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

Average Error: 14.3 → 2.0

Time: 20.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -1.6142066513649255 \cdot 10^{-137}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;\frac{y}{z} \le 1.9933305190113944 \cdot 10^{-307}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{y}{z} \le 9.807437992397751 \cdot 10^{+137}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r4257529 = x;
        double r4257530 = y;
        double r4257531 = z;
        double r4257532 = r4257530 / r4257531;
        double r4257533 = t;
        double r4257534 = r4257532 * r4257533;
        double r4257535 = r4257534 / r4257533;
        double r4257536 = r4257529 * r4257535;
        return r4257536;
}

↓

double f(double x, double y, double z, double __attribute__((unused)) t) {
        double r4257537 = y;
        double r4257538 = z;
        double r4257539 = r4257537 / r4257538;
        double r4257540 = -1.6142066513649255e-137;
        bool r4257541 = r4257539 <= r4257540;
        double r4257542 = x;
        double r4257543 = r4257539 * r4257542;
        double r4257544 = 1.9933305190113944e-307;
        bool r4257545 = r4257539 <= r4257544;
        double r4257546 = r4257542 * r4257537;
        double r4257547 = r4257546 / r4257538;
        double r4257548 = 9.807437992397751e+137;
        bool r4257549 = r4257539 <= r4257548;
        double r4257550 = r4257542 / r4257538;
        double r4257551 = r4257550 * r4257537;
        double r4257552 = r4257549 ? r4257543 : r4257551;
        double r4257553 = r4257545 ? r4257547 : r4257552;
        double r4257554 = r4257541 ? r4257543 : r4257553;
        return r4257554;
}

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) < -1.6142066513649255e-137 or 1.9933305190113944e-307 < (/ y z) < 9.807437992397751e+137

   1. Initial program 11.0

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

   2. Simplified 8.6

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

   4. Applied div-inv 8.7

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

   5. Applied associate-*l* 2.4

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

   6. Simplified 2.3

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

   if -1.6142066513649255e-137 < (/ y z) < 1.9933305190113944e-307

   1. Initial program 16.8

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

   2. Simplified 1.1

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

   4. Applied div-inv 1.2

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

   5. Applied associate-*l* 11.2

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

   6. Simplified 11.1

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

   7. Taylor expanded around inf 1.0

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

   if 9.807437992397751e+137 < (/ y z)

   1. Initial program 30.1

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

   2. Simplified 2.5

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

4. Final simplification 2.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -1.6142066513649255 \cdot 10^{-137}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;\frac{y}{z} \le 1.9933305190113944 \cdot 10^{-307}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{y}{z} \le 9.807437992397751 \cdot 10^{+137}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2019135 +o rules:numerics
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1"
  (* x (/ (* (/ y z) t) t)))