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

Average Error: 14.8 → 1.8

Time: 52.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -4.801230547067030745875236962804965666353 \cdot 10^{-232}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{y}{z} \le 7.40416252088828699055648515267267277537 \cdot 10^{-288}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;\frac{y}{z} \le 1.694302386726195194013907163636463756663 \cdot 10^{191}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r32027722 = x;
        double r32027723 = y;
        double r32027724 = z;
        double r32027725 = r32027723 / r32027724;
        double r32027726 = t;
        double r32027727 = r32027725 * r32027726;
        double r32027728 = r32027727 / r32027726;
        double r32027729 = r32027722 * r32027728;
        return r32027729;
}

↓

double f(double x, double y, double z, double __attribute__((unused)) t) {
        double r32027730 = y;
        double r32027731 = z;
        double r32027732 = r32027730 / r32027731;
        double r32027733 = -4.801230547067031e-232;
        bool r32027734 = r32027732 <= r32027733;
        double r32027735 = x;
        double r32027736 = r32027735 * r32027732;
        double r32027737 = 7.404162520888287e-288;
        bool r32027738 = r32027732 <= r32027737;
        double r32027739 = r32027735 / r32027731;
        double r32027740 = r32027739 * r32027730;
        double r32027741 = 1.6943023867261952e+191;
        bool r32027742 = r32027732 <= r32027741;
        double r32027743 = r32027735 * r32027730;
        double r32027744 = r32027743 / r32027731;
        double r32027745 = r32027742 ? r32027736 : r32027744;
        double r32027746 = r32027738 ? r32027740 : r32027745;
        double r32027747 = r32027734 ? r32027736 : r32027746;
        return r32027747;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	14.8
Target	1.5
Herbie	1.8

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

Derivation

1. Split input into 3 regimes

2. if (/ y z) < -4.801230547067031e-232 or 7.404162520888287e-288 < (/ y z) < 1.6943023867261952e+191

   1. Initial program 11.5

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

   2. Simplified 2.3

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

   if -4.801230547067031e-232 < (/ y z) < 7.404162520888287e-288

   1. Initial program 18.4

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

   2. Simplified 13.6

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

   4. Applied div-inv 13.6

      \[\leadsto \color{blue}{\left(y \cdot \frac{1}{z}\right)} \cdot x\]

   5. Applied associate-*l* 0.2

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{z} \cdot x\right)}\]

   6. Simplified 0.2

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

   if 1.6943023867261952e+191 < (/ y z)

   1. Initial program 40.3

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

   2. Simplified 25.2

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

   3. Taylor expanded around 0 1.2

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

4. Final simplification 1.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -4.801230547067030745875236962804965666353 \cdot 10^{-232}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{y}{z} \le 7.40416252088828699055648515267267277537 \cdot 10^{-288}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;\frac{y}{z} \le 1.694302386726195194013907163636463756663 \cdot 10^{191}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, B"

  :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)))