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

Average Error: 14.5 → 0.5

Time: 3.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -6.122731566035475311671418161909773301552 \cdot 10^{157}:\\ \;\;\;\;{\left(\frac{1}{\frac{z}{x \cdot y}}\right)}^{1}\\ \mathbf{elif}\;\frac{y}{z} \le -2.242439350440375267697031348628742827796 \cdot 10^{-179}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{y}{z} \le 5.129556799950046333853278890491110988102 \cdot 10^{-271}:\\ \;\;\;\;{\left(\frac{x \cdot y}{z}\right)}^{1}\\ \mathbf{elif}\;\frac{y}{z} \le 1.795095217187765936797771641264103283314 \cdot 10^{286}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x \cdot y}{z}\right)}^{1}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r102638 = x;
        double r102639 = y;
        double r102640 = z;
        double r102641 = r102639 / r102640;
        double r102642 = t;
        double r102643 = r102641 * r102642;
        double r102644 = r102643 / r102642;
        double r102645 = r102638 * r102644;
        return r102645;
}

↓

double f(double x, double y, double z, double __attribute__((unused)) t) {
        double r102646 = y;
        double r102647 = z;
        double r102648 = r102646 / r102647;
        double r102649 = -6.122731566035475e+157;
        bool r102650 = r102648 <= r102649;
        double r102651 = 1.0;
        double r102652 = x;
        double r102653 = r102652 * r102646;
        double r102654 = r102647 / r102653;
        double r102655 = r102651 / r102654;
        double r102656 = pow(r102655, r102651);
        double r102657 = -2.2424393504403753e-179;
        bool r102658 = r102648 <= r102657;
        double r102659 = r102652 * r102648;
        double r102660 = 5.129556799950046e-271;
        bool r102661 = r102648 <= r102660;
        double r102662 = r102653 / r102647;
        double r102663 = pow(r102662, r102651);
        double r102664 = 1.795095217187766e+286;
        bool r102665 = r102648 <= r102664;
        double r102666 = r102665 ? r102659 : r102663;
        double r102667 = r102661 ? r102663 : r102666;
        double r102668 = r102658 ? r102659 : r102667;
        double r102669 = r102650 ? r102656 : r102668;
        return r102669;
}

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) < -6.122731566035475e+157

   1. Initial program 33.9

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

   2. Simplified 18.6

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

   4. Applied *-un-lft-identity 18.6

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

   5. Applied add-cube-cbrt 19.4

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

   6. Applied times-frac 19.4

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

   7. Applied associate-*r* 6.1

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

   8. Simplified 6.1

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

   10. Applied pow1 6.1

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

   11. Applied pow1 6.1

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

   12. Applied pow1 6.1

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

   13. Applied pow1 6.1

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

   14. Applied pow-prod-down 6.1

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

   15. Applied pow-prod-down 6.1

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

   16. Applied pow-prod-down 6.1

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

   17. Simplified 2.3

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

   19. Applied clear-num 2.4

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

   if -6.122731566035475e+157 < (/ y z) < -2.2424393504403753e-179 or 5.129556799950046e-271 < (/ y z) < 1.795095217187766e+286

   1. Initial program 9.0

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

   2. Simplified 0.2

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

   if -2.2424393504403753e-179 < (/ y z) < 5.129556799950046e-271 or 1.795095217187766e+286 < (/ y z)

   1. Initial program 21.5

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

   2. Simplified 15.3

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

   4. Applied *-un-lft-identity 15.3

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

   5. Applied add-cube-cbrt 15.6

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

   6. Applied times-frac 15.6

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

   7. Applied associate-*r* 3.8

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

   8. Simplified 3.8

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

   10. Applied pow1 3.8

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

   11. Applied pow1 3.8

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

   12. Applied pow1 3.8

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

   13. Applied pow1 3.8

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

   14. Applied pow-prod-down 3.8

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

   15. Applied pow-prod-down 3.8

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

   16. Applied pow-prod-down 3.8

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

   17. Simplified 0.6

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -6.122731566035475311671418161909773301552 \cdot 10^{157}:\\ \;\;\;\;{\left(\frac{1}{\frac{z}{x \cdot y}}\right)}^{1}\\ \mathbf{elif}\;\frac{y}{z} \le -2.242439350440375267697031348628742827796 \cdot 10^{-179}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{y}{z} \le 5.129556799950046333853278890491110988102 \cdot 10^{-271}:\\ \;\;\;\;{\left(\frac{x \cdot y}{z}\right)}^{1}\\ \mathbf{elif}\;\frac{y}{z} \le 1.795095217187765936797771641264103283314 \cdot 10^{286}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x \cdot y}{z}\right)}^{1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 
(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)))