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

Average Error: 14.3 → 2.0

Time: 18.2s

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 r2992388 = x;
        double r2992389 = y;
        double r2992390 = z;
        double r2992391 = r2992389 / r2992390;
        double r2992392 = t;
        double r2992393 = r2992391 * r2992392;
        double r2992394 = r2992393 / r2992392;
        double r2992395 = r2992388 * r2992394;
        return r2992395;
}

↓

double f(double x, double y, double z, double __attribute__((unused)) t) {
        double r2992396 = y;
        double r2992397 = z;
        double r2992398 = r2992396 / r2992397;
        double r2992399 = -1.6142066513649255e-137;
        bool r2992400 = r2992398 <= r2992399;
        double r2992401 = x;
        double r2992402 = r2992398 * r2992401;
        double r2992403 = 1.9933305190113944e-307;
        bool r2992404 = r2992398 <= r2992403;
        double r2992405 = r2992401 * r2992396;
        double r2992406 = r2992405 / r2992397;
        double r2992407 = 9.807437992397751e+137;
        bool r2992408 = r2992398 <= r2992407;
        double r2992409 = r2992401 / r2992397;
        double r2992410 = r2992409 * r2992396;
        double r2992411 = r2992408 ? r2992402 : r2992410;
        double r2992412 = r2992404 ? r2992406 : r2992411;
        double r2992413 = r2992400 ? r2992402 : r2992412;
        return r2992413;
}

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 
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1"
  (* x (/ (* (/ y z) t) t)))