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

Average Error: 7.6 → 4.5

Time: 4.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \le -2.72059074699743426 \cdot 10^{236} \lor \neg \left(\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \le 2.40235735934087887 \cdot 10^{45}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r804376 = x;
        double r804377 = y;
        double r804378 = z;
        double r804379 = r804377 * r804378;
        double r804380 = r804379 - r804376;
        double r804381 = t;
        double r804382 = r804381 * r804378;
        double r804383 = r804382 - r804376;
        double r804384 = r804380 / r804383;
        double r804385 = r804376 + r804384;
        double r804386 = 1.0;
        double r804387 = r804376 + r804386;
        double r804388 = r804385 / r804387;
        return r804388;
}

↓

double f(double x, double y, double z, double t) {
        double r804389 = x;
        double r804390 = y;
        double r804391 = z;
        double r804392 = r804390 * r804391;
        double r804393 = r804392 - r804389;
        double r804394 = t;
        double r804395 = r804394 * r804391;
        double r804396 = r804395 - r804389;
        double r804397 = r804393 / r804396;
        double r804398 = r804389 + r804397;
        double r804399 = 1.0;
        double r804400 = r804389 + r804399;
        double r804401 = r804398 / r804400;
        double r804402 = -2.7205907469974343e+236;
        bool r804403 = r804401 <= r804402;
        double r804404 = 2.402357359340879e+45;
        bool r804405 = r804401 <= r804404;
        double r804406 = !r804405;
        bool r804407 = r804403 || r804406;
        double r804408 = r804390 / r804394;
        double r804409 = r804389 + r804408;
        double r804410 = r804409 / r804400;
        double r804411 = r804407 ? r804410 : r804401;
        return r804411;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	7.6
Target	0.4
Herbie	4.5

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

Derivation

1. Split input into 2 regimes

2. if (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)) < -2.7205907469974343e+236 or 2.402357359340879e+45 < (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0))

   1. Initial program 37.4

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

   2. Taylor expanded around inf 20.6

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

   if -2.7205907469974343e+236 < (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)) < 2.402357359340879e+45

   1. Initial program 0.8

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

4. Final simplification 4.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \le -2.72059074699743426 \cdot 10^{236} \lor \neg \left(\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \le 2.40235735934087887 \cdot 10^{45}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 
(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)))