Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, A

Average Error: 10.9 → 1.3

Time: 15.5s

Precision: 64

Math

\[x + \frac{y \cdot \left(z - t\right)}{z - a}\]

↓

\[\begin{array}{l} \mathbf{if}\;z \le -6.964036638581209574726043986735525528163 \cdot 10^{-102} \lor \neg \left(z \le 9.403031276241487005540332044897243661846 \cdot 10^{-164}\right):\\ \;\;\;\;x + y \cdot \frac{z - t}{z - a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\sqrt[3]{z - a} \cdot \sqrt[3]{z - a}} \cdot \frac{z - t}{\sqrt[3]{z - a}}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r363609 = x;
        double r363610 = y;
        double r363611 = z;
        double r363612 = t;
        double r363613 = r363611 - r363612;
        double r363614 = r363610 * r363613;
        double r363615 = a;
        double r363616 = r363611 - r363615;
        double r363617 = r363614 / r363616;
        double r363618 = r363609 + r363617;
        return r363618;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r363619 = z;
        double r363620 = -6.9640366385812096e-102;
        bool r363621 = r363619 <= r363620;
        double r363622 = 9.403031276241487e-164;
        bool r363623 = r363619 <= r363622;
        double r363624 = !r363623;
        bool r363625 = r363621 || r363624;
        double r363626 = x;
        double r363627 = y;
        double r363628 = t;
        double r363629 = r363619 - r363628;
        double r363630 = a;
        double r363631 = r363619 - r363630;
        double r363632 = r363629 / r363631;
        double r363633 = r363627 * r363632;
        double r363634 = r363626 + r363633;
        double r363635 = cbrt(r363631);
        double r363636 = r363635 * r363635;
        double r363637 = r363627 / r363636;
        double r363638 = r363629 / r363635;
        double r363639 = r363637 * r363638;
        double r363640 = r363626 + r363639;
        double r363641 = r363625 ? r363634 : r363640;
        return r363641;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	10.9
Target	1.4
Herbie	1.3

\[x + \frac{y}{\frac{z - a}{z - t}}\]

Derivation

1. Split input into 2 regimes

2. if z < -6.9640366385812096e-102 or 9.403031276241487e-164 < z

   1. Initial program 13.5

      \[x + \frac{y \cdot \left(z - t\right)}{z - a}\]
   2. Using strategy rm

   3. Applied *-un-lft-identity 13.5

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

   4. Applied times-frac 0.5

      \[\leadsto x + \color{blue}{\frac{y}{1} \cdot \frac{z - t}{z - a}}\]

   5. Simplified 0.5

      \[\leadsto x + \color{blue}{y} \cdot \frac{z - t}{z - a}\]

   if -6.9640366385812096e-102 < z < 9.403031276241487e-164

   1. Initial program 3.3

      \[x + \frac{y \cdot \left(z - t\right)}{z - a}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 3.8

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

   4. Applied times-frac 3.4

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

4. Final simplification 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -6.964036638581209574726043986735525528163 \cdot 10^{-102} \lor \neg \left(z \le 9.403031276241487005540332044897243661846 \cdot 10^{-164}\right):\\ \;\;\;\;x + y \cdot \frac{z - t}{z - a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\sqrt[3]{z - a} \cdot \sqrt[3]{z - a}} \cdot \frac{z - t}{\sqrt[3]{z - a}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, A"
  :precision binary64

  :herbie-target
  (+ x (/ y (/ (- z a) (- z t))))

  (+ x (/ (* y (- z t)) (- z a))))