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

Average Error: 14.3 → 2.2

Time: 10.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -1.0532464204699888 \cdot 10^{+209}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \le -6.588547053565423 \cdot 10^{-193}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;\frac{y}{z} \le 2.2740014914314757 \cdot 10^{-89}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r1647410 = x;
        double r1647411 = y;
        double r1647412 = z;
        double r1647413 = r1647411 / r1647412;
        double r1647414 = t;
        double r1647415 = r1647413 * r1647414;
        double r1647416 = r1647415 / r1647414;
        double r1647417 = r1647410 * r1647416;
        return r1647417;
}

↓

double f(double x, double y, double z, double __attribute__((unused)) t) {
        double r1647418 = y;
        double r1647419 = z;
        double r1647420 = r1647418 / r1647419;
        double r1647421 = -1.0532464204699888e+209;
        bool r1647422 = r1647420 <= r1647421;
        double r1647423 = x;
        double r1647424 = r1647423 / r1647419;
        double r1647425 = r1647418 * r1647424;
        double r1647426 = -6.588547053565423e-193;
        bool r1647427 = r1647420 <= r1647426;
        double r1647428 = r1647420 * r1647423;
        double r1647429 = 2.2740014914314757e-89;
        bool r1647430 = r1647420 <= r1647429;
        double r1647431 = r1647423 * r1647418;
        double r1647432 = r1647431 / r1647419;
        double r1647433 = r1647430 ? r1647432 : r1647428;
        double r1647434 = r1647427 ? r1647428 : r1647433;
        double r1647435 = r1647422 ? r1647425 : r1647434;
        return r1647435;
}

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) < -1.0532464204699888e+209

   1. Initial program 40.4

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

   2. Simplified 0.4

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

   if -1.0532464204699888e+209 < (/ y z) < -6.588547053565423e-193 or 2.2740014914314757e-89 < (/ y z)

   1. Initial program 11.1

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

   2. Simplified 8.8

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

   4. Applied associate-*r/ 9.1

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

   6. Applied associate-/l* 8.7

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

   8. Applied associate-/r/ 2.8

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

   if -6.588547053565423e-193 < (/ y z) < 2.2740014914314757e-89

   1. Initial program 15.7

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

   2. Simplified 1.4

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

   4. Applied associate-*r/ 1.6

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

   6. Applied associate-/l* 1.4

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

   8. Applied associate-/r/ 8.1

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

   9. Taylor expanded around 0 1.6

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

4. Final simplification 2.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -1.0532464204699888 \cdot 10^{+209}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \le -6.588547053565423 \cdot 10^{-193}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;\frac{y}{z} \le 2.2740014914314757 \cdot 10^{-89}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1"
  (* x (/ (* (/ y z) t) t)))