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

Average Error: 16.2 → 10.1

Time: 5.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -5.346964750333344 \cdot 10^{27} \lor \neg \left(t \le 1.08451189140463412 \cdot 10^{202}\right):\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - \left(\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \left(\sqrt[3]{\frac{y}{\sqrt[3]{a - t}}} \cdot \sqrt[3]{\frac{y}{\sqrt[3]{a - t}}}\right)\right) \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{a - t}}}\right)\right)\\ \end{array}\]

C

double code(double x, double y, double z, double t, double a) {
	return ((double) (((double) (x + y)) - ((double) (((double) (((double) (z - t)) * y)) / ((double) (a - t))))));
}

↓

double code(double x, double y, double z, double t, double a) {
	double VAR;
	if (((t <= -5.346964750333344e+27) || !(t <= 1.0845118914046341e+202))) {
		VAR = ((double) (((double) (((double) (z * y)) / t)) + x));
	} else {
		VAR = ((double) (x + ((double) (y - ((double) (((double) (((double) (((double) (z - t)) / ((double) (((double) cbrt(((double) (a - t)))) * ((double) cbrt(((double) (a - t)))))))) * ((double) (((double) cbrt(((double) (y / ((double) cbrt(((double) (a - t)))))))) * ((double) cbrt(((double) (y / ((double) cbrt(((double) (a - t)))))))))))) * ((double) (((double) cbrt(y)) * ((double) cbrt(((double) (1.0 / ((double) cbrt(((double) (a - t))))))))))))))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	16.2
Target	8.2
Herbie	10.1

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

Derivation

1. Split input into 2 regimes

2. if t < -5.346964750333344e27 or 1.08451189140463412e202 < t

   1. Initial program 28.9

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

   2. Taylor expanded around inf 17.0

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

   if -5.346964750333344e27 < t < 1.08451189140463412e202

   1. Initial program 9.7

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

   3. Applied add-cube-cbrt 9.9

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

   4. Applied times-frac 6.6

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

   6. Applied add-cube-cbrt 6.7

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

   7. Applied associate-*r* 6.7

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

   9. Applied div-inv 6.7

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

   10. Applied cbrt-prod 6.7

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

   12. Applied associate--l+ 6.6

       \[\leadsto \color{blue}{x + \left(y - \left(\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \left(\sqrt[3]{\frac{y}{\sqrt[3]{a - t}}} \cdot \sqrt[3]{\frac{y}{\sqrt[3]{a - t}}}\right)\right) \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{a - t}}}\right)\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 10.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -5.346964750333344 \cdot 10^{27} \lor \neg \left(t \le 1.08451189140463412 \cdot 10^{202}\right):\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - \left(\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \left(\sqrt[3]{\frac{y}{\sqrt[3]{a - t}}} \cdot \sqrt[3]{\frac{y}{\sqrt[3]{a - t}}}\right)\right) \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{a - t}}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020149 
(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) (/ 1.0 (- 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) (/ 1.0 (- a t))) y))))

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