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

Average Error: 10.2 → 0.6

Time: 2.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \le -1.9809258355162253 \cdot 10^{-133} \lor \neg \left(\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \le 7.844822943688851 \cdot 10^{-106}\right):\\ \;\;\;\;\frac{x}{z} \cdot \left(y + 1\right) - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y + 1}{z} - x\\ \end{array}\]

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)) <= -1.9809258355162253e-133) || !(((double) (((double) (x * ((double) (((double) (y - z)) + 1.0)))) / z)) <= 7.844822943688851e-106))) {
		VAR = ((double) (((double) (((double) (x / z)) * ((double) (y + 1.0)))) - x));
	} else {
		VAR = ((double) (((double) (x * ((double) (((double) (y + 1.0)) / z)))) - x));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	10.2
Target	0.5
Herbie	0.6

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

Derivation

1. Split input into 2 regimes

2. if (/ (* x (+ (- y z) 1.0)) z) < -1.9809258355162253e-133 or 7.844822943688851e-106 < (/ (* x (+ (- y z) 1.0)) z)

   1. Initial program 12.7

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

   2. Taylor expanded around 0 4.1

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

   3. Simplified 0.7

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

   if -1.9809258355162253e-133 < (/ (* x (+ (- y z) 1.0)) z) < 7.844822943688851e-106

   1. Initial program 0.1

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

   2. Simplified 0.1

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \le -1.9809258355162253 \cdot 10^{-133} \lor \neg \left(\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \le 7.844822943688851 \cdot 10^{-106}\right):\\ \;\;\;\;\frac{x}{z} \cdot \left(y + 1\right) - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y + 1}{z} - x\\ \end{array}\]

Reproduce

herbie shell --seed 2020185 
(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))