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

Average Error: 14.8 → 0.3

Time: 3.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} = -\infty:\\ \;\;\;\;\frac{1}{\frac{z}{x \cdot y}}\\ \mathbf{elif}\;\frac{y}{z} \le -1.434471595776521596384923059287499479756 \cdot 10^{-254}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;\frac{y}{z} \le 1.707355746264835359667361500171699768251 \cdot 10^{-300}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{y}{z} \le 7.166584424187464259231246965323639075754 \cdot 10^{225}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r618188 = x;
        double r618189 = y;
        double r618190 = z;
        double r618191 = r618189 / r618190;
        double r618192 = t;
        double r618193 = r618191 * r618192;
        double r618194 = r618193 / r618192;
        double r618195 = r618188 * r618194;
        return r618195;
}

↓

double f(double x, double y, double z, double __attribute__((unused)) t) {
        double r618196 = y;
        double r618197 = z;
        double r618198 = r618196 / r618197;
        double r618199 = -inf.0;
        bool r618200 = r618198 <= r618199;
        double r618201 = 1.0;
        double r618202 = x;
        double r618203 = r618202 * r618196;
        double r618204 = r618197 / r618203;
        double r618205 = r618201 / r618204;
        double r618206 = -1.4344715957765216e-254;
        bool r618207 = r618198 <= r618206;
        double r618208 = r618197 / r618196;
        double r618209 = r618202 / r618208;
        double r618210 = 1.7073557462648354e-300;
        bool r618211 = r618198 <= r618210;
        double r618212 = r618203 / r618197;
        double r618213 = 7.166584424187464e+225;
        bool r618214 = r618198 <= r618213;
        double r618215 = r618202 * r618198;
        double r618216 = r618214 ? r618215 : r618212;
        double r618217 = r618211 ? r618212 : r618216;
        double r618218 = r618207 ? r618209 : r618217;
        double r618219 = r618200 ? r618205 : r618218;
        return r618219;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	14.8
Target	1.6
Herbie	0.3

\[\begin{array}{l} \mathbf{if}\;\frac{\frac{y}{z} \cdot t}{t} \lt -1.206722051230450047215521150762600712224 \cdot 10^{245}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} \lt -5.90752223693390632993316700759382836344 \cdot 10^{-275}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} \lt 5.658954423153415216825328199697215652986 \cdot 10^{-65}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} \lt 2.008718050240713347941382056648619307142 \cdot 10^{217}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array}\]

Derivation

1. Split input into 4 regimes

2. if (/ y z) < -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/ 0.3

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

   6. Applied clear-num 0.4

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

   if -inf.0 < (/ y z) < -1.4344715957765216e-254

   1. Initial program 10.4

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

   2. Simplified 0.2

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

   4. Applied associate-*r/ 8.3

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

   6. Applied associate-/l* 0.2

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

   if -1.4344715957765216e-254 < (/ y z) < 1.7073557462648354e-300 or 7.166584424187464e+225 < (/ y z)

   1. Initial program 25.4

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

   2. Simplified 19.4

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

   4. Applied associate-*r/ 0.3

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

   if 1.7073557462648354e-300 < (/ y z) < 7.166584424187464e+225

   1. Initial program 9.1

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

   2. Simplified 0.2

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} = -\infty:\\ \;\;\;\;\frac{1}{\frac{z}{x \cdot y}}\\ \mathbf{elif}\;\frac{y}{z} \le -1.434471595776521596384923059287499479756 \cdot 10^{-254}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;\frac{y}{z} \le 1.707355746264835359667361500171699768251 \cdot 10^{-300}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{y}{z} \le 7.166584424187464259231246965323639075754 \cdot 10^{225}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 +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)))