Diagrams.TwoD.Segment.Bernstein:evaluateBernstein from diagrams-lib-1.3.0.3

Average Error: 10.4 → 0.1

Time: 2.6s

Precision: 64

Math

\[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\]

↓

\[\begin{array}{l} \mathbf{if}\;z \le -7.4248231966556819 \cdot 10^{-23} \lor \neg \left(z \le 2.6213843378382894 \cdot 10^{-38}\right):\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{1}{\left(y - z\right) + 1}}\\ \end{array}\]

C

double code(double x, double y, double z) {
	return ((double) (((double) (x * ((double) (((double) (y - z)) + 1.0)))) / z));
}

↓

double code(double x, double y, double z) {
	double VAR;
	if (((z <= -7.424823196655682e-23) || !(z <= 2.6213843378382894e-38))) {
		VAR = ((double) (x / ((double) (z / ((double) (((double) (y - z)) + 1.0))))));
	} else {
		VAR = ((double) (((double) (x / z)) / ((double) (1.0 / ((double) (((double) (y - z)) + 1.0))))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	10.4
Target	0.5
Herbie	0.1

\[\begin{array}{l} \mathbf{if}\;x \lt -2.7148310671343599 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x \lt 3.87410881643954616 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if z < -7.424823196655682e-23 or 2.6213843378382894e-38 < z

   1. Initial program 15.6

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\]
   2. Using strategy rm

   3. Applied associate-/l* 0.1

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

   if -7.424823196655682e-23 < z < 2.6213843378382894e-38

   1. Initial program 0.1

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\]
   2. Using strategy rm

   3. Applied associate-/l* 8.6

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{\left(y - z\right) + 1}}}\]
   4. Using strategy rm

   5. Applied div-inv 8.6

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

   6. Applied associate-/r* 0.2

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -7.4248231966556819 \cdot 10^{-23} \lor \neg \left(z \le 2.6213843378382894 \cdot 10^{-38}\right):\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{1}{\left(y - z\right) + 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020123 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.TwoD.Segment.Bernstein:evaluateBernstein from diagrams-lib-1.3.0.3"
  :precision binary64

  :herbie-target
  (if (< x -2.71483106713436e-162) (- (* (+ 1 y) (/ x z)) x) (if (< x 3.874108816439546e-197) (* (* x (+ (- y z) 1)) (/ 1 z)) (- (* (+ 1 y) (/ x z)) x)))

  (/ (* x (+ (- y z) 1)) z))