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

Average Error: 16.6 → 7.7

Time: 5.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \leq -1.960689538275324 \cdot 10^{-45}:\\ \;\;\;\;x + \left(\sqrt[3]{y + y \cdot \frac{z - t}{t - a}} \cdot \sqrt[3]{y + y \cdot \frac{z - t}{t - a}}\right) \cdot \sqrt[3]{y + \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(\frac{z - t}{t - a} \cdot \sqrt[3]{y}\right)}\\ \mathbf{elif}\;t \leq 1.3569370442215147 \cdot 10^{-304}:\\ \;\;\;\;x + \left(y + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{t - a}\right)\\ \mathbf{elif}\;t \leq 3.021594397083245 \cdot 10^{+158} \lor \neg \left(t \leq 3.311116811777091 \cdot 10^{+277}\right):\\ \;\;\;\;x + \left(\sqrt[3]{y + y \cdot \frac{z - t}{t - a}} \cdot \sqrt[3]{y + y \cdot \frac{z - t}{t - a}}\right) \cdot \sqrt[3]{y + \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(\frac{z - t}{t - a} \cdot \sqrt[3]{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double VAR;
	if ((t <= -1.960689538275324e-45)) {
		VAR = ((double) (x + ((double) (((double) (((double) cbrt(((double) (y + ((double) (y * (((double) (z - t)) / ((double) (t - a))))))))) * ((double) cbrt(((double) (y + ((double) (y * (((double) (z - t)) / ((double) (t - a))))))))))) * ((double) cbrt(((double) (y + ((double) (((double) (((double) cbrt(y)) * ((double) cbrt(y)))) * ((double) ((((double) (z - t)) / ((double) (t - a))) * ((double) cbrt(y))))))))))))));
	} else {
		double VAR_1;
		if ((t <= 1.3569370442215147e-304)) {
			VAR_1 = ((double) (x + ((double) (y + ((double) (((double) (y * ((double) (z - t)))) * (1.0 / ((double) (t - a)))))))));
		} else {
			double VAR_2;
			if (((t <= 3.021594397083245e+158) || !(t <= 3.311116811777091e+277))) {
				VAR_2 = ((double) (x + ((double) (((double) (((double) cbrt(((double) (y + ((double) (y * (((double) (z - t)) / ((double) (t - a))))))))) * ((double) cbrt(((double) (y + ((double) (y * (((double) (z - t)) / ((double) (t - a))))))))))) * ((double) cbrt(((double) (y + ((double) (((double) (((double) cbrt(y)) * ((double) cbrt(y)))) * ((double) ((((double) (z - t)) / ((double) (t - a))) * ((double) cbrt(y))))))))))))));
			} else {
				VAR_2 = ((double) (x + ((double) (y * (z / 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	16.6
Target	8.2
Herbie	7.7

\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} < -1.3664970889390727 \cdot 10^{-07}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} < 1.4754293444577233 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if t < -1.9606895382753241e-45 or 1.35693704422151472e-304 < t < 3.02159439708324501e158 or 3.31111681177709092e277 < t

   1. Initial program 17.6

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

   2. Simplified 7.6

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

   4. Applied add-cube-cbrt 8.0

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

   6. Applied add-cube-cbrt 8.0

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

   7. Applied associate-*l* 8.0

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

   8. Simplified 8.0

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

   if -1.9606895382753241e-45 < t < 1.35693704422151472e-304

   1. Initial program 5.5

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

   2. Simplified 4.1

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

   4. Applied div-inv 4.3

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

   5. Applied associate-*r* 5.3

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

   if 3.02159439708324501e158 < t < 3.31111681177709092e277

   1. Initial program 30.2

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

   2. Simplified 11.9

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

   3. Taylor expanded around inf 16.3

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

   4. Simplified 10.5

      \[\leadsto x + \color{blue}{\frac{z}{t} \cdot y}\]
3. Recombined 3 regimes into one program.

4. Final simplification 7.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.960689538275324 \cdot 10^{-45}:\\ \;\;\;\;x + \left(\sqrt[3]{y + y \cdot \frac{z - t}{t - a}} \cdot \sqrt[3]{y + y \cdot \frac{z - t}{t - a}}\right) \cdot \sqrt[3]{y + \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(\frac{z - t}{t - a} \cdot \sqrt[3]{y}\right)}\\ \mathbf{elif}\;t \leq 1.3569370442215147 \cdot 10^{-304}:\\ \;\;\;\;x + \left(y + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{t - a}\right)\\ \mathbf{elif}\;t \leq 3.021594397083245 \cdot 10^{+158} \lor \neg \left(t \leq 3.311116811777091 \cdot 10^{+277}\right):\\ \;\;\;\;x + \left(\sqrt[3]{y + y \cdot \frac{z - t}{t - a}} \cdot \sqrt[3]{y + y \cdot \frac{z - t}{t - a}}\right) \cdot \sqrt[3]{y + \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(\frac{z - t}{t - a} \cdot \sqrt[3]{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2020199 
(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))))