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

Average Error: 14.6 → 2.1

Time: 16.9s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -2.726224425942599138418068187989578050083 \cdot 10^{-214} \lor \neg \left(\frac{y}{z} \le 7.182752616858212935481303668920257428991 \cdot 10^{-116}\right):\\ \;\;\;\;\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z}} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r125708 = x;
        double r125709 = y;
        double r125710 = z;
        double r125711 = r125709 / r125710;
        double r125712 = t;
        double r125713 = r125711 * r125712;
        double r125714 = r125713 / r125712;
        double r125715 = r125708 * r125714;
        return r125715;
}

↓

double f(double x, double y, double z, double __attribute__((unused)) t) {
        double r125716 = y;
        double r125717 = z;
        double r125718 = r125716 / r125717;
        double r125719 = -2.726224425942599e-214;
        bool r125720 = r125718 <= r125719;
        double r125721 = 7.182752616858213e-116;
        bool r125722 = r125718 <= r125721;
        double r125723 = !r125722;
        bool r125724 = r125720 || r125723;
        double r125725 = cbrt(r125716);
        double r125726 = r125725 * r125725;
        double r125727 = cbrt(r125717);
        double r125728 = r125727 * r125727;
        double r125729 = r125726 / r125728;
        double r125730 = r125725 / r125727;
        double r125731 = x;
        double r125732 = r125730 * r125731;
        double r125733 = r125729 * r125732;
        double r125734 = r125731 * r125716;
        double r125735 = r125734 / r125717;
        double r125736 = r125724 ? r125733 : r125735;
        return r125736;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Derivation

1. Split input into 2 regimes

2. if (/ y z) < -2.726224425942599e-214 or 7.182752616858213e-116 < (/ y z)

   1. Initial program 13.9

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

   2. Simplified 4.8

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

   4. Applied add-cube-cbrt 5.8

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

   5. Applied add-cube-cbrt 6.0

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

   6. Applied times-frac 6.0

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

   7. Applied associate-*l* 2.3

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

   if -2.726224425942599e-214 < (/ y z) < 7.182752616858213e-116

   1. Initial program 15.9

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

   2. Simplified 8.3

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

   4. Applied *-un-lft-identity 8.3

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

   5. Applied associate-*l* 8.3

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

   6. Simplified 1.6

      \[\leadsto 1 \cdot \color{blue}{\frac{x \cdot y}{z}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 2.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -2.726224425942599138418068187989578050083 \cdot 10^{-214} \lor \neg \left(\frac{y}{z} \le 7.182752616858212935481303668920257428991 \cdot 10^{-116}\right):\\ \;\;\;\;\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z}} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 
(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)))