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

Average Error: 16.3 → 9.3

Time: 7.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -1.01675393957722618 \cdot 10^{-117}:\\ \;\;\;\;\left(x + y\right) - \left(\left(\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}\right) \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right) \cdot \frac{\sqrt[3]{z - t}}{\frac{a - t}{\sqrt[3]{y}}}\\ \mathbf{elif}\;a \le 4.2967849802929143 \cdot 10^{-186}:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \frac{z - t}{\frac{a - t}{\sqrt[3]{y}}}\\ \end{array}\]

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 <= -1.0167539395772262e-117)) {
		VAR = ((double) (((double) (x + y)) - ((double) (((double) (((double) (((double) cbrt(((double) (z - t)))) * ((double) cbrt(((double) (z - t)))))) * ((double) (((double) cbrt(y)) * ((double) cbrt(y)))))) * ((double) (((double) cbrt(((double) (z - t)))) / ((double) (((double) (a - t)) / ((double) cbrt(y))))))))));
	} else {
		double VAR_1;
		if ((a <= 4.296784980292914e-186)) {
			VAR_1 = ((double) (((double) (((double) (z * y)) / t)) + x));
		} else {
			VAR_1 = ((double) (((double) (x + y)) - ((double) (((double) (((double) cbrt(y)) * ((double) cbrt(y)))) * ((double) (((double) (z - t)) / ((double) (((double) (a - t)) / ((double) cbrt(y))))))))));
		}
		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	16.3
Target	8.1
Herbie	9.3

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

Derivation

1. Split input into 3 regimes

2. if a < -1.01675393957722618e-117

   1. Initial program 14.9

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

   3. Applied associate-/l* 8.9

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

   5. Applied add-cube-cbrt 9.0

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

   6. Applied *-un-lft-identity 9.0

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

   7. Applied times-frac 9.0

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

   8. Applied add-cube-cbrt 9.0

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

   9. Applied times-frac 9.4

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

   10. Simplified 9.4

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

   if -1.01675393957722618e-117 < a < 4.2967849802929143e-186

   1. Initial program 20.6

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

   2. Taylor expanded around inf 9.4

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

   if 4.2967849802929143e-186 < 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 associate-/l* 9.9

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

   5. Applied add-cube-cbrt 10.1

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

   6. Applied *-un-lft-identity 10.1

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

   7. Applied times-frac 10.1

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

   8. Applied *-un-lft-identity 10.1

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

   9. Applied times-frac 9.1

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

   10. Simplified 9.1

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

4. Final simplification 9.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -1.01675393957722618 \cdot 10^{-117}:\\ \;\;\;\;\left(x + y\right) - \left(\left(\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}\right) \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right) \cdot \frac{\sqrt[3]{z - t}}{\frac{a - t}{\sqrt[3]{y}}}\\ \mathbf{elif}\;a \le 4.2967849802929143 \cdot 10^{-186}:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \frac{z - t}{\frac{a - t}{\sqrt[3]{y}}}\\ \end{array}\]

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))))