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

Average Error: 24.9 → 8.1

Time: 19.5s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

• could not determine a ground truth for program body

  1. x = 8.463071591988411e-41
  2. y = 5.028654687011449e-267
  3. z = 2.7208697733301495e+159
  4. t = 5.176829154239403e-35

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;e^{z} \le 0.0:\\ \;\;\;\;x - \log \left(\left(1 - y\right) + y \cdot e^{z}\right) \cdot \frac{1}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \left(1 \cdot \left(\left(z \cdot y\right) \cdot \frac{1}{t}\right) + \left(\frac{\log 1}{t} + 0.5 \cdot \frac{{z}^{2} \cdot y}{t}\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r413192 = x;
        double r413193 = 1.0;
        double r413194 = y;
        double r413195 = r413193 - r413194;
        double r413196 = z;
        double r413197 = exp(r413196);
        double r413198 = r413194 * r413197;
        double r413199 = r413195 + r413198;
        double r413200 = log(r413199);
        double r413201 = t;
        double r413202 = r413200 / r413201;
        double r413203 = r413192 - r413202;
        return r413203;
}

↓

double f(double x, double y, double z, double t) {
        double r413204 = z;
        double r413205 = exp(r413204);
        double r413206 = 0.0;
        bool r413207 = r413205 <= r413206;
        double r413208 = x;
        double r413209 = 1.0;
        double r413210 = y;
        double r413211 = r413209 - r413210;
        double r413212 = r413210 * r413205;
        double r413213 = r413211 + r413212;
        double r413214 = log(r413213);
        double r413215 = 1.0;
        double r413216 = t;
        double r413217 = r413215 / r413216;
        double r413218 = r413214 * r413217;
        double r413219 = r413208 - r413218;
        double r413220 = r413204 * r413210;
        double r413221 = r413220 * r413217;
        double r413222 = r413209 * r413221;
        double r413223 = log(r413209);
        double r413224 = r413223 / r413216;
        double r413225 = 0.5;
        double r413226 = 2.0;
        double r413227 = pow(r413204, r413226);
        double r413228 = r413227 * r413210;
        double r413229 = r413228 / r413216;
        double r413230 = r413225 * r413229;
        double r413231 = r413224 + r413230;
        double r413232 = r413222 + r413231;
        double r413233 = r413208 - r413232;
        double r413234 = r413207 ? r413219 : r413233;
        return r413234;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	24.9
Target	15.6
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 (exp z) < 0.0

   1. Initial program 11.0

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

   3. Applied div-inv 11.1

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

   if 0.0 < (exp z)

   1. Initial program 30.5

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

   2. Taylor expanded around 0 6.8

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

   4. Applied div-inv 6.9

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

4. Final simplification 8.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \le 0.0:\\ \;\;\;\;x - \log \left(\left(1 - y\right) + y \cdot e^{z}\right) \cdot \frac{1}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \left(1 \cdot \left(\left(z \cdot y\right) \cdot \frac{1}{t}\right) + \left(\frac{\log 1}{t} + 0.5 \cdot \frac{{z}^{2} \cdot y}{t}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(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)))