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

Average Error: 16.3 → 10.2

Time: 6.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -1.53218274451368 \cdot 10^{89}:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - \left(\left(\left(z - t\right) \cdot \left(\sqrt[3]{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \sqrt[3]{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right)\right) \cdot \left(\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right)\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{a - t}}\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double VAR;
	if ((t <= -1.53218274451368e+89)) {
		VAR = (((z * y) / t) + x);
	} else {
		VAR = ((x + y) - ((((z - t) * (cbrt(((cbrt(y) * cbrt(y)) / (cbrt((a - t)) * cbrt((a - t))))) * cbrt(((cbrt(y) * cbrt(y)) / (cbrt((a - t)) * cbrt((a - t))))))) * (cbrt((cbrt(y) * cbrt(y))) * cbrt((1.0 / (cbrt((a - t)) * cbrt((a - t))))))) * (cbrt(y) / cbrt((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	16.3
Target	8.1
Herbie	10.2

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

Derivation

1. Split input into 2 regimes

2. if t < -1.53218274451368e+89

   1. Initial program 28.7

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

   2. Taylor expanded around inf 16.5

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

   if -1.53218274451368e+89 < t

   1. Initial program 13.4

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

   3. Applied *-un-lft-identity 13.4

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

   4. Applied times-frac 9.9

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

   5. Simplified 9.9

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

   7. Applied add-cube-cbrt 10.0

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

   8. Applied add-cube-cbrt 10.1

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

   9. Applied times-frac 10.1

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

   10. Applied associate-*r* 8.7

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

   12. Applied add-cube-cbrt 8.7

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

   13. Applied associate-*r* 8.7

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

   15. Applied div-inv 8.7

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

   16. Applied cbrt-prod 8.7

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

4. Final simplification 10.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.53218274451368 \cdot 10^{89}:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - \left(\left(\left(z - t\right) \cdot \left(\sqrt[3]{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \sqrt[3]{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right)\right) \cdot \left(\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right)\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{a - t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020075 
(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))))