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

Average Error: 14.6 → 2.1

Time: 16.3s

Precision: 64

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

Math

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

↓

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

C

double f(double x, double y, double z, double t) {
        double r103369 = x;
        double r103370 = y;
        double r103371 = z;
        double r103372 = r103370 / r103371;
        double r103373 = t;
        double r103374 = r103372 * r103373;
        double r103375 = r103374 / r103373;
        double r103376 = r103369 * r103375;
        return r103376;
}

↓

double f(double x, double y, double z, double __attribute__((unused)) t) {
        double r103377 = y;
        double r103378 = z;
        double r103379 = r103377 / r103378;
        double r103380 = -2.726224425942599e-214;
        bool r103381 = r103379 <= r103380;
        double r103382 = 7.182752616858213e-116;
        bool r103383 = r103379 <= r103382;
        double r103384 = !r103383;
        bool r103385 = r103381 || r103384;
        double r103386 = cbrt(r103377);
        double r103387 = r103386 * r103386;
        double r103388 = cbrt(r103378);
        double r103389 = r103388 * r103388;
        double r103390 = r103387 / r103389;
        double r103391 = r103386 / r103388;
        double r103392 = x;
        double r103393 = r103391 * r103392;
        double r103394 = r103390 * r103393;
        double r103395 = r103377 * r103392;
        double r103396 = r103395 / r103378;
        double r103397 = r103385 ? r103394 : r103396;
        return r103397;
}

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 add-cube-cbrt 8.6

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

   5. Applied add-cube-cbrt 8.8

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

   6. Applied times-frac 8.8

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

   7. Applied associate-*l* 1.6

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

   9. Applied add-cube-cbrt 1.7

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

   11. Applied associate-*l/ 2.0

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

   12. Applied frac-times 2.4

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

   13. Simplified 2.1

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

   14. Simplified 1.6

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

4. Final simplification 2.1

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

Reproduce

herbie shell --seed 2019350 +o rules:numerics
(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)))