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

Average Error: 16.9 → 11.0

Time: 11.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -2.2562619026297881 \cdot 10^{70}:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{elif}\;t \le 3.15078871528651268 \cdot 10^{145}:\\ \;\;\;\;\left(x + y\right) - \left(\sqrt[3]{\frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \left(\frac{\sqrt[3]{z - t}}{\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \sqrt[3]{\sqrt[3]{a - t}}} \cdot \frac{y}{\sqrt[3]{a - t}}\right)} \cdot \sqrt[3]{\frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \left(\frac{\sqrt[3]{z - t}}{\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \sqrt[3]{\sqrt[3]{a - t}}} \cdot \frac{y}{\sqrt[3]{a - t}}\right)}\right) \cdot \sqrt[3]{\frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \left(\frac{\sqrt[3]{z - t}}{\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \sqrt[3]{\sqrt[3]{a - t}}} \cdot \frac{y}{\sqrt[3]{a - t}}\right)}\\ \mathbf{elif}\;t \le 2.6363510366221429 \cdot 10^{207}:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\left(x + y\right) - \frac{z - t}{\sqrt[3]{a - t} \cdot \left(\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \sqrt[3]{\sqrt[3]{a - t}}\right)} \cdot \frac{y}{\sqrt[3]{a - t}}} \cdot \sqrt[3]{\left(x + y\right) - \frac{z - t}{\sqrt[3]{a - t} \cdot \left(\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \sqrt[3]{\sqrt[3]{a - t}}\right)} \cdot \frac{y}{\sqrt[3]{a - t}}}\right) \cdot \sqrt[3]{\left(x + y\right) - \frac{z - t}{\sqrt[3]{a - t} \cdot \left(\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \sqrt[3]{\sqrt[3]{a - t}}\right)} \cdot \frac{y}{\sqrt[3]{a - t}}}\\ \end{array}\]

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 <= -2.256261902629788e+70)) {
		VAR = ((double) (((double) (((double) (z * y)) / t)) + x));
	} else {
		double VAR_1;
		if ((t <= 3.1507887152865127e+145)) {
			VAR_1 = ((double) (((double) (x + y)) - ((double) (((double) (((double) cbrt(((double) (((double) (((double) (((double) cbrt(((double) (z - t)))) * ((double) cbrt(((double) (z - t)))))) / ((double) cbrt(((double) (a - t)))))) * ((double) (((double) (((double) cbrt(((double) (z - t)))) / ((double) (((double) cbrt(((double) (((double) cbrt(((double) (a - t)))) * ((double) cbrt(((double) (a - t)))))))) * ((double) cbrt(((double) cbrt(((double) (a - t)))))))))) * ((double) (y / ((double) cbrt(((double) (a - t)))))))))))) * ((double) cbrt(((double) (((double) (((double) (((double) cbrt(((double) (z - t)))) * ((double) cbrt(((double) (z - t)))))) / ((double) cbrt(((double) (a - t)))))) * ((double) (((double) (((double) cbrt(((double) (z - t)))) / ((double) (((double) cbrt(((double) (((double) cbrt(((double) (a - t)))) * ((double) cbrt(((double) (a - t)))))))) * ((double) cbrt(((double) cbrt(((double) (a - t)))))))))) * ((double) (y / ((double) cbrt(((double) (a - t)))))))))))))) * ((double) cbrt(((double) (((double) (((double) (((double) cbrt(((double) (z - t)))) * ((double) cbrt(((double) (z - t)))))) / ((double) cbrt(((double) (a - t)))))) * ((double) (((double) (((double) cbrt(((double) (z - t)))) / ((double) (((double) cbrt(((double) (((double) cbrt(((double) (a - t)))) * ((double) cbrt(((double) (a - t)))))))) * ((double) cbrt(((double) cbrt(((double) (a - t)))))))))) * ((double) (y / ((double) cbrt(((double) (a - t))))))))))))))));
		} else {
			double VAR_2;
			if ((t <= 2.636351036622143e+207)) {
				VAR_2 = ((double) (((double) (((double) (z * y)) / t)) + x));
			} else {
				VAR_2 = ((double) (((double) (((double) cbrt(((double) (((double) (x + y)) - ((double) (((double) (((double) (z - t)) / ((double) (((double) cbrt(((double) (a - t)))) * ((double) (((double) cbrt(((double) (((double) cbrt(((double) (a - t)))) * ((double) cbrt(((double) (a - t)))))))) * ((double) cbrt(((double) cbrt(((double) (a - t)))))))))))) * ((double) (y / ((double) cbrt(((double) (a - t)))))))))))) * ((double) cbrt(((double) (((double) (x + y)) - ((double) (((double) (((double) (z - t)) / ((double) (((double) cbrt(((double) (a - t)))) * ((double) (((double) cbrt(((double) (((double) cbrt(((double) (a - t)))) * ((double) cbrt(((double) (a - t)))))))) * ((double) cbrt(((double) cbrt(((double) (a - t)))))))))))) * ((double) (y / ((double) cbrt(((double) (a - t)))))))))))))) * ((double) cbrt(((double) (((double) (x + y)) - ((double) (((double) (((double) (z - t)) / ((double) (((double) cbrt(((double) (a - t)))) * ((double) (((double) cbrt(((double) (((double) cbrt(((double) (a - t)))) * ((double) cbrt(((double) (a - t)))))))) * ((double) cbrt(((double) cbrt(((double) (a - t)))))))))))) * ((double) (y / ((double) cbrt(((double) (a - t))))))))))))));
			}
			VAR_1 = VAR_2;
		}
		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	16.9
Target	8.7
Herbie	11.0

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

Derivation

1. Split input into 3 regimes

2. if t < -2.2562619026297881e70 or 3.15078871528651268e145 < t < 2.6363510366221429e207

   1. Initial program 29.6

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

   2. Taylor expanded around inf 18.8

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

   if -2.2562619026297881e70 < t < 3.15078871528651268e145

   1. Initial program 9.4

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

   3. Applied add-cube-cbrt 9.6

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

   4. Applied times-frac 6.7

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

   6. Applied add-cube-cbrt 6.7

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

   7. Applied cbrt-prod 6.7

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

   9. Applied add-cube-cbrt 6.8

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

   10. Applied times-frac 6.8

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

   11. Applied associate-*l* 5.8

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

   13. Applied add-cube-cbrt 5.9

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

   if 2.6363510366221429e207 < t

   1. Initial program 36.7

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

   3. Applied add-cube-cbrt 36.7

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

   4. Applied times-frac 26.3

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

   6. Applied add-cube-cbrt 26.3

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

   7. Applied cbrt-prod 26.3

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

   9. Applied add-cube-cbrt 27.0

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

4. Final simplification 11.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.2562619026297881 \cdot 10^{70}:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{elif}\;t \le 3.15078871528651268 \cdot 10^{145}:\\ \;\;\;\;\left(x + y\right) - \left(\sqrt[3]{\frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \left(\frac{\sqrt[3]{z - t}}{\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \sqrt[3]{\sqrt[3]{a - t}}} \cdot \frac{y}{\sqrt[3]{a - t}}\right)} \cdot \sqrt[3]{\frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \left(\frac{\sqrt[3]{z - t}}{\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \sqrt[3]{\sqrt[3]{a - t}}} \cdot \frac{y}{\sqrt[3]{a - t}}\right)}\right) \cdot \sqrt[3]{\frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \left(\frac{\sqrt[3]{z - t}}{\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \sqrt[3]{\sqrt[3]{a - t}}} \cdot \frac{y}{\sqrt[3]{a - t}}\right)}\\ \mathbf{elif}\;t \le 2.6363510366221429 \cdot 10^{207}:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\left(x + y\right) - \frac{z - t}{\sqrt[3]{a - t} \cdot \left(\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \sqrt[3]{\sqrt[3]{a - t}}\right)} \cdot \frac{y}{\sqrt[3]{a - t}}} \cdot \sqrt[3]{\left(x + y\right) - \frac{z - t}{\sqrt[3]{a - t} \cdot \left(\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \sqrt[3]{\sqrt[3]{a - t}}\right)} \cdot \frac{y}{\sqrt[3]{a - t}}}\right) \cdot \sqrt[3]{\left(x + y\right) - \frac{z - t}{\sqrt[3]{a - t} \cdot \left(\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \sqrt[3]{\sqrt[3]{a - t}}\right)} \cdot \frac{y}{\sqrt[3]{a - t}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020162 
(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))))