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

Average Error: 15.0 → 1.6

Time: 5.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} = -\infty:\\ \;\;\;\;\frac{1}{\frac{z}{x \cdot y}}\\ \mathbf{elif}\;\frac{y}{z} \le -4.63381832549882 \cdot 10^{-264}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;\frac{y}{z} \le 7.18495937326560576 \cdot 10^{-264}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r2758 = x;
        double r2759 = y;
        double r2760 = z;
        double r2761 = r2759 / r2760;
        double r2762 = t;
        double r2763 = r2761 * r2762;
        double r2764 = r2763 / r2762;
        double r2765 = r2758 * r2764;
        return r2765;
}

↓

double f(double x, double y, double z, double __attribute__((unused)) t) {
        double r2766 = y;
        double r2767 = z;
        double r2768 = r2766 / r2767;
        double r2769 = -inf.0;
        bool r2770 = r2768 <= r2769;
        double r2771 = 1.0;
        double r2772 = x;
        double r2773 = r2772 * r2766;
        double r2774 = r2767 / r2773;
        double r2775 = r2771 / r2774;
        double r2776 = -4.63381832549882e-264;
        bool r2777 = r2768 <= r2776;
        double r2778 = r2767 / r2766;
        double r2779 = r2772 / r2778;
        double r2780 = 7.184959373265606e-264;
        bool r2781 = r2768 <= r2780;
        double r2782 = r2773 / r2767;
        double r2783 = r2781 ? r2782 : r2779;
        double r2784 = r2777 ? r2779 : r2783;
        double r2785 = r2770 ? r2775 : r2784;
        return r2785;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	15.0
Target	1.5
Herbie	1.6

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

Derivation

1. Split input into 3 regimes

2. if (/ y z) < -inf.0

   1. Initial program 64.0

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

   2. Simplified 64.0

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

   4. Applied associate-*r/ 0.2

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

   6. Applied clear-num 0.3

      \[\leadsto \color{blue}{\frac{1}{\frac{z}{x \cdot y}}}\]

   if -inf.0 < (/ y z) < -4.63381832549882e-264 or 7.184959373265606e-264 < (/ y z)

   1. Initial program 12.4

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

   2. Simplified 2.1

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

   4. Applied associate-*r/ 7.9

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

   6. Applied associate-/l* 2.0

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

   if -4.63381832549882e-264 < (/ y z) < 7.184959373265606e-264

   1. Initial program 19.8

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

   2. Simplified 15.5

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

   4. Applied associate-*r/ 0.1

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

4. Final simplification 1.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} = -\infty:\\ \;\;\;\;\frac{1}{\frac{z}{x \cdot y}}\\ \mathbf{elif}\;\frac{y}{z} \le -4.63381832549882 \cdot 10^{-264}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;\frac{y}{z} \le 7.18495937326560576 \cdot 10^{-264}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 +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)))