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

Average Error: 14.5 → 1.2

Time: 10.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -4.539793361071769873969043675808442911102 \cdot 10^{214}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \le -2.530604856090351627042526853504735687815 \cdot 10^{-166}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;\frac{y}{z} \le 6.053074699504574675208589942494399087398 \cdot 10^{-63}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{y}{z} \le 6.289758256242324172285873340578120865516 \cdot 10^{161}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r383724 = x;
        double r383725 = y;
        double r383726 = z;
        double r383727 = r383725 / r383726;
        double r383728 = t;
        double r383729 = r383727 * r383728;
        double r383730 = r383729 / r383728;
        double r383731 = r383724 * r383730;
        return r383731;
}

↓

double f(double x, double y, double z, double __attribute__((unused)) t) {
        double r383732 = y;
        double r383733 = z;
        double r383734 = r383732 / r383733;
        double r383735 = -4.53979336107177e+214;
        bool r383736 = r383734 <= r383735;
        double r383737 = x;
        double r383738 = r383737 / r383733;
        double r383739 = r383732 * r383738;
        double r383740 = -2.5306048560903516e-166;
        bool r383741 = r383734 <= r383740;
        double r383742 = r383734 * r383737;
        double r383743 = 6.053074699504575e-63;
        bool r383744 = r383734 <= r383743;
        double r383745 = r383733 / r383737;
        double r383746 = r383732 / r383745;
        double r383747 = 6.289758256242324e+161;
        bool r383748 = r383734 <= r383747;
        double r383749 = r383732 * r383737;
        double r383750 = r383749 / r383733;
        double r383751 = r383748 ? r383742 : r383750;
        double r383752 = r383744 ? r383746 : r383751;
        double r383753 = r383741 ? r383742 : r383752;
        double r383754 = r383736 ? r383739 : r383753;
        return r383754;
}

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	1.2

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

Derivation

1. Split input into 4 regimes

2. if (/ y z) < -4.53979336107177e+214

   1. Initial program 42.7

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

   2. Simplified 0.6

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

   4. Applied *-un-lft-identity 0.6

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

   5. Applied times-frac 0.4

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

   6. Simplified 0.4

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

   if -4.53979336107177e+214 < (/ y z) < -2.5306048560903516e-166 or 6.053074699504575e-63 < (/ y z) < 6.289758256242324e+161

   1. Initial program 7.3

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

   2. Simplified 11.4

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

   4. Applied associate-/l* 10.3

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

   6. Applied associate-/r/ 0.2

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

   if -2.5306048560903516e-166 < (/ y z) < 6.053074699504575e-63

   1. Initial program 14.9

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

   2. Simplified 2.5

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

   4. Applied associate-/l* 2.2

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

   if 6.289758256242324e+161 < (/ y z)

   1. Initial program 34.1

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

   2. Simplified 2.3

      \[\leadsto \color{blue}{\frac{y \cdot x}{z}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 1.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -4.539793361071769873969043675808442911102 \cdot 10^{214}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \le -2.530604856090351627042526853504735687815 \cdot 10^{-166}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;\frac{y}{z} \le 6.053074699504574675208589942494399087398 \cdot 10^{-63}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{y}{z} \le 6.289758256242324172285873340578120865516 \cdot 10^{161}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019305 +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) -1.20672205123045005e245) (/ y (/ z x)) (if (< (/ (* (/ y z) t) t) -5.90752223693390633e-275) (* x (/ y z)) (if (< (/ (* (/ y z) t) t) 5.65895442315341522e-65) (/ y (/ z x)) (if (< (/ (* (/ y z) t) t) 2.0087180502407133e217) (* x (/ y z)) (/ (* y x) z)))))

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