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

Average Error: 14.8 → 1.7

Time: 9.1s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -2.0989985598366828 \cdot 10^{-236}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;\frac{y}{z} \le 4.4303806781760739 \cdot 10^{-235}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{y}{z} \le 6.4284572525223976 \cdot 10^{303}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{z}{x \cdot y}}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r618809 = x;
        double r618810 = y;
        double r618811 = z;
        double r618812 = r618810 / r618811;
        double r618813 = t;
        double r618814 = r618812 * r618813;
        double r618815 = r618814 / r618813;
        double r618816 = r618809 * r618815;
        return r618816;
}

↓

double f(double x, double y, double z, double __attribute__((unused)) t) {
        double r618817 = y;
        double r618818 = z;
        double r618819 = r618817 / r618818;
        double r618820 = -2.098998559836683e-236;
        bool r618821 = r618819 <= r618820;
        double r618822 = x;
        double r618823 = r618818 / r618817;
        double r618824 = r618822 / r618823;
        double r618825 = 4.430380678176074e-235;
        bool r618826 = r618819 <= r618825;
        double r618827 = r618822 * r618817;
        double r618828 = r618827 / r618818;
        double r618829 = 6.428457252522398e+303;
        bool r618830 = r618819 <= r618829;
        double r618831 = 1.0;
        double r618832 = r618818 / r618827;
        double r618833 = r618831 / r618832;
        double r618834 = r618830 ? r618824 : r618833;
        double r618835 = r618826 ? r618828 : r618834;
        double r618836 = r618821 ? r618824 : r618835;
        return r618836;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	14.8
Target	1.5
Herbie	1.7

\[\begin{array}{l} \mathbf{if}\;\frac{\frac{y}{z} \cdot t}{t} \lt -1.20672205123045005 \cdot 10^{245}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} \lt -5.90752223693390633 \cdot 10^{-275}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} \lt 5.65895442315341522 \cdot 10^{-65}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} \lt 2.0087180502407133 \cdot 10^{217}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (/ y z) < -2.098998559836683e-236 or 4.430380678176074e-235 < (/ y z) < 6.428457252522398e+303

   1. Initial program 11.8

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

   2. Simplified 2.3

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

   4. Applied add-cube-cbrt 3.3

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

   5. Applied *-un-lft-identity 3.3

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

   6. Applied times-frac 3.3

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

   7. Applied associate-*r* 5.8

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

   8. Simplified 5.8

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

   10. Applied add-cube-cbrt 6.1

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

   11. Taylor expanded around 0 8.2

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

   13. Applied associate-/l* 2.2

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

   if -2.098998559836683e-236 < (/ y z) < 4.430380678176074e-235

   1. Initial program 18.7

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

   2. Simplified 12.1

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

   4. Applied add-cube-cbrt 12.3

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

   5. Applied *-un-lft-identity 12.3

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

   6. Applied times-frac 12.3

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

   7. Applied associate-*r* 2.9

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

   8. Simplified 2.9

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

   10. Applied add-cube-cbrt 3.0

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

   11. Taylor expanded around 0 0.4

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

   if 6.428457252522398e+303 < (/ y z)

   1. Initial program 64.0

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

   2. Simplified 61.9

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

   4. Applied add-cube-cbrt 62.0

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

   5. Applied *-un-lft-identity 62.0

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

   6. Applied times-frac 62.0

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

   7. Applied associate-*r* 13.7

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

   8. Simplified 13.7

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

   10. Applied add-cube-cbrt 14.1

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

   11. Taylor expanded around 0 0.2

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

   13. Applied clear-num 0.4

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

4. Final simplification 1.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -2.0989985598366828 \cdot 10^{-236}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;\frac{y}{z} \le 4.4303806781760739 \cdot 10^{-235}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{y}{z} \le 6.4284572525223976 \cdot 10^{303}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{z}{x \cdot y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, B"
  :precision binary64

  :herbie-target
  (if (< (/ (* (/ y z) t) t) -1.20672205123045e+245) (/ y (/ z x)) (if (< (/ (* (/ y z) t) t) -5.907522236933906e-275) (* x (/ y z)) (if (< (/ (* (/ y z) t) t) 5.658954423153415e-65) (/ y (/ z x)) (if (< (/ (* (/ y z) t) t) 2.0087180502407133e+217) (* x (/ y z)) (/ (* y x) z)))))

  (* x (/ (* (/ y z) t) t)))