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

Average Error: 14.5 → 0.5

Time: 3.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -6.122731566035475311671418161909773301552 \cdot 10^{157}:\\ \;\;\;\;\frac{1}{\frac{z}{x \cdot y}}\\ \mathbf{elif}\;\frac{y}{z} \le -2.242439350440375267697031348628742827796 \cdot 10^{-179}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{y}{z} \le 5.129556799950046333853278890491110988102 \cdot 10^{-271}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{y}{z} \le 1.795095217187765936797771641264103283314 \cdot 10^{286}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r716098 = x;
        double r716099 = y;
        double r716100 = z;
        double r716101 = r716099 / r716100;
        double r716102 = t;
        double r716103 = r716101 * r716102;
        double r716104 = r716103 / r716102;
        double r716105 = r716098 * r716104;
        return r716105;
}

↓

double f(double x, double y, double z, double __attribute__((unused)) t) {
        double r716106 = y;
        double r716107 = z;
        double r716108 = r716106 / r716107;
        double r716109 = -6.122731566035475e+157;
        bool r716110 = r716108 <= r716109;
        double r716111 = 1.0;
        double r716112 = x;
        double r716113 = r716112 * r716106;
        double r716114 = r716107 / r716113;
        double r716115 = r716111 / r716114;
        double r716116 = -2.2424393504403753e-179;
        bool r716117 = r716108 <= r716116;
        double r716118 = r716112 * r716108;
        double r716119 = 5.129556799950046e-271;
        bool r716120 = r716108 <= r716119;
        double r716121 = r716113 / r716107;
        double r716122 = 1.795095217187766e+286;
        bool r716123 = r716108 <= r716122;
        double r716124 = r716123 ? r716118 : r716121;
        double r716125 = r716120 ? r716121 : r716124;
        double r716126 = r716117 ? r716118 : r716125;
        double r716127 = r716110 ? r716115 : r716126;
        return r716127;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	14.5
Target	1.4
Herbie	0.5

\[\begin{array}{l} \mathbf{if}\;\frac{\frac{y}{z} \cdot t}{t} \lt -1.206722051230450047215521150762600712224 \cdot 10^{245}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} \lt -5.90752223693390632993316700759382836344 \cdot 10^{-275}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} \lt 5.658954423153415216825328199697215652986 \cdot 10^{-65}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} \lt 2.008718050240713347941382056648619307142 \cdot 10^{217}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (/ y z) < -6.122731566035475e+157

   1. Initial program 33.9

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

   2. Simplified 18.6

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

   4. Applied associate-*r/ 2.3

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

   6. Applied clear-num 2.4

      \[\leadsto \color{blue}{\frac{1}{\frac{z}{x \cdot y}}}\]

   if -6.122731566035475e+157 < (/ y z) < -2.2424393504403753e-179 or 5.129556799950046e-271 < (/ y z) < 1.795095217187766e+286

   1. Initial program 9.0

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

   2. Simplified 0.2

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

   if -2.2424393504403753e-179 < (/ y z) < 5.129556799950046e-271 or 1.795095217187766e+286 < (/ y z)

   1. Initial program 21.5

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

   2. Simplified 15.3

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

   4. Applied associate-*r/ 0.6

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -6.122731566035475311671418161909773301552 \cdot 10^{157}:\\ \;\;\;\;\frac{1}{\frac{z}{x \cdot y}}\\ \mathbf{elif}\;\frac{y}{z} \le -2.242439350440375267697031348628742827796 \cdot 10^{-179}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{y}{z} \le 5.129556799950046333853278890491110988102 \cdot 10^{-271}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{y}{z} \le 1.795095217187765936797771641264103283314 \cdot 10^{286}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 +o rules:numerics
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, B"
  :precision binary64

  :herbie-target
  (if (< (/ (* (/ y z) t) t) -1.20672205123045e+245) (/ y (/ z x)) (if (< (/ (* (/ y z) t) t) -5.907522236933906e-275) (* x (/ y z)) (if (< (/ (* (/ y z) t) t) 5.658954423153415e-65) (/ y (/ z x)) (if (< (/ (* (/ y z) t) t) 2.0087180502407133e+217) (* x (/ y z)) (/ (* y x) z)))))

  (* x (/ (* (/ y z) t) t)))