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

Average Error: 14.8 → 0.5

Time: 2.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -4.6183489977482823 \cdot 10^{236}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \mathbf{elif}\;\frac{y}{z} \le -6.8792362608384231 \cdot 10^{-264}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{y}{z} \le 1.06689610291439 \cdot 10^{-310}:\\ \;\;\;\;\frac{1}{\frac{z}{x \cdot y}}\\ \mathbf{elif}\;\frac{y}{z} \le 8.69850988828769301 \cdot 10^{221}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r583889 = x;
        double r583890 = y;
        double r583891 = z;
        double r583892 = r583890 / r583891;
        double r583893 = t;
        double r583894 = r583892 * r583893;
        double r583895 = r583894 / r583893;
        double r583896 = r583889 * r583895;
        return r583896;
}

↓

double f(double x, double y, double z, double __attribute__((unused)) t) {
        double r583897 = y;
        double r583898 = z;
        double r583899 = r583897 / r583898;
        double r583900 = -4.618348997748282e+236;
        bool r583901 = r583899 <= r583900;
        double r583902 = x;
        double r583903 = r583902 * r583897;
        double r583904 = 1.0;
        double r583905 = r583904 / r583898;
        double r583906 = r583903 * r583905;
        double r583907 = -6.879236260838423e-264;
        bool r583908 = r583899 <= r583907;
        double r583909 = r583902 * r583899;
        double r583910 = 1.0668961029144e-310;
        bool r583911 = r583899 <= r583910;
        double r583912 = r583898 / r583903;
        double r583913 = r583904 / r583912;
        double r583914 = 8.698509888287693e+221;
        bool r583915 = r583899 <= r583914;
        double r583916 = r583898 / r583897;
        double r583917 = r583902 / r583916;
        double r583918 = r583915 ? r583917 : r583906;
        double r583919 = r583911 ? r583913 : r583918;
        double r583920 = r583908 ? r583909 : r583919;
        double r583921 = r583901 ? r583906 : r583920;
        return r583921;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	14.8
Target	1.7
Herbie	0.5

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

Derivation

1. Split input into 4 regimes

2. if (/ y z) < -4.618348997748282e+236 or 8.698509888287693e+221 < (/ y z)

   1. Initial program 45.8

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

   2. Simplified 33.2

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

   4. Applied div-inv 33.3

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

   5. Applied associate-*r* 1.1

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

   if -4.618348997748282e+236 < (/ y z) < -6.879236260838423e-264

   1. Initial program 9.7

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

   2. Simplified 0.2

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

   if -6.879236260838423e-264 < (/ y z) < 1.0668961029144e-310

   1. Initial program 19.2

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

   2. Simplified 16.5

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

   4. Applied associate-*r/ 0.1

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

   6. Applied clear-num 0.9

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

   if 1.0668961029144e-310 < (/ y z) < 8.698509888287693e+221

   1. Initial program 9.0

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

   2. Simplified 0.2

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

   4. Applied associate-*r/ 8.6

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

   6. Applied associate-/l* 0.3

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{y}}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -4.6183489977482823 \cdot 10^{236}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \mathbf{elif}\;\frac{y}{z} \le -6.8792362608384231 \cdot 10^{-264}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{y}{z} \le 1.06689610291439 \cdot 10^{-310}:\\ \;\;\;\;\frac{1}{\frac{z}{x \cdot y}}\\ \mathbf{elif}\;\frac{y}{z} \le 8.69850988828769301 \cdot 10^{221}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020064 +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)))