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

Average Error: 10.7 → 0.2

Time: 2.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \le -130116796999.60809 \lor \neg \left(\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \le 4.4559355348650056 \cdot 10^{-74}\right):\\ \;\;\;\;\frac{x}{z} \cdot \left(1 + y\right) - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \end{array}\]

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 (((((double) (((double) (x * ((double) (((double) (y - z)) + 1.0)))) / z)) <= -130116796999.6081) || !(((double) (((double) (x * ((double) (((double) (y - z)) + 1.0)))) / z)) <= 4.455935534865006e-74))) {
		VAR = ((double) (((double) (((double) (x / z)) * ((double) (1.0 + y)))) - x));
	} else {
		VAR = ((double) (x / ((double) (z / ((double) (((double) (y - z)) + 1.0))))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	10.7
Target	0.4
Herbie	0.2

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

Derivation

1. Split input into 2 regimes

2. if (/ (* x (+ (- y z) 1.0)) z) < -130116796999.60809 or 4.4559355348650056e-74 < (/ (* x (+ (- y z) 1.0)) z)

   1. Initial program 16.2

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

   3. Applied associate-/l* 5.0

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

   5. Applied associate-/r/ 0.6

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

   6. Taylor expanded around 0 5.4

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

   7. Simplified 0.3

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

   if -130116796999.60809 < (/ (* x (+ (- y z) 1.0)) z) < 4.4559355348650056e-74

   1. Initial program 0.1

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

   3. Applied associate-/l* 0.1

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \le -130116796999.60809 \lor \neg \left(\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \le 4.4559355348650056 \cdot 10^{-74}\right):\\ \;\;\;\;\frac{x}{z} \cdot \left(1 + y\right) - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020147 
(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) (- (* (+ 1.0 y) (/ x z)) x) (if (< x 3.874108816439546e-197) (* (* x (+ (- y z) 1.0)) (/ 1.0 z)) (- (* (+ 1.0 y) (/ x z)) x)))

  (/ (* x (+ (- y z) 1.0)) z))