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

Average Error: 16.1 → 9.5

Time: 13.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -6.3087141949950974 \cdot 10^{-146}:\\ \;\;\;\;\left(x + y\right) - \left(z - t\right) \cdot \frac{\frac{\frac{y}{\sqrt[3]{a - t}}}{\sqrt[3]{a - t}}}{\sqrt[3]{a - t}}\\ \mathbf{elif}\;a \le 6.9294151038589965 \cdot 10^{-69}:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - {\left(\frac{\frac{{\left(\sqrt[3]{z - t}\right)}^{3}}{\sqrt[3]{a - t}}}{\frac{\sqrt[3]{a - t}}{\frac{y}{\sqrt[3]{a - t}}}}\right)}^{1}\\ \end{array}\]

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 ((a <= -6.308714194995097e-146)) {
		VAR = ((double) (((double) (x + y)) - ((double) (((double) (z - t)) * ((double) (((double) (((double) (y / ((double) cbrt(((double) (a - t)))))) / ((double) cbrt(((double) (a - t)))))) / ((double) cbrt(((double) (a - t))))))))));
	} else {
		double VAR_1;
		if ((a <= 6.929415103858996e-69)) {
			VAR_1 = ((double) (((double) (((double) (z * y)) / t)) + x));
		} else {
			VAR_1 = ((double) (((double) (x + y)) - ((double) pow(((double) (((double) (((double) pow(((double) cbrt(((double) (z - t)))), 3.0)) / ((double) cbrt(((double) (a - t)))))) / ((double) (((double) cbrt(((double) (a - t)))) / ((double) (y / ((double) cbrt(((double) (a - t)))))))))), 1.0))));
		}
		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	16.1
Target	8.3
Herbie	9.5

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

Derivation

1. Split input into 3 regimes

2. if a < -6.3087141949950974e-146

   1. Initial program 14.9

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

   3. Applied add-cube-cbrt 15.0

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

   4. Applied times-frac 8.7

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

   6. Applied div-inv 8.7

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

   7. Applied associate-*l* 9.4

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

   8. Simplified 9.4

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

   if -6.3087141949950974e-146 < a < 6.9294151038589965e-69

   1. Initial program 18.8

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

   2. Taylor expanded around inf 11.0

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

   if 6.9294151038589965e-69 < a

   1. Initial program 14.9

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

   3. Applied add-cube-cbrt 15.0

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

   4. Applied times-frac 7.8

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

   6. Applied add-cube-cbrt 7.9

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

   7. Applied times-frac 7.9

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

   8. Applied associate-*l* 7.8

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

   10. Applied pow1 7.8

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

   11. Applied pow1 7.8

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

   12. Applied pow-prod-down 7.8

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

   13. Applied pow1 7.8

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

   14. Applied pow-prod-down 7.8

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

   15. Simplified 8.2

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

4. Final simplification 9.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -6.3087141949950974 \cdot 10^{-146}:\\ \;\;\;\;\left(x + y\right) - \left(z - t\right) \cdot \frac{\frac{\frac{y}{\sqrt[3]{a - t}}}{\sqrt[3]{a - t}}}{\sqrt[3]{a - t}}\\ \mathbf{elif}\;a \le 6.9294151038589965 \cdot 10^{-69}:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - {\left(\frac{\frac{{\left(\sqrt[3]{z - t}\right)}^{3}}{\sqrt[3]{a - t}}}{\frac{\sqrt[3]{a - t}}{\frac{y}{\sqrt[3]{a - t}}}}\right)}^{1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020171 
(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))))