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

Average Error: 14.5 → 1.2

Time: 11.0s

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 r58636 = x;
        double r58637 = y;
        double r58638 = z;
        double r58639 = r58637 / r58638;
        double r58640 = t;
        double r58641 = r58639 * r58640;
        double r58642 = r58641 / r58640;
        double r58643 = r58636 * r58642;
        return r58643;
}

↓

double f(double x, double y, double z, double __attribute__((unused)) t) {
        double r58644 = y;
        double r58645 = z;
        double r58646 = r58644 / r58645;
        double r58647 = -4.53979336107177e+214;
        bool r58648 = r58646 <= r58647;
        double r58649 = x;
        double r58650 = r58649 / r58645;
        double r58651 = r58644 * r58650;
        double r58652 = -2.5306048560903516e-166;
        bool r58653 = r58646 <= r58652;
        double r58654 = r58646 * r58649;
        double r58655 = 6.053074699504575e-63;
        bool r58656 = r58646 <= r58655;
        double r58657 = r58645 / r58649;
        double r58658 = r58644 / r58657;
        double r58659 = 6.289758256242324e+161;
        bool r58660 = r58646 <= r58659;
        double r58661 = r58644 * r58649;
        double r58662 = r58661 / r58645;
        double r58663 = r58660 ? r58654 : r58662;
        double r58664 = r58656 ? r58658 : r58663;
        double r58665 = r58653 ? r58654 : r58664;
        double r58666 = r58648 ? r58651 : r58665;
        return r58666;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

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 
(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)))