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

Average Error: 16.3 → 8.7

Time: 6.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le -3.88334044086139269 \cdot 10^{-188}:\\ \;\;\;\;\left(x + y\right) - \left(z - t\right) \cdot \frac{y}{a - t}\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le 0.0:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - \left(\sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \left(\sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \frac{y}{\sqrt[3]{a - t}}\right)\\ \end{array}\]

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 ((((double) (((double) (x + y)) - ((double) (((double) (((double) (z - t)) * y)) / ((double) (a - t)))))) <= -3.883340440861393e-188)) {
		VAR = ((double) (((double) (x + y)) - ((double) (((double) (z - t)) * ((double) (y / ((double) (a - t))))))));
	} else {
		double VAR_1;
		if ((((double) (((double) (x + y)) - ((double) (((double) (((double) (z - t)) * y)) / ((double) (a - t)))))) <= 0.0)) {
			VAR_1 = ((double) (((double) (((double) (z * y)) / t)) + x));
		} else {
			VAR_1 = ((double) (((double) (x + y)) - ((double) (((double) (((double) cbrt(((double) (((double) (z - t)) / ((double) (((double) cbrt(((double) (a - t)))) * ((double) cbrt(((double) (a - t)))))))))) * ((double) cbrt(((double) (((double) (z - t)) / ((double) (((double) cbrt(((double) (a - t)))) * ((double) cbrt(((double) (a - t)))))))))))) * ((double) (((double) cbrt(((double) (((double) (z - t)) / ((double) (((double) cbrt(((double) (a - t)))) * ((double) cbrt(((double) (a - t)))))))))) * ((double) (y / ((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	16.3
Target	8.4
Herbie	8.7

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

Derivation

1. Split input into 3 regimes

2. if (- (+ x y) (/ (* (- z t) y) (- a t))) < -3.88334044086139269e-188

   1. Initial program 12.7

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

   3. Applied *-un-lft-identity 12.7

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

   4. Applied times-frac 7.9

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

   5. Simplified 7.9

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

   if -3.88334044086139269e-188 < (- (+ x y) (/ (* (- z t) y) (- a t))) < 0.0

   1. Initial program 53.6

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

   2. Taylor expanded around inf 19.7

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

   if 0.0 < (- (+ x y) (/ (* (- z t) y) (- a t)))

   1. Initial program 12.4

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

   3. Applied add-cube-cbrt 12.7

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

   4. Applied times-frac 7.2

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

   6. Applied add-cube-cbrt 7.3

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

   7. Applied associate-*l* 7.3

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

4. Final simplification 8.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le -3.88334044086139269 \cdot 10^{-188}:\\ \;\;\;\;\left(x + y\right) - \left(z - t\right) \cdot \frac{y}{a - t}\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le 0.0:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - \left(\sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \left(\sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \frac{y}{\sqrt[3]{a - t}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020161 
(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))))