Diagrams.Trail:splitAtParam from diagrams-lib-1.3.0.3, A

Average Error: 7.4 → 3.7

Time: 4.8s

Precision: 64

Math

\[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]

↓

\[\begin{array}{l} \mathbf{if}\;z \le -6.72915306137204471 \cdot 10^{92} \lor \neg \left(z \le 6.17993228896471188 \cdot 10^{40}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{1}{\frac{t \cdot z - x}{y \cdot z - x}}}{x + 1}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r663339 = x;
        double r663340 = y;
        double r663341 = z;
        double r663342 = r663340 * r663341;
        double r663343 = r663342 - r663339;
        double r663344 = t;
        double r663345 = r663344 * r663341;
        double r663346 = r663345 - r663339;
        double r663347 = r663343 / r663346;
        double r663348 = r663339 + r663347;
        double r663349 = 1.0;
        double r663350 = r663339 + r663349;
        double r663351 = r663348 / r663350;
        return r663351;
}

↓

double f(double x, double y, double z, double t) {
        double r663352 = z;
        double r663353 = -6.729153061372045e+92;
        bool r663354 = r663352 <= r663353;
        double r663355 = 6.179932288964712e+40;
        bool r663356 = r663352 <= r663355;
        double r663357 = !r663356;
        bool r663358 = r663354 || r663357;
        double r663359 = x;
        double r663360 = y;
        double r663361 = t;
        double r663362 = r663360 / r663361;
        double r663363 = r663359 + r663362;
        double r663364 = 1.0;
        double r663365 = r663359 + r663364;
        double r663366 = r663363 / r663365;
        double r663367 = 1.0;
        double r663368 = r663361 * r663352;
        double r663369 = r663368 - r663359;
        double r663370 = r663360 * r663352;
        double r663371 = r663370 - r663359;
        double r663372 = r663369 / r663371;
        double r663373 = r663367 / r663372;
        double r663374 = r663359 + r663373;
        double r663375 = r663374 / r663365;
        double r663376 = r663358 ? r663366 : r663375;
        return r663376;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	7.4
Target	0.3
Herbie	3.7

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

Derivation

1. Split input into 2 regimes

2. if z < -6.729153061372045e+92 or 6.179932288964712e+40 < z

   1. Initial program 18.4

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]

   2. Taylor expanded around inf 8.6

      \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1}\]

   if -6.729153061372045e+92 < z < 6.179932288964712e+40

   1. Initial program 0.8

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]
   2. Using strategy rm

   3. Applied clear-num 0.8

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

4. Final simplification 3.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -6.72915306137204471 \cdot 10^{92} \lor \neg \left(z \le 6.17993228896471188 \cdot 10^{40}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{1}{\frac{t \cdot z - x}{y \cdot z - x}}}{x + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 
(FPCore (x y z t)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, A"
  :precision binary64

  :herbie-target
  (/ (+ x (- (/ y (- t (/ x z))) (/ x (- (* t z) x)))) (+ x 1))

  (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1)))