Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, A

Average Error: 10.1 → 1.1

Time: 2.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -2.3556285618525312 \cdot 10^{-32}:\\ \;\;\;\;x + \frac{y}{z - a} \cdot \left(z - t\right)\\ \mathbf{elif}\;y \le 1.21539727317836126 \cdot 10^{-115}:\\ \;\;\;\;x + \frac{1}{\frac{z - a}{y \cdot \left(z - t\right)}}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(\frac{z}{z - a} - \frac{t}{z - a}\right)\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double VAR;
	if ((y <= -2.3556285618525312e-32)) {
		VAR = ((double) (x + ((double) (((double) (y / ((double) (z - a)))) * ((double) (z - t))))));
	} else {
		double VAR_1;
		if ((y <= 1.2153972731783613e-115)) {
			VAR_1 = ((double) (x + ((double) (1.0 / ((double) (((double) (z - a)) / ((double) (y * ((double) (z - t))))))))));
		} else {
			VAR_1 = ((double) (x + ((double) (y * ((double) (((double) (z / ((double) (z - a)))) - ((double) (t / ((double) (z - a))))))))));
		}
		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	10.1
Target	1.3
Herbie	1.1

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

Derivation

1. Split input into 3 regimes

2. if y < -2.3556285618525312e-32

   1. Initial program 20.0

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

   3. Applied *-un-lft-identity 20.0

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

   4. Applied times-frac 0.5

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

   5. Simplified 0.5

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

   7. Applied div-sub 0.5

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

   9. Applied div-inv 0.5

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

   10. Applied div-inv 0.6

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

   11. Applied distribute-rgt-out-- 0.6

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

   12. Applied associate-*r* 2.6

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

   13. Simplified 2.5

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

   if -2.3556285618525312e-32 < y < 1.2153972731783613e-115

   1. Initial program 0.5

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

   3. Applied clear-num 0.5

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

   if 1.2153972731783613e-115 < y

   1. Initial program 14.8

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

   3. Applied *-un-lft-identity 14.8

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

   4. Applied times-frac 0.8

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

   5. Simplified 0.8

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

   7. Applied div-sub 0.8

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

4. Final simplification 1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -2.3556285618525312 \cdot 10^{-32}:\\ \;\;\;\;x + \frac{y}{z - a} \cdot \left(z - t\right)\\ \mathbf{elif}\;y \le 1.21539727317836126 \cdot 10^{-115}:\\ \;\;\;\;x + \frac{1}{\frac{z - a}{y \cdot \left(z - t\right)}}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(\frac{z}{z - a} - \frac{t}{z - a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020122 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, A"
  :precision binary64

  :herbie-target
  (+ x (/ y (/ (- z a) (- z t))))

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