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

Average Error: 14.1 → 2.4

Time: 1.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -3.00847826157384981 \cdot 10^{150} \lor \neg \left(\frac{y}{z} \le -9.59916746614835709 \cdot 10^{-156} \lor \neg \left(\frac{y}{z} \le 1.38338 \cdot 10^{-322}\right)\right):\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r511502 = x;
        double r511503 = y;
        double r511504 = z;
        double r511505 = r511503 / r511504;
        double r511506 = t;
        double r511507 = r511505 * r511506;
        double r511508 = r511507 / r511506;
        double r511509 = r511502 * r511508;
        return r511509;
}

↓

double f(double x, double y, double z, double __attribute__((unused)) t) {
        double r511510 = y;
        double r511511 = z;
        double r511512 = r511510 / r511511;
        double r511513 = -3.0084782615738498e+150;
        bool r511514 = r511512 <= r511513;
        double r511515 = -9.599167466148357e-156;
        bool r511516 = r511512 <= r511515;
        double r511517 = 1.3833838083555e-322;
        bool r511518 = r511512 <= r511517;
        double r511519 = !r511518;
        bool r511520 = r511516 || r511519;
        double r511521 = !r511520;
        bool r511522 = r511514 || r511521;
        double r511523 = x;
        double r511524 = r511523 * r511510;
        double r511525 = r511524 / r511511;
        double r511526 = r511511 / r511510;
        double r511527 = r511523 / r511526;
        double r511528 = r511522 ? r511525 : r511527;
        return r511528;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	14.1
Target	1.4
Herbie	2.4

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

Derivation

1. Split input into 2 regimes

2. if (/ y z) < -3.0084782615738498e+150 or -9.599167466148357e-156 < (/ y z) < 1.3833838083555e-322

   1. Initial program 21.1

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

   2. Simplified 13.0

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

   4. Applied associate-*r/ 1.3

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

   if -3.0084782615738498e+150 < (/ y z) < -9.599167466148357e-156 or 1.3833838083555e-322 < (/ y z)

   1. Initial program 10.8

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

   2. Simplified 2.9

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

   4. Applied associate-*r/ 8.5

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

   6. Applied associate-/l* 2.9

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

4. Final simplification 2.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -3.00847826157384981 \cdot 10^{150} \lor \neg \left(\frac{y}{z} \le -9.59916746614835709 \cdot 10^{-156} \lor \neg \left(\frac{y}{z} \le 1.38338 \cdot 10^{-322}\right)\right):\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 +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)))