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

Average Error: 14.4 → 1.3

Time: 29.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -2.5398666775420435 \cdot 10^{+227}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{y}{z} \le -3.293755038166146 \cdot 10^{-302}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \left(y \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \sqrt[3]{\sqrt[3]{z}}}\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r4885096 = x;
        double r4885097 = y;
        double r4885098 = z;
        double r4885099 = r4885097 / r4885098;
        double r4885100 = t;
        double r4885101 = r4885099 * r4885100;
        double r4885102 = r4885101 / r4885100;
        double r4885103 = r4885096 * r4885102;
        return r4885103;
}

↓

double f(double x, double y, double z, double __attribute__((unused)) t) {
        double r4885104 = y;
        double r4885105 = z;
        double r4885106 = r4885104 / r4885105;
        double r4885107 = -2.5398666775420435e+227;
        bool r4885108 = r4885106 <= r4885107;
        double r4885109 = x;
        double r4885110 = r4885109 * r4885104;
        double r4885111 = r4885110 / r4885105;
        double r4885112 = -3.293755038166146e-302;
        bool r4885113 = r4885106 <= r4885112;
        double r4885114 = r4885106 * r4885109;
        double r4885115 = cbrt(r4885109);
        double r4885116 = r4885115 * r4885115;
        double r4885117 = cbrt(r4885105);
        double r4885118 = r4885117 * r4885117;
        double r4885119 = r4885116 / r4885118;
        double r4885120 = cbrt(r4885118);
        double r4885121 = cbrt(r4885117);
        double r4885122 = r4885120 * r4885121;
        double r4885123 = r4885115 / r4885122;
        double r4885124 = r4885104 * r4885123;
        double r4885125 = r4885119 * r4885124;
        double r4885126 = r4885113 ? r4885114 : r4885125;
        double r4885127 = r4885108 ? r4885111 : r4885126;
        return r4885127;
}

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) < -2.5398666775420435e+227

   1. Initial program 42.6

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

   2. Simplified 0.6

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

   4. Applied add-cube-cbrt 1.8

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

   5. Applied add-cube-cbrt 2.0

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

   6. Applied times-frac 2.0

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

   7. Applied associate-*l* 1.8

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

   8. Taylor expanded around inf 0.8

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

   if -2.5398666775420435e+227 < (/ y z) < -3.293755038166146e-302

   1. Initial program 9.3

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

   2. Simplified 8.2

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

   4. Applied div-inv 8.3

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

   5. Applied associate-*l* 0.3

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

   6. Simplified 0.2

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

   if -3.293755038166146e-302 < (/ y z)

   1. Initial program 15.1

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

   2. Simplified 5.6

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

   4. Applied add-cube-cbrt 6.4

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

   5. Applied add-cube-cbrt 6.5

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

   6. Applied times-frac 6.5

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

   7. Applied associate-*l* 1.9

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

   9. Applied add-cube-cbrt 2.0

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

   10. Applied cbrt-prod 2.0

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

4. Final simplification 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -2.5398666775420435 \cdot 10^{+227}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{y}{z} \le -3.293755038166146 \cdot 10^{-302}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \left(y \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \sqrt[3]{\sqrt[3]{z}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019149 
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1"
  (* x (/ (* (/ y z) t) t)))