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

Average Error: 15.1 → 5.9

Time: 12.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -4.057062710333388 \cdot 10^{-256}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;z \le 1.0531420844975264 \cdot 10^{-216}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r17623348 = x;
        double r17623349 = y;
        double r17623350 = z;
        double r17623351 = r17623349 / r17623350;
        double r17623352 = t;
        double r17623353 = r17623351 * r17623352;
        double r17623354 = r17623353 / r17623352;
        double r17623355 = r17623348 * r17623354;
        return r17623355;
}

↓

double f(double x, double y, double z, double __attribute__((unused)) t) {
        double r17623356 = z;
        double r17623357 = -4.057062710333388e-256;
        bool r17623358 = r17623356 <= r17623357;
        double r17623359 = x;
        double r17623360 = y;
        double r17623361 = r17623359 * r17623360;
        double r17623362 = r17623361 / r17623356;
        double r17623363 = 1.0531420844975264e-216;
        bool r17623364 = r17623356 <= r17623363;
        double r17623365 = r17623356 / r17623360;
        double r17623366 = r17623359 / r17623365;
        double r17623367 = r17623364 ? r17623366 : r17623362;
        double r17623368 = r17623358 ? r17623362 : r17623367;
        return r17623368;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	15.1
Target	1.5
Herbie	5.9

\[\begin{array}{l} \mathbf{if}\;\frac{\frac{y}{z} \cdot t}{t} \lt -1.20672205123045 \cdot 10^{+245}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} \lt -5.907522236933906 \cdot 10^{-275}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} \lt 5.658954423153415 \cdot 10^{-65}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} \lt 2.0087180502407133 \cdot 10^{+217}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if z < -4.057062710333388e-256 or 1.0531420844975264e-216 < z

   1. Initial program 14.3

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

   2. Simplified 5.3

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

   3. Taylor expanded around 0 5.3

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

   5. Applied div-inv 5.4

      \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \frac{1}{z}}\]
   6. Using strategy rm

   7. Applied un-div-inv 5.3

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

   if -4.057062710333388e-256 < z < 1.0531420844975264e-216

   1. Initial program 23.2

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

   2. Simplified 13.9

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

   3. Taylor expanded around 0 14.2

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

   5. Applied associate-/l* 12.2

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

4. Final simplification 5.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -4.057062710333388 \cdot 10^{-256}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;z \le 1.0531420844975264 \cdot 10^{-216}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, B"

  :herbie-target
  (if (< (/ (* (/ y z) t) t) -1.20672205123045e+245) (/ y (/ z x)) (if (< (/ (* (/ y z) t) t) -5.907522236933906e-275) (* x (/ y z)) (if (< (/ (* (/ y z) t) t) 5.658954423153415e-65) (/ y (/ z x)) (if (< (/ (* (/ y z) t) t) 2.0087180502407133e+217) (* x (/ y z)) (/ (* y x) z)))))

  (* x (/ (* (/ y z) t) t)))