Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, B

Average Error: 16.6 → 9.0

Time: 6.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -9.34367984237828866 \cdot 10^{-178} \lor \neg \left(a \le 4.5065014455694435 \cdot 10^{-66}\right):\\ \;\;\;\;\left(x + y\right) - \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{z - t}{\frac{\sqrt[3]{a - t}}{\sqrt[3]{y}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r643749 = x;
        double r643750 = y;
        double r643751 = r643749 + r643750;
        double r643752 = z;
        double r643753 = t;
        double r643754 = r643752 - r643753;
        double r643755 = r643754 * r643750;
        double r643756 = a;
        double r643757 = r643756 - r643753;
        double r643758 = r643755 / r643757;
        double r643759 = r643751 - r643758;
        return r643759;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r643760 = a;
        double r643761 = -9.343679842378289e-178;
        bool r643762 = r643760 <= r643761;
        double r643763 = 4.5065014455694435e-66;
        bool r643764 = r643760 <= r643763;
        double r643765 = !r643764;
        bool r643766 = r643762 || r643765;
        double r643767 = x;
        double r643768 = y;
        double r643769 = r643767 + r643768;
        double r643770 = cbrt(r643768);
        double r643771 = r643770 * r643770;
        double r643772 = t;
        double r643773 = r643760 - r643772;
        double r643774 = cbrt(r643773);
        double r643775 = r643774 * r643774;
        double r643776 = r643771 / r643775;
        double r643777 = z;
        double r643778 = r643777 - r643772;
        double r643779 = r643774 / r643770;
        double r643780 = r643778 / r643779;
        double r643781 = r643776 * r643780;
        double r643782 = r643769 - r643781;
        double r643783 = r643777 * r643768;
        double r643784 = r643783 / r643772;
        double r643785 = r643784 + r643767;
        double r643786 = r643766 ? r643782 : r643785;
        return r643786;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	16.6
Target	8.5
Herbie	9.0

\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt -1.3664970889390727 \cdot 10^{-7}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt 1.47542934445772333 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if a < -9.343679842378289e-178 or 4.5065014455694435e-66 < a

   1. Initial program 15.0

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

   3. Applied associate-/l* 9.1

      \[\leadsto \left(x + y\right) - \color{blue}{\frac{z - t}{\frac{a - t}{y}}}\]
   4. Using strategy rm

   5. Applied add-cube-cbrt 9.2

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

   6. Applied add-cube-cbrt 9.2

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

   7. Applied times-frac 9.2

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

   8. Applied *-un-lft-identity 9.2

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

   9. Applied times-frac 8.3

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

   10. Simplified 8.3

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

   if -9.343679842378289e-178 < a < 4.5065014455694435e-66

   1. Initial program 20.7

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

   2. Taylor expanded around inf 10.7

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

4. Final simplification 9.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -9.34367984237828866 \cdot 10^{-178} \lor \neg \left(a \le 4.5065014455694435 \cdot 10^{-66}\right):\\ \;\;\;\;\left(x + y\right) - \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{z - t}{\frac{\sqrt[3]{a - t}}{\sqrt[3]{y}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2020027 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, B"
  :precision binary64

  :herbie-target
  (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) -1.3664970889390727e-07) (- (+ y x) (* (* (- z t) (/ 1 (- a t))) y)) (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) 1.4754293444577233e-239) (/ (- (* y (- a z)) (* x t)) (- a t)) (- (+ y x) (* (* (- z t) (/ 1 (- a t))) y))))

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