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

Average Error: 14.5 → 0.7

Time: 44.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -2.7825123622347264 \cdot 10^{+238}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \le -6.83182473818689 \cdot 10^{-115}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;\frac{y}{z} \le 4.5454039417395 \cdot 10^{-322}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \le 1.2584134571556272 \cdot 10^{+188}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r3192420 = x;
        double r3192421 = y;
        double r3192422 = z;
        double r3192423 = r3192421 / r3192422;
        double r3192424 = t;
        double r3192425 = r3192423 * r3192424;
        double r3192426 = r3192425 / r3192424;
        double r3192427 = r3192420 * r3192426;
        return r3192427;
}

↓

double f(double x, double y, double z, double __attribute__((unused)) t) {
        double r3192428 = y;
        double r3192429 = z;
        double r3192430 = r3192428 / r3192429;
        double r3192431 = -2.7825123622347264e+238;
        bool r3192432 = r3192430 <= r3192431;
        double r3192433 = x;
        double r3192434 = r3192433 / r3192429;
        double r3192435 = r3192428 * r3192434;
        double r3192436 = -6.83182473818689e-115;
        bool r3192437 = r3192430 <= r3192436;
        double r3192438 = r3192430 * r3192433;
        double r3192439 = 4.5454039417395e-322;
        bool r3192440 = r3192430 <= r3192439;
        double r3192441 = 1.2584134571556272e+188;
        bool r3192442 = r3192430 <= r3192441;
        double r3192443 = r3192442 ? r3192438 : r3192435;
        double r3192444 = r3192440 ? r3192435 : r3192443;
        double r3192445 = r3192437 ? r3192438 : r3192444;
        double r3192446 = r3192432 ? r3192435 : r3192445;
        return r3192446;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Derivation

1. Split input into 2 regimes

2. if (/ y z) < -2.7825123622347264e+238 or -6.83182473818689e-115 < (/ y z) < 4.5454039417395e-322 or 1.2584134571556272e+188 < (/ y z)

   1. Initial program 23.4

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

   2. Simplified 1.5

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

   if -2.7825123622347264e+238 < (/ y z) < -6.83182473818689e-115 or 4.5454039417395e-322 < (/ y z) < 1.2584134571556272e+188

   1. Initial program 8.9

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

   2. Simplified 8.8

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

   3. Taylor expanded around -inf 9.2

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

   5. Applied *-un-lft-identity 9.2

      \[\leadsto \frac{x \cdot y}{\color{blue}{1 \cdot z}}\]

   6. Applied times-frac 0.3

      \[\leadsto \color{blue}{\frac{x}{1} \cdot \frac{y}{z}}\]

   7. Simplified 0.3

      \[\leadsto \color{blue}{x} \cdot \frac{y}{z}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -2.7825123622347264 \cdot 10^{+238}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \le -6.83182473818689 \cdot 10^{-115}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;\frac{y}{z} \le 4.5454039417395 \cdot 10^{-322}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \le 1.2584134571556272 \cdot 10^{+188}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019134 +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)))