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

Average Error: 14.8 → 1.8

Time: 29.9s

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 r5229196 = x;
        double r5229197 = y;
        double r5229198 = z;
        double r5229199 = r5229197 / r5229198;
        double r5229200 = t;
        double r5229201 = r5229199 * r5229200;
        double r5229202 = r5229201 / r5229200;
        double r5229203 = r5229196 * r5229202;
        return r5229203;
}

↓

double f(double x, double y, double z, double __attribute__((unused)) t) {
        double r5229204 = y;
        double r5229205 = z;
        double r5229206 = r5229204 / r5229205;
        double r5229207 = -4.801230547067031e-232;
        bool r5229208 = r5229206 <= r5229207;
        double r5229209 = x;
        double r5229210 = r5229209 * r5229206;
        double r5229211 = 7.404162520888287e-288;
        bool r5229212 = r5229206 <= r5229211;
        double r5229213 = r5229209 / r5229205;
        double r5229214 = r5229213 * r5229204;
        double r5229215 = 1.6943023867261952e+191;
        bool r5229216 = r5229206 <= r5229215;
        double r5229217 = r5229209 * r5229204;
        double r5229218 = r5229217 / r5229205;
        double r5229219 = r5229216 ? r5229210 : r5229218;
        double r5229220 = r5229212 ? r5229214 : r5229219;
        double r5229221 = r5229208 ? r5229210 : r5229220;
        return r5229221;
}

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) < -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"
  (* x (/ (* (/ y z) t) t)))