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

Average Error: 14.5 → 2.2

Time: 4.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -1.42110258613913981 \cdot 10^{-87} \lor \neg \left(\frac{y}{z} \le 1.1014995100088346 \cdot 10^{-297} \lor \neg \left(\frac{y}{z} \le 1.84201517993839439 \cdot 10^{192}\right)\right):\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r680508 = x;
        double r680509 = y;
        double r680510 = z;
        double r680511 = r680509 / r680510;
        double r680512 = t;
        double r680513 = r680511 * r680512;
        double r680514 = r680513 / r680512;
        double r680515 = r680508 * r680514;
        return r680515;
}

↓

double f(double x, double y, double z, double __attribute__((unused)) t) {
        double r680516 = y;
        double r680517 = z;
        double r680518 = r680516 / r680517;
        double r680519 = -1.4211025861391398e-87;
        bool r680520 = r680518 <= r680519;
        double r680521 = 1.1014995100088346e-297;
        bool r680522 = r680518 <= r680521;
        double r680523 = 1.8420151799383944e+192;
        bool r680524 = r680518 <= r680523;
        double r680525 = !r680524;
        bool r680526 = r680522 || r680525;
        double r680527 = !r680526;
        bool r680528 = r680520 || r680527;
        double r680529 = x;
        double r680530 = r680529 * r680518;
        double r680531 = r680529 * r680516;
        double r680532 = r680531 / r680517;
        double r680533 = r680528 ? r680530 : r680532;
        return r680533;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	14.5
Target	1.6
Herbie	2.2

\[\begin{array}{l} \mathbf{if}\;\frac{\frac{y}{z} \cdot t}{t} \lt -1.20672205123045005 \cdot 10^{245}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} \lt -5.90752223693390633 \cdot 10^{-275}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} \lt 5.65895442315341522 \cdot 10^{-65}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} \lt 2.0087180502407133 \cdot 10^{217}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (/ y z) < -1.4211025861391398e-87 or 1.1014995100088346e-297 < (/ y z) < 1.8420151799383944e+192

   1. Initial program 11.1

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

   2. Simplified 2.5

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

   if -1.4211025861391398e-87 < (/ y z) < 1.1014995100088346e-297 or 1.8420151799383944e+192 < (/ y z)

   1. Initial program 20.6

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

   2. Simplified 12.1

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

   4. Applied add-cube-cbrt 12.6

      \[\leadsto \color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)} \cdot \frac{y}{z}\]

   5. Applied associate-*l* 12.6

      \[\leadsto \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \frac{y}{z}\right)}\]
   6. Using strategy rm

   7. Applied associate-*r/ 6.8

      \[\leadsto \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \color{blue}{\frac{\sqrt[3]{x} \cdot y}{z}}\]

   8. Applied associate-*r/ 2.4

      \[\leadsto \color{blue}{\frac{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot y\right)}{z}}\]

   9. Simplified 1.8

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

4. Final simplification 2.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -1.42110258613913981 \cdot 10^{-87} \lor \neg \left(\frac{y}{z} \le 1.1014995100088346 \cdot 10^{-297} \lor \neg \left(\frac{y}{z} \le 1.84201517993839439 \cdot 10^{192}\right)\right):\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, B"
  :precision binary64

  :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)))