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

Average Error: 15.1 → 1.2

Time: 21.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} = -\infty:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \le -1.418027389015775738156568367664862717017 \cdot 10^{-287}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{\sqrt[3]{z}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{x}}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{z}}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r4261218 = x;
        double r4261219 = y;
        double r4261220 = z;
        double r4261221 = r4261219 / r4261220;
        double r4261222 = t;
        double r4261223 = r4261221 * r4261222;
        double r4261224 = r4261223 / r4261222;
        double r4261225 = r4261218 * r4261224;
        return r4261225;
}

↓

double f(double x, double y, double z, double __attribute__((unused)) t) {
        double r4261226 = y;
        double r4261227 = z;
        double r4261228 = r4261226 / r4261227;
        double r4261229 = -inf.0;
        bool r4261230 = r4261228 <= r4261229;
        double r4261231 = x;
        double r4261232 = r4261231 / r4261227;
        double r4261233 = r4261226 * r4261232;
        double r4261234 = -1.4180273890157757e-287;
        bool r4261235 = r4261228 <= r4261234;
        double r4261236 = r4261227 / r4261226;
        double r4261237 = r4261231 / r4261236;
        double r4261238 = cbrt(r4261227);
        double r4261239 = cbrt(r4261231);
        double r4261240 = r4261238 / r4261239;
        double r4261241 = r4261240 * r4261240;
        double r4261242 = r4261226 / r4261241;
        double r4261243 = r4261239 / r4261238;
        double r4261244 = r4261242 * r4261243;
        double r4261245 = r4261235 ? r4261237 : r4261244;
        double r4261246 = r4261230 ? r4261233 : r4261245;
        return r4261246;
}

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 z) < -inf.0

   1. Initial program 64.0

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

   2. Simplified 0.3

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

   if -inf.0 < (/ y z) < -1.4180273890157757e-287

   1. Initial program 10.4

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

   2. Simplified 7.8

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

   4. Applied add-cube-cbrt 8.6

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

   5. Applied *-un-lft-identity 8.6

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

   6. Applied times-frac 8.7

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

   7. Applied associate-*r* 2.7

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

   8. Simplified 2.7

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

   10. Applied *-un-lft-identity 2.7

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

   11. Applied cbrt-prod 2.7

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

   12. Applied *-un-lft-identity 2.7

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

   13. Applied times-frac 2.7

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

   14. Applied associate-*r* 2.7

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

   15. Simplified 2.7

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

   17. Applied *-un-lft-identity 2.7

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

   18. Applied associate-*l* 2.7

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

   19. Simplified 0.2

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

   if -1.4180273890157757e-287 < (/ y z)

   1. Initial program 16.1

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

   2. Simplified 5.0

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

   4. Applied add-cube-cbrt 5.7

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

   5. Applied *-un-lft-identity 5.7

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

   6. Applied times-frac 5.7

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

   7. Applied associate-*r* 6.0

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

   8. Simplified 6.0

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

   10. Applied *-un-lft-identity 6.0

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

   11. Applied cbrt-prod 6.0

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

   12. Applied *-un-lft-identity 6.0

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

   13. Applied times-frac 6.0

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

   14. Applied associate-*r* 6.0

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

   15. Simplified 6.0

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

   17. Applied *-un-lft-identity 6.0

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

   18. Applied cbrt-prod 6.0

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

   19. Applied add-cube-cbrt 6.1

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

   20. Applied times-frac 6.1

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

   21. Applied associate-*r* 5.2

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

   22. Simplified 1.8

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

4. Final simplification 1.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} = -\infty:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \le -1.418027389015775738156568367664862717017 \cdot 10^{-287}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{\sqrt[3]{z}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{x}}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1"
  (* x (/ (* (/ y z) t) t)))