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

Average Error: 16.4 → 11.0

Time: 7.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le 3.1924211206661897 \cdot 10^{196}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{1}, \left(\sqrt[3]{\frac{\sqrt[3]{y}}{a - t}} \cdot \sqrt[3]{\frac{\sqrt[3]{y}}{a - t}}\right) \cdot \left(\sqrt[3]{\frac{\sqrt[3]{y}}{a - t}} \cdot \left(t - z\right)\right), x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

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 <= 3.1924211206661897e+196)) {
		VAR = ((double) fma(((double) (((double) (((double) cbrt(y)) * ((double) cbrt(y)))) / 1.0)), ((double) (((double) (((double) cbrt(((double) (((double) cbrt(y)) / ((double) (a - t)))))) * ((double) cbrt(((double) (((double) cbrt(y)) / ((double) (a - t)))))))) * ((double) (((double) cbrt(((double) (((double) cbrt(y)) / ((double) (a - t)))))) * ((double) (t - z)))))), ((double) (x + y))));
	} else {
		VAR = x;
	}
	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	16.4
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 2 regimes

2. if t < 3.1924211206661897e+196

   1. Initial program 14.4

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

   2. Simplified 10.8

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

   4. Applied fma-udef 10.8

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

   6. Applied *-un-lft-identity 10.8

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

   7. Applied add-cube-cbrt 11.0

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

   8. Applied times-frac 11.0

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

   9. Applied associate-*l* 10.1

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

   11. Applied fma-def 10.1

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{1}, \frac{\sqrt[3]{y}}{a - t} \cdot \left(t - z\right), x + y\right)}\]
   12. Using strategy rm

   13. Applied add-cube-cbrt 10.2

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

   14. Applied associate-*l* 10.2

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

   if 3.1924211206661897e+196 < t

   1. Initial program 34.0

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

   2. Simplified 25.7

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

   3. Taylor expanded around 0 18.5

      \[\leadsto \color{blue}{x}\]
3. Recombined 2 regimes into one program.

4. Final simplification 11.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 3.1924211206661897 \cdot 10^{196}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{1}, \left(\sqrt[3]{\frac{\sqrt[3]{y}}{a - t}} \cdot \sqrt[3]{\frac{\sqrt[3]{y}}{a - t}}\right) \cdot \left(\sqrt[3]{\frac{\sqrt[3]{y}}{a - t}} \cdot \left(t - z\right)\right), x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Reproduce

herbie shell --seed 2020120 +o rules:numerics
(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 (- 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 (- a t))) y))))

  (- (+ x y) (/ (* (- z t) y) (- a t))))