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

Average Error: 14.3 → 0.3

Time: 16.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -5.49778287371169 \cdot 10^{+261}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \le -1.983516013696309 \cdot 10^{-226}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;\frac{y}{z} \le 9.35914650043368 \cdot 10^{-309}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \le 8.460748976825988 \cdot 10^{+248}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r3447399 = x;
        double r3447400 = y;
        double r3447401 = z;
        double r3447402 = r3447400 / r3447401;
        double r3447403 = t;
        double r3447404 = r3447402 * r3447403;
        double r3447405 = r3447404 / r3447403;
        double r3447406 = r3447399 * r3447405;
        return r3447406;
}

↓

double f(double x, double y, double z, double __attribute__((unused)) t) {
        double r3447407 = y;
        double r3447408 = z;
        double r3447409 = r3447407 / r3447408;
        double r3447410 = -5.49778287371169e+261;
        bool r3447411 = r3447409 <= r3447410;
        double r3447412 = x;
        double r3447413 = r3447412 / r3447408;
        double r3447414 = r3447407 * r3447413;
        double r3447415 = -1.983516013696309e-226;
        bool r3447416 = r3447409 <= r3447415;
        double r3447417 = r3447408 / r3447407;
        double r3447418 = r3447412 / r3447417;
        double r3447419 = 9.35914650043368e-309;
        bool r3447420 = r3447409 <= r3447419;
        double r3447421 = 8.460748976825988e+248;
        bool r3447422 = r3447409 <= r3447421;
        double r3447423 = r3447412 * r3447407;
        double r3447424 = r3447423 / r3447408;
        double r3447425 = r3447422 ? r3447418 : r3447424;
        double r3447426 = r3447420 ? r3447414 : r3447425;
        double r3447427 = r3447416 ? r3447418 : r3447426;
        double r3447428 = r3447411 ? r3447414 : r3447427;
        return r3447428;
}

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) < -5.49778287371169e+261 or -1.983516013696309e-226 < (/ y z) < 9.35914650043368e-309

   1. Initial program 23.3

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

   2. Simplified 0.3

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

   if -5.49778287371169e+261 < (/ y z) < -1.983516013696309e-226 or 9.35914650043368e-309 < (/ y z) < 8.460748976825988e+248

   1. Initial program 9.5

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

   2. Simplified 8.5

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

   3. Taylor expanded around 0 8.2

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

   5. Applied associate-/l* 0.2

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

   if 8.460748976825988e+248 < (/ y z)

   1. Initial program 47.6

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

   2. Simplified 0.4

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

   3. Taylor expanded around 0 0.2

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -5.49778287371169 \cdot 10^{+261}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \le -1.983516013696309 \cdot 10^{-226}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;\frac{y}{z} \le 9.35914650043368 \cdot 10^{-309}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \le 8.460748976825988 \cdot 10^{+248}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +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)))