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

Average Error: 13.8 → 5.9

Time: 19.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -2.399464687274639 \cdot 10^{+92}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;y \le 3.0447027671553217 \cdot 10^{-120}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r22340305 = x;
        double r22340306 = y;
        double r22340307 = z;
        double r22340308 = r22340306 / r22340307;
        double r22340309 = t;
        double r22340310 = r22340308 * r22340309;
        double r22340311 = r22340310 / r22340309;
        double r22340312 = r22340305 * r22340311;
        return r22340312;
}

↓

double f(double x, double y, double z, double __attribute__((unused)) t) {
        double r22340313 = y;
        double r22340314 = -2.399464687274639e+92;
        bool r22340315 = r22340313 <= r22340314;
        double r22340316 = x;
        double r22340317 = z;
        double r22340318 = r22340317 / r22340313;
        double r22340319 = r22340316 / r22340318;
        double r22340320 = 3.0447027671553217e-120;
        bool r22340321 = r22340313 <= r22340320;
        double r22340322 = r22340317 / r22340316;
        double r22340323 = r22340313 / r22340322;
        double r22340324 = r22340316 * r22340313;
        double r22340325 = r22340324 / r22340317;
        double r22340326 = r22340321 ? r22340323 : r22340325;
        double r22340327 = r22340315 ? r22340319 : r22340326;
        return r22340327;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Derivation

1. Split input into 3 regimes

2. if y < -2.399464687274639e+92

   1. Initial program 19.2

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

   2. Simplified 9.9

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

   4. Applied *-un-lft-identity 9.9

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

   5. Applied add-cube-cbrt 10.8

      \[\leadsto x \cdot \frac{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}{1 \cdot z}\]

   6. Applied times-frac 10.8

      \[\leadsto x \cdot \color{blue}{\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{1} \cdot \frac{\sqrt[3]{y}}{z}\right)}\]

   7. Applied associate-*r* 7.7

      \[\leadsto \color{blue}{\left(x \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{1}\right) \cdot \frac{\sqrt[3]{y}}{z}}\]

   8. Taylor expanded around 0 8.8

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

   10. Applied associate-/l* 9.5

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

   if -2.399464687274639e+92 < y < 3.0447027671553217e-120

   1. Initial program 12.1

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

   2. Simplified 5.2

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

   4. Applied *-un-lft-identity 5.2

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

   5. Applied add-cube-cbrt 6.0

      \[\leadsto x \cdot \frac{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}{1 \cdot z}\]

   6. Applied times-frac 6.0

      \[\leadsto x \cdot \color{blue}{\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{1} \cdot \frac{\sqrt[3]{y}}{z}\right)}\]

   7. Applied associate-*r* 5.2

      \[\leadsto \color{blue}{\left(x \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{1}\right) \cdot \frac{\sqrt[3]{y}}{z}}\]
   8. Using strategy rm

   9. Applied associate-*r/ 5.2

      \[\leadsto \color{blue}{\frac{x \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)}{1}} \cdot \frac{\sqrt[3]{y}}{z}\]

   10. Applied associate-*l/ 5.2

       \[\leadsto \color{blue}{\frac{\left(x \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right) \cdot \frac{\sqrt[3]{y}}{z}}{1}}\]

   11. Simplified 5.0

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

   if 3.0447027671553217e-120 < y

   1. Initial program 14.4

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

   2. Simplified 5.1

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

   4. Applied *-un-lft-identity 5.1

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

   5. Applied add-cube-cbrt 6.0

      \[\leadsto x \cdot \frac{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}{1 \cdot z}\]

   6. Applied times-frac 6.0

      \[\leadsto x \cdot \color{blue}{\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{1} \cdot \frac{\sqrt[3]{y}}{z}\right)}\]

   7. Applied associate-*r* 5.5

      \[\leadsto \color{blue}{\left(x \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{1}\right) \cdot \frac{\sqrt[3]{y}}{z}}\]

   8. Taylor expanded around 0 5.8

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

4. Final simplification 5.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -2.399464687274639 \cdot 10^{+92}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;y \le 3.0447027671553217 \cdot 10^{-120}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019120 
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1"
  (* x (/ (* (/ y z) t) t)))