Graphics.Rendering.Chart.Plot.AreaSpots:renderAreaSpots4D from Chart-1.5.3

Average Error: 11.5 → 1.5

Time: 5.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y - z\right)}{t - z} = -\infty:\\ \;\;\;\;\frac{x}{t - z} \cdot \left(y - z\right)\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{t - z} \le 5.316577800148401 \cdot 10^{177}:\\ \;\;\;\;\frac{x \cdot \left(y - z\right)}{t - z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{t - z} - \frac{z}{t - z}\right)\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t) {
	double temp;
	if ((((x * (y - z)) / (t - z)) <= -inf.0)) {
		temp = ((x / (t - z)) * (y - z));
	} else {
		double temp_1;
		if ((((x * (y - z)) / (t - z)) <= 5.316577800148401e+177)) {
			temp_1 = ((x * (y - z)) / (t - z));
		} else {
			temp_1 = (x * ((y / (t - z)) - (z / (t - z))));
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	11.5
Target	2.2
Herbie	1.5

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

Derivation

1. Split input into 3 regimes

2. if (/ (* x (- y z)) (- t z)) < -inf.0

   1. Initial program 64.0

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

   3. Applied *-un-lft-identity 64.0

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

   4. Applied times-frac 0.1

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

   5. Simplified 0.1

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

   7. Applied div-sub 0.1

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

   9. Applied div-inv 0.2

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

   10. Applied div-inv 0.2

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

   11. Applied distribute-rgt-out-- 0.2

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

   12. Applied associate-*r* 0.4

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

   13. Simplified 0.2

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

   if -inf.0 < (/ (* x (- y z)) (- t z)) < 5.316577800148401e+177

   1. Initial program 1.4

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

   if 5.316577800148401e+177 < (/ (* x (- y z)) (- t z))

   1. Initial program 43.7

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

   3. Applied *-un-lft-identity 43.7

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

   4. Applied times-frac 2.7

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

   5. Simplified 2.7

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

   7. Applied div-sub 2.7

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

4. Final simplification 1.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y - z\right)}{t - z} = -\infty:\\ \;\;\;\;\frac{x}{t - z} \cdot \left(y - z\right)\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{t - z} \le 5.316577800148401 \cdot 10^{177}:\\ \;\;\;\;\frac{x \cdot \left(y - z\right)}{t - z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{t - z} - \frac{z}{t - z}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderAreaSpots4D from Chart-1.5.3"
  :precision binary64

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

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