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

Average Error: 1.4 → 0.5

Time: 4.5s

Precision: 64

Math

\[x + y \cdot \frac{z - t}{a - t}\]

↓

\[\begin{array}{l} \mathbf{if}\;y \le -6.2914023978356098 \cdot 10^{35} \lor \neg \left(y \le 8.8693386644516809 \cdot 10^{-85}\right):\\ \;\;\;\;x + \frac{z - t}{a - t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x + 1 \cdot \frac{\left(z - t\right) \cdot y}{a - t}\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double temp;
	if (((y <= -6.29140239783561e+35) || !(y <= 8.869338664451681e-85))) {
		temp = (x + (((z - t) / (a - t)) * y));
	} else {
		temp = (x + (1.0 * (((z - t) * y) / (a - t))));
	}
	return temp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	1.4
Target	0.4
Herbie	0.5

\[\begin{array}{l} \mathbf{if}\;y \lt -8.50808486055124107 \cdot 10^{-17}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;y \lt 2.8944268627920891 \cdot 10^{-49}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if y < -6.29140239783561e+35 or 8.869338664451681e-85 < y

   1. Initial program 0.5

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

   3. Applied *-commutative 0.5

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

   if -6.29140239783561e+35 < y < 8.869338664451681e-85

   1. Initial program 2.3

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

   3. Applied *-commutative 2.3

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

   5. Applied *-un-lft-identity 2.3

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

   6. Applied add-cube-cbrt 2.6

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

   7. Applied times-frac 2.6

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

   8. Applied associate-*l* 1.9

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

   10. Applied *-un-lft-identity 1.9

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

   11. Applied associate-*l* 1.9

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

   12. Simplified 0.5

       \[\leadsto x + 1 \cdot \color{blue}{\frac{\left(z - t\right) \cdot y}{a - t}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -6.2914023978356098 \cdot 10^{35} \lor \neg \left(y \le 8.8693386644516809 \cdot 10^{-85}\right):\\ \;\;\;\;x + \frac{z - t}{a - t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x + 1 \cdot \frac{\left(z - t\right) \cdot y}{a - t}\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, B"
  :precision binary64

  :herbie-target
  (if (< y -8.508084860551241e-17) (+ x (* y (/ (- z t) (- a t)))) (if (< y 2.894426862792089e-49) (+ x (* (* y (- z t)) (/ 1 (- a t)))) (+ x (* y (/ (- z t) (- a t))))))

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