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

Average Error: 24.7 → 8.8

Time: 9.2s

Precision: 64

• could not determine a ground truth for program body

  1. x = 2.2108085494553437e+120
  2. y = 7.734281002290599e-114
  3. z = 3.1637604227072965e+33
  4. t = 1.0286577002489468e+63

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -47013041841263568:\\ \;\;\;\;x - \frac{1}{\frac{t}{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log 1 + y \cdot \left(0.5 \cdot {z}^{2} + 1 \cdot z\right)}{t}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r338390 = x;
        double r338391 = 1.0;
        double r338392 = y;
        double r338393 = r338391 - r338392;
        double r338394 = z;
        double r338395 = exp(r338394);
        double r338396 = r338392 * r338395;
        double r338397 = r338393 + r338396;
        double r338398 = log(r338397);
        double r338399 = t;
        double r338400 = r338398 / r338399;
        double r338401 = r338390 - r338400;
        return r338401;
}

↓

double f(double x, double y, double z, double t) {
        double r338402 = z;
        double r338403 = -4.701304184126357e+16;
        bool r338404 = r338402 <= r338403;
        double r338405 = x;
        double r338406 = 1.0;
        double r338407 = t;
        double r338408 = 1.0;
        double r338409 = y;
        double r338410 = r338408 - r338409;
        double r338411 = exp(r338402);
        double r338412 = r338409 * r338411;
        double r338413 = r338410 + r338412;
        double r338414 = log(r338413);
        double r338415 = r338407 / r338414;
        double r338416 = r338406 / r338415;
        double r338417 = r338405 - r338416;
        double r338418 = log(r338408);
        double r338419 = 0.5;
        double r338420 = 2.0;
        double r338421 = pow(r338402, r338420);
        double r338422 = r338419 * r338421;
        double r338423 = r338408 * r338402;
        double r338424 = r338422 + r338423;
        double r338425 = r338409 * r338424;
        double r338426 = r338418 + r338425;
        double r338427 = r338426 / r338407;
        double r338428 = r338405 - r338427;
        double r338429 = r338404 ? r338417 : r338428;
        return r338429;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	24.7
Target	16.3
Herbie	8.8

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

Derivation

1. Split input into 2 regimes

2. if z < -4.701304184126357e+16

   1. Initial program 11.3

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

   3. Applied clear-num 11.3

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

   if -4.701304184126357e+16 < 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 7.8

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

   3. Simplified 7.8

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

4. Final simplification 8.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -47013041841263568:\\ \;\;\;\;x - \frac{1}{\frac{t}{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log 1 + y \cdot \left(0.5 \cdot {z}^{2} + 1 \cdot z\right)}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(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)))