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

Average Error: 14.0 → 2.3

Time: 3.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -1.45613981804384912 \cdot 10^{278}:\\ \;\;\;\;{\left(\frac{x \cdot y}{z}\right)}^{1}\\ \mathbf{elif}\;\frac{y}{z} \le -5.31650938160314506 \cdot 10^{-21}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{y}{z} \le 3.3409790596422747 \cdot 10^{-221}:\\ \;\;\;\;{\left(\frac{x \cdot y}{z}\right)}^{1}\\ \mathbf{elif}\;\frac{y}{z} \le 7.8490367773301062 \cdot 10^{81}:\\ \;\;\;\;{\left(\frac{x}{\frac{z}{y}}\right)}^{1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x \cdot y}{z}\right)}^{1}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r84346 = x;
        double r84347 = y;
        double r84348 = z;
        double r84349 = r84347 / r84348;
        double r84350 = t;
        double r84351 = r84349 * r84350;
        double r84352 = r84351 / r84350;
        double r84353 = r84346 * r84352;
        return r84353;
}

↓

double f(double x, double y, double z, double __attribute__((unused)) t) {
        double r84354 = y;
        double r84355 = z;
        double r84356 = r84354 / r84355;
        double r84357 = -1.456139818043849e+278;
        bool r84358 = r84356 <= r84357;
        double r84359 = x;
        double r84360 = r84359 * r84354;
        double r84361 = r84360 / r84355;
        double r84362 = 1.0;
        double r84363 = pow(r84361, r84362);
        double r84364 = -5.316509381603145e-21;
        bool r84365 = r84356 <= r84364;
        double r84366 = r84359 * r84356;
        double r84367 = 3.3409790596422747e-221;
        bool r84368 = r84356 <= r84367;
        double r84369 = 7.849036777330106e+81;
        bool r84370 = r84356 <= r84369;
        double r84371 = r84355 / r84354;
        double r84372 = r84359 / r84371;
        double r84373 = pow(r84372, r84362);
        double r84374 = r84370 ? r84373 : r84363;
        double r84375 = r84368 ? r84363 : r84374;
        double r84376 = r84365 ? r84366 : r84375;
        double r84377 = r84358 ? r84363 : r84376;
        return r84377;
}

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) < -1.456139818043849e+278 or -5.316509381603145e-21 < (/ y z) < 3.3409790596422747e-221 or 7.849036777330106e+81 < (/ y z)

   1. Initial program 18.6

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

   2. Simplified 10.7

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

   4. Applied *-un-lft-identity 10.7

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

   5. Applied add-cube-cbrt 11.3

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

   6. Applied times-frac 11.3

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

   7. Applied associate-*r* 4.8

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

   8. Simplified 4.8

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

   10. Applied pow1 4.8

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

   11. Applied pow1 4.8

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

   12. Applied pow1 4.8

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

   13. Applied pow1 4.8

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

   14. Applied pow-prod-down 4.8

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

   15. Applied pow-prod-down 4.8

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

   16. Applied pow-prod-down 4.8

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

   17. Simplified 3.9

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

   if -1.456139818043849e+278 < (/ y z) < -5.316509381603145e-21

   1. Initial program 10.4

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

   2. Simplified 0.3

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

   if 3.3409790596422747e-221 < (/ y z) < 7.849036777330106e+81

   1. Initial program 6.3

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

   2. Simplified 0.2

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

   4. Applied *-un-lft-identity 0.2

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

   5. Applied add-cube-cbrt 1.3

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

   6. Applied times-frac 1.3

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

   7. Applied associate-*r* 6.8

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

   8. Simplified 6.8

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

   10. Applied pow1 6.8

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

   11. Applied pow1 6.8

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

   12. Applied pow1 6.8

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

   13. Applied pow1 6.8

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

   14. Applied pow-prod-down 6.8

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

   15. Applied pow-prod-down 6.8

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

   16. Applied pow-prod-down 6.8

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

   17. Simplified 10.7

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

   19. Applied associate-/l* 0.2

       \[\leadsto {\color{blue}{\left(\frac{x}{\frac{z}{y}}\right)}}^{1}\]
3. Recombined 3 regimes into one program.

4. Final simplification 2.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -1.45613981804384912 \cdot 10^{278}:\\ \;\;\;\;{\left(\frac{x \cdot y}{z}\right)}^{1}\\ \mathbf{elif}\;\frac{y}{z} \le -5.31650938160314506 \cdot 10^{-21}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{y}{z} \le 3.3409790596422747 \cdot 10^{-221}:\\ \;\;\;\;{\left(\frac{x \cdot y}{z}\right)}^{1}\\ \mathbf{elif}\;\frac{y}{z} \le 7.8490367773301062 \cdot 10^{81}:\\ \;\;\;\;{\left(\frac{x}{\frac{z}{y}}\right)}^{1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x \cdot y}{z}\right)}^{1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1"
  :precision binary64
  (* x (/ (* (/ y z) t) t)))