + y \cdot \left(\frac{1}{2} \cdot {z}^{2} + z\right)\right)}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2020162 
(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 (/ (/ (neg 0.5) (* y t)) (* z z))) (* (/ (neg 0.5) (* y t)) (/ (/ 2.0 z) (* z z)))) (- x (/ (log (+ 1.0 (* z y))) t)))

  (- x (/ (log (+ (- 1.0 y) (* y (exp z)))) t)))