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

Average Error: 10.4 → 0.2

Time: 5.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \leq -1.448487285633533 \cdot 10^{+67} \lor \neg \left(\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \leq 2.241693807638564 \cdot 10^{-39}\right):\\ \;\;\;\;\left(\frac{x}{z} + y \cdot \frac{x}{z}\right) - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y + 1\right) - z}}\\ \end{array}\]

FPCore

(FPCore (x y z) :precision binary64 (/ (* x (+ (- y z) 1.0)) z))

↓

(FPCore (x y z)
 :precision binary64
 (if (or (<= (/ (* x (+ (- y z) 1.0)) z) -1.448487285633533e+67)
         (not (<= (/ (* x (+ (- y z) 1.0)) z) 2.241693807638564e-39)))
   (- (+ (/ x z) (* y (/ x z))) x)
   (/ x (/ z (- (+ y 1.0) z)))))

C

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

↓

double code(double x, double y, double z) {
	double tmp;
	if ((((x * ((y - z) + 1.0)) / z) <= -1.448487285633533e+67) || !(((x * ((y - z) + 1.0)) / z) <= 2.241693807638564e-39)) {
		tmp = ((x / z) + (y * (x / z))) - x;
	} else {
		tmp = x / (z / ((y + 1.0) - z));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	10.4
Target	0.4
Herbie	0.2

\[\begin{array}{l} \mathbf{if}\;x < -2.71483106713436 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x < 3.874108816439546 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 (-.f64 y z) 1)) z) < -1.44848728563353314e67 or 2.241693807638564e-39 < (/.f64 (*.f64 x (+.f64 (-.f64 y z) 1)) z)

   1. Initial program 17.9

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

   2. Taylor expanded around 0 6.4

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

   3. Simplified 0.1

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

   if -1.44848728563353314e67 < (/.f64 (*.f64 x (+.f64 (-.f64 y z) 1)) z) < 2.241693807638564e-39

   1. Initial program 0.1

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

   3. Applied associate-/l*_binary64_16732 0.2

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

   4. Simplified 0.2

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \leq -1.448487285633533 \cdot 10^{+67} \lor \neg \left(\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \leq 2.241693807638564 \cdot 10^{-39}\right):\\ \;\;\;\;\left(\frac{x}{z} + y \cdot \frac{x}{z}\right) - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y + 1\right) - z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021015 
(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))