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

Average Error: 14.7 → 1.8

Time: 4.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -4.55389167634140265 \cdot 10^{-149}:\\ \;\;\;\;{\left(\frac{x}{\frac{z}{y}}\right)}^{1}\\ \mathbf{elif}\;\frac{y}{z} \le 1.4704572909860111 \cdot 10^{-239}:\\ \;\;\;\;{\left(\frac{x \cdot y}{z}\right)}^{1}\\ \mathbf{elif}\;\frac{y}{z} \le 7.35260291402330968 \cdot 10^{277}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{\frac{z}{x \cdot y}}\right)}^{1}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r90588 = x;
        double r90589 = y;
        double r90590 = z;
        double r90591 = r90589 / r90590;
        double r90592 = t;
        double r90593 = r90591 * r90592;
        double r90594 = r90593 / r90592;
        double r90595 = r90588 * r90594;
        return r90595;
}

↓

double f(double x, double y, double z, double __attribute__((unused)) t) {
        double r90596 = y;
        double r90597 = z;
        double r90598 = r90596 / r90597;
        double r90599 = -4.553891676341403e-149;
        bool r90600 = r90598 <= r90599;
        double r90601 = x;
        double r90602 = r90597 / r90596;
        double r90603 = r90601 / r90602;
        double r90604 = 1.0;
        double r90605 = pow(r90603, r90604);
        double r90606 = 1.470457290986011e-239;
        bool r90607 = r90598 <= r90606;
        double r90608 = r90601 * r90596;
        double r90609 = r90608 / r90597;
        double r90610 = pow(r90609, r90604);
        double r90611 = 7.35260291402331e+277;
        bool r90612 = r90598 <= r90611;
        double r90613 = r90601 * r90598;
        double r90614 = r90597 / r90608;
        double r90615 = r90604 / r90614;
        double r90616 = pow(r90615, r90604);
        double r90617 = r90612 ? r90613 : r90616;
        double r90618 = r90607 ? r90610 : r90617;
        double r90619 = r90600 ? r90605 : r90618;
        return r90619;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Derivation

1. Split input into 4 regimes

2. if (/ y z) < -4.553891676341403e-149

   1. Initial program 13.5

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

   2. Simplified 5.0

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

   4. Applied add-cube-cbrt 6.0

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

   5. Applied *-un-lft-identity 6.0

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

   6. Applied times-frac 6.0

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

   7. Applied associate-*r* 7.4

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

   8. Simplified 7.4

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

   10. Applied pow1 7.4

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

   11. Applied pow1 7.4

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

   12. Applied pow-prod-down 7.4

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

   13. Simplified 9.7

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

   15. Applied associate-/l* 4.4

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

   if -4.553891676341403e-149 < (/ y z) < 1.470457290986011e-239

   1. Initial program 17.2

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

   2. Simplified 10.4

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

   4. Applied add-cube-cbrt 10.7

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

   5. Applied *-un-lft-identity 10.7

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

   6. Applied times-frac 10.7

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

   7. Applied associate-*r* 2.6

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

   8. Simplified 2.6

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

   10. Applied pow1 2.6

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

   11. Applied pow1 2.6

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

   12. Applied pow-prod-down 2.6

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

   13. Simplified 0.9

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

   if 1.470457290986011e-239 < (/ y z) < 7.35260291402331e+277

   1. Initial program 9.8

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

   2. Simplified 0.2

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

   if 7.35260291402331e+277 < (/ y z)

   1. Initial program 55.4

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

   2. Simplified 49.4

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

   4. Applied add-cube-cbrt 49.7

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

   5. Applied *-un-lft-identity 49.7

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

   6. Applied times-frac 49.7

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

   7. Applied associate-*r* 12.6

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

   8. Simplified 12.5

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

   10. Applied pow1 12.5

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

   11. Applied pow1 12.5

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

   12. Applied pow-prod-down 12.5

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

   13. Simplified 0.3

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

   15. Applied clear-num 0.4

       \[\leadsto {\color{blue}{\left(\frac{1}{\frac{z}{x \cdot y}}\right)}}^{1}\]
3. Recombined 4 regimes into one program.

4. Final simplification 1.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -4.55389167634140265 \cdot 10^{-149}:\\ \;\;\;\;{\left(\frac{x}{\frac{z}{y}}\right)}^{1}\\ \mathbf{elif}\;\frac{y}{z} \le 1.4704572909860111 \cdot 10^{-239}:\\ \;\;\;\;{\left(\frac{x \cdot y}{z}\right)}^{1}\\ \mathbf{elif}\;\frac{y}{z} \le 7.35260291402330968 \cdot 10^{277}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{\frac{z}{x \cdot y}}\right)}^{1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 
(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)))