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

Average Error: 25.2 → 8.1

Time: 7.6s

Precision: 64

• could not determine a ground truth for program body

  1. x = -2.0579672449480405e-99
  2. y = 7.629327140293554e+242
  3. z = 7.84958463149562e+22
  4. t = -9.116935142050009e-235

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;e^{z} \le 0.0:\\ \;\;\;\;x - \frac{\log \left(\left(1 - y\right) + \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(\sqrt[3]{y} \cdot e^{z}\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \left(1 \cdot \left(\left(z \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{t}}\right) + \left(\frac{\log 1}{t} + 0.5 \cdot \frac{{z}^{2} \cdot y}{t}\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r315267 = x;
        double r315268 = 1.0;
        double r315269 = y;
        double r315270 = r315268 - r315269;
        double r315271 = z;
        double r315272 = exp(r315271);
        double r315273 = r315269 * r315272;
        double r315274 = r315270 + r315273;
        double r315275 = log(r315274);
        double r315276 = t;
        double r315277 = r315275 / r315276;
        double r315278 = r315267 - r315277;
        return r315278;
}

↓

double f(double x, double y, double z, double t) {
        double r315279 = z;
        double r315280 = exp(r315279);
        double r315281 = 0.0;
        bool r315282 = r315280 <= r315281;
        double r315283 = x;
        double r315284 = 1.0;
        double r315285 = y;
        double r315286 = r315284 - r315285;
        double r315287 = cbrt(r315285);
        double r315288 = r315287 * r315287;
        double r315289 = r315287 * r315280;
        double r315290 = r315288 * r315289;
        double r315291 = r315286 + r315290;
        double r315292 = log(r315291);
        double r315293 = t;
        double r315294 = r315292 / r315293;
        double r315295 = r315283 - r315294;
        double r315296 = cbrt(r315293);
        double r315297 = r315296 * r315296;
        double r315298 = r315288 / r315297;
        double r315299 = r315279 * r315298;
        double r315300 = r315287 / r315296;
        double r315301 = r315299 * r315300;
        double r315302 = r315284 * r315301;
        double r315303 = log(r315284);
        double r315304 = r315303 / r315293;
        double r315305 = 0.5;
        double r315306 = 2.0;
        double r315307 = pow(r315279, r315306);
        double r315308 = r315307 * r315285;
        double r315309 = r315308 / r315293;
        double r315310 = r315305 * r315309;
        double r315311 = r315304 + r315310;
        double r315312 = r315302 + r315311;
        double r315313 = r315283 - r315312;
        double r315314 = r315282 ? r315295 : r315313;
        return r315314;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	25.2
Target	16.6
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 11.5

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

   3. Applied add-cube-cbrt 11.5

      \[\leadsto x - \frac{\log \left(\left(1 - y\right) + \color{blue}{\left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}\right)} \cdot e^{z}\right)}{t}\]

   4. Applied associate-*l* 11.5

      \[\leadsto x - \frac{\log \left(\left(1 - y\right) + \color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(\sqrt[3]{y} \cdot e^{z}\right)}\right)}{t}\]

   if 0.0 < (exp z)

   1. Initial program 31.2

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

   2. Taylor expanded around 0 7.1

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

   4. Applied *-un-lft-identity 7.1

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

   5. Applied times-frac 9.4

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

   6. Simplified 9.4

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

   8. Applied add-cube-cbrt 9.6

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

   9. Applied add-cube-cbrt 9.6

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

   10. Applied times-frac 9.6

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

   11. Applied associate-*r* 6.6

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

4. Final simplification 8.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \le 0.0:\\ \;\;\;\;x - \frac{\log \left(\left(1 - y\right) + \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(\sqrt[3]{y} \cdot e^{z}\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \left(1 \cdot \left(\left(z \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{t}}\right) + \left(\frac{\log 1}{t} + 0.5 \cdot \frac{{z}^{2} \cdot y}{t}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020064 
(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)))