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

Average Error: 24.7 → 8.1

Time: 6.2s

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

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -226906882.284806222:\\ \;\;\;\;x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \left(1 \cdot \frac{y}{\frac{t}{z}} + \frac{\log 1}{t}\right)\\ \end{array}\]

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 <= -226906882.28480622)) {
		VAR = (x - (log(((1.0 - y) + (y * exp(z)))) / t));
	} else {
		VAR = (x - ((1.0 * (y / (t / z))) + (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	8.1

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

Derivation

1. Split input into 2 regimes

2. if z < -226906882.28480622

   1. Initial program 10.9

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

   if -226906882.28480622 < z

   1. Initial program 30.2

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

   2. Taylor expanded around 0 7.5

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

   3. Simplified 7.5

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

   4. Taylor expanded around 0 7.6

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

   6. Applied *-commutative 7.6

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

   7. Applied associate-/l* 6.9

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

4. Final simplification 8.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -226906882.284806222:\\ \;\;\;\;x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \left(1 \cdot \frac{y}{\frac{t}{z}} + \frac{\log 1}{t}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 
(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)))