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

Average Error: 14.4 → 0.5

Time: 4.2m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -9.916325328902394 \cdot 10^{+225}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \le -8.08619711399183 \cdot 10^{-156}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;\frac{y}{z} \le 4.2595672029945 \cdot 10^{-318}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \le 2.0905090728700514 \cdot 10^{+191}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r4985480 = x;
        double r4985481 = y;
        double r4985482 = z;
        double r4985483 = r4985481 / r4985482;
        double r4985484 = t;
        double r4985485 = r4985483 * r4985484;
        double r4985486 = r4985485 / r4985484;
        double r4985487 = r4985480 * r4985486;
        return r4985487;
}

↓

double f(double x, double y, double z, double __attribute__((unused)) t) {
        double r4985488 = y;
        double r4985489 = z;
        double r4985490 = r4985488 / r4985489;
        double r4985491 = -9.916325328902394e+225;
        bool r4985492 = r4985490 <= r4985491;
        double r4985493 = x;
        double r4985494 = r4985493 / r4985489;
        double r4985495 = r4985488 * r4985494;
        double r4985496 = -8.08619711399183e-156;
        bool r4985497 = r4985490 <= r4985496;
        double r4985498 = r4985490 * r4985493;
        double r4985499 = 4.2595672029945e-318;
        bool r4985500 = r4985490 <= r4985499;
        double r4985501 = 2.0905090728700514e+191;
        bool r4985502 = r4985490 <= r4985501;
        double r4985503 = r4985502 ? r4985498 : r4985495;
        double r4985504 = r4985500 ? r4985495 : r4985503;
        double r4985505 = r4985497 ? r4985498 : r4985504;
        double r4985506 = r4985492 ? r4985495 : r4985505;
        return r4985506;
}

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) < -9.916325328902394e+225 or -8.08619711399183e-156 < (/ y z) < 4.2595672029945e-318 or 2.0905090728700514e+191 < (/ y z)

   1. Initial program 25.9

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

   2. Simplified 1.0

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

   if -9.916325328902394e+225 < (/ y z) < -8.08619711399183e-156 or 4.2595672029945e-318 < (/ y z) < 2.0905090728700514e+191

   1. Initial program 8.3

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

   2. Simplified 9.0

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

   3. Taylor expanded around 0 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 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -9.916325328902394 \cdot 10^{+225}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \le -8.08619711399183 \cdot 10^{-156}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;\frac{y}{z} \le 4.2595672029945 \cdot 10^{-318}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \le 2.0905090728700514 \cdot 10^{+191}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1"
  (* x (/ (* (/ y z) t) t)))