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

Average Error: 15.0 → 1.6

Time: 3.4s

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 r812852 = x;
        double r812853 = y;
        double r812854 = z;
        double r812855 = r812853 / r812854;
        double r812856 = t;
        double r812857 = r812855 * r812856;
        double r812858 = r812857 / r812856;
        double r812859 = r812852 * r812858;
        return r812859;
}

↓

double f(double x, double y, double z, double __attribute__((unused)) t) {
        double r812860 = y;
        double r812861 = z;
        double r812862 = r812860 / r812861;
        double r812863 = -inf.0;
        bool r812864 = r812862 <= r812863;
        double r812865 = 1.0;
        double r812866 = x;
        double r812867 = r812866 * r812860;
        double r812868 = r812861 / r812867;
        double r812869 = r812865 / r812868;
        double r812870 = -4.63381832549882e-264;
        bool r812871 = r812862 <= r812870;
        double r812872 = r812861 / r812860;
        double r812873 = r812866 / r812872;
        double r812874 = 7.184959373265606e-264;
        bool r812875 = r812862 <= r812874;
        double r812876 = r812867 / r812861;
        double r812877 = r812875 ? r812876 : r812873;
        double r812878 = r812871 ? r812873 : r812877;
        double r812879 = r812864 ? r812869 : r812878;
        return r812879;
}

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)))