Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1

Average Error: 15.1 → 3.7

Time: 10.9s

Precision: 64

Math

\[x \cdot \frac{\frac{y}{z} \cdot t}{t}\]

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -1.63086994211637596483956921546385031623 \cdot 10^{66}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;\frac{y}{z} \le -6.545487680777495021156682531032583832353 \cdot 10^{-230}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;\frac{y}{z} \le 4.814036022956164419199695264571075550494 \cdot 10^{-12}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r77499 = x;
        double r77500 = y;
        double r77501 = z;
        double r77502 = r77500 / r77501;
        double r77503 = t;
        double r77504 = r77502 * r77503;
        double r77505 = r77504 / r77503;
        double r77506 = r77499 * r77505;
        return r77506;
}

↓

double f(double x, double y, double z, double __attribute__((unused)) t) {
        double r77507 = y;
        double r77508 = z;
        double r77509 = r77507 / r77508;
        double r77510 = -1.630869942116376e+66;
        bool r77511 = r77509 <= r77510;
        double r77512 = x;
        double r77513 = r77507 * r77512;
        double r77514 = r77513 / r77508;
        double r77515 = -6.545487680777495e-230;
        bool r77516 = r77509 <= r77515;
        double r77517 = r77509 * r77512;
        double r77518 = 4.8140360229561644e-12;
        bool r77519 = r77509 <= r77518;
        double r77520 = r77508 / r77512;
        double r77521 = r77507 / r77520;
        double r77522 = r77519 ? r77514 : r77521;
        double r77523 = r77516 ? r77517 : r77522;
        double r77524 = r77511 ? r77514 : r77523;
        return r77524;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Derivation

1. Split input into 3 regimes

2. if (/ y z) < -1.630869942116376e+66 or -6.545487680777495e-230 < (/ y z) < 4.8140360229561644e-12

   1. Initial program 17.6

      \[x \cdot \frac{\frac{y}{z} \cdot t}{t}\]

   2. Simplified 3.9

      \[\leadsto \color{blue}{\frac{y \cdot x}{z}}\]

   if -1.630869942116376e+66 < (/ y z) < -6.545487680777495e-230

   1. Initial program 6.5

      \[x \cdot \frac{\frac{y}{z} \cdot t}{t}\]

   2. Simplified 9.8

      \[\leadsto \color{blue}{\frac{y \cdot x}{z}}\]
   3. Using strategy rm

   4. Applied associate-/l* 9.1

      \[\leadsto \color{blue}{\frac{y}{\frac{z}{x}}}\]
   5. Using strategy rm

   6. Applied associate-/r/ 0.2

      \[\leadsto \color{blue}{\frac{y}{z} \cdot x}\]

   if 4.8140360229561644e-12 < (/ y z)

   1. Initial program 18.7

      \[x \cdot \frac{\frac{y}{z} \cdot t}{t}\]

   2. Simplified 8.3

      \[\leadsto \color{blue}{\frac{y \cdot x}{z}}\]
   3. Using strategy rm

   4. Applied associate-/l* 7.3

      \[\leadsto \color{blue}{\frac{y}{\frac{z}{x}}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 3.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -1.63086994211637596483956921546385031623 \cdot 10^{66}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;\frac{y}{z} \le -6.545487680777495021156682531032583832353 \cdot 10^{-230}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;\frac{y}{z} \le 4.814036022956164419199695264571075550494 \cdot 10^{-12}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019208 +o rules:numerics
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1"
  :precision binary64
  (* x (/ (* (/ y z) t) t)))