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

Average Error: 1.3 → 0.7

Time: 8.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \cdot \frac{z - t}{a - t} = -inf.0:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a - t}\\ \mathbf{elif}\;y \cdot \frac{z - t}{a - t} \le 2.9907223722640885 \cdot 10^{276}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \frac{\frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t}}}{\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \frac{\frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}}}{\sqrt[3]{\sqrt[3]{a - t}}}\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double VAR;
	if ((((double) (y * ((double) (((double) (z - t)) / ((double) (a - t)))))) <= -inf.0)) {
		VAR = ((double) (x + ((double) (((double) (y * ((double) (z - t)))) / ((double) (a - t))))));
	} else {
		double VAR_1;
		if ((((double) (y * ((double) (((double) (z - t)) / ((double) (a - t)))))) <= 2.9907223722640885e+276)) {
			VAR_1 = ((double) (x + ((double) (y * ((double) (((double) (z - t)) / ((double) (a - t))))))));
		} else {
			VAR_1 = ((double) (x + ((double) (((double) (y * ((double) (((double) (((double) (((double) cbrt(((double) (z - t)))) * ((double) cbrt(((double) (z - t)))))) / ((double) cbrt(((double) (a - t)))))) / ((double) cbrt(((double) (((double) cbrt(((double) (a - t)))) * ((double) cbrt(((double) (a - t)))))))))))) * ((double) (((double) (((double) cbrt(((double) (z - t)))) / ((double) cbrt(((double) (a - t)))))) / ((double) cbrt(((double) cbrt(((double) (a - t))))))))))));
		}
		VAR = VAR_1;
	}
	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	1.3
Target	0.5
Herbie	0.7

\[\begin{array}{l} \mathbf{if}\;y \lt -8.50808486055124107 \cdot 10^{-17}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;y \lt 2.8944268627920891 \cdot 10^{-49}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (* y (/ (- z t) (- a t))) < -inf.0

   1. Initial program 64.0

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

   3. Applied associate-*r/ 0.2

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

   if -inf.0 < (* y (/ (- z t) (- a t))) < 2.9907223722640885e+276

   1. Initial program 0.3

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

   if 2.9907223722640885e+276 < (* y (/ (- z t) (- a t)))

   1. Initial program 19.4

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

   3. Applied add-cube-cbrt 20.1

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

   4. Applied associate-/r* 20.2

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

   6. Applied add-cube-cbrt 20.2

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

   7. Applied cbrt-prod 20.3

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

   8. Applied add-cube-cbrt 20.2

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

   9. Applied times-frac 20.2

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

   10. Applied times-frac 20.1

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

   11. Applied associate-*r* 14.4

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \frac{z - t}{a - t} = -inf.0:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a - t}\\ \mathbf{elif}\;y \cdot \frac{z - t}{a - t} \le 2.9907223722640885 \cdot 10^{276}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \frac{\frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t}}}{\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \frac{\frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}}}{\sqrt[3]{\sqrt[3]{a - t}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020140 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, B"
  :precision binary64

  :herbie-target
  (if (< y -8.508084860551241e-17) (+ x (* y (/ (- z t) (- a t)))) (if (< y 2.894426862792089e-49) (+ x (* (* y (- z t)) (/ 1.0 (- a t)))) (+ x (* y (/ (- z t) (- a t))))))

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