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

Average Error: 14.6 → 2.9

Time: 2.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -1.68209557109898762 \cdot 10^{256} \lor \neg \left(\frac{y}{z} \le -8.7578361305297263 \cdot 10^{-224}\right):\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r1308508 = x;
        double r1308509 = y;
        double r1308510 = z;
        double r1308511 = r1308509 / r1308510;
        double r1308512 = t;
        double r1308513 = r1308511 * r1308512;
        double r1308514 = r1308513 / r1308512;
        double r1308515 = r1308508 * r1308514;
        return r1308515;
}

↓

double f(double x, double y, double z, double __attribute__((unused)) t) {
        double r1308516 = y;
        double r1308517 = z;
        double r1308518 = r1308516 / r1308517;
        double r1308519 = -1.6820955710989876e+256;
        bool r1308520 = r1308518 <= r1308519;
        double r1308521 = -8.757836130529726e-224;
        bool r1308522 = r1308518 <= r1308521;
        double r1308523 = !r1308522;
        bool r1308524 = r1308520 || r1308523;
        double r1308525 = x;
        double r1308526 = r1308525 * r1308516;
        double r1308527 = 1.0;
        double r1308528 = r1308527 / r1308517;
        double r1308529 = r1308526 * r1308528;
        double r1308530 = r1308517 / r1308516;
        double r1308531 = r1308525 / r1308530;
        double r1308532 = r1308524 ? r1308529 : r1308531;
        return r1308532;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	14.6
Target	1.7
Herbie	2.9

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

Derivation

1. Split input into 2 regimes

2. if (/ y z) < -1.6820955710989876e+256 or -8.757836130529726e-224 < (/ y z)

   1. Initial program 17.3

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

   2. Simplified 9.1

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

   4. Applied div-inv 9.2

      \[\leadsto x \cdot \color{blue}{\left(y \cdot \frac{1}{z}\right)}\]

   5. Applied associate-*r* 4.3

      \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \frac{1}{z}}\]

   if -1.6820955710989876e+256 < (/ y z) < -8.757836130529726e-224

   1. Initial program 9.3

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

   2. Simplified 0.2

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

   4. Applied associate-*r/ 8.9

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

   6. Applied associate-/l* 0.2

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

4. Final simplification 2.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -1.68209557109898762 \cdot 10^{256} \lor \neg \left(\frac{y}{z} \le -8.7578361305297263 \cdot 10^{-224}\right):\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020034 +o rules:numerics
(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)))