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

Average Error: 10.2 → 0.2

Time: 23.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -2.020892345540895412577476308946463194127 \cdot 10^{-80} \lor \neg \left(x \le 4.594502781479872539407817327237920661379 \cdot 10^{-108}\right):\\ \;\;\;\;\frac{x}{z} \cdot 1 + \left(\frac{x}{z} \cdot y - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r426617 = x;
        double r426618 = y;
        double r426619 = z;
        double r426620 = r426618 - r426619;
        double r426621 = 1.0;
        double r426622 = r426620 + r426621;
        double r426623 = r426617 * r426622;
        double r426624 = r426623 / r426619;
        return r426624;
}

↓

double f(double x, double y, double z) {
        double r426625 = x;
        double r426626 = -2.0208923455408954e-80;
        bool r426627 = r426625 <= r426626;
        double r426628 = 4.5945027814798725e-108;
        bool r426629 = r426625 <= r426628;
        double r426630 = !r426629;
        bool r426631 = r426627 || r426630;
        double r426632 = z;
        double r426633 = r426625 / r426632;
        double r426634 = 1.0;
        double r426635 = r426633 * r426634;
        double r426636 = y;
        double r426637 = r426633 * r426636;
        double r426638 = r426637 - r426625;
        double r426639 = r426635 + r426638;
        double r426640 = r426636 - r426632;
        double r426641 = r426640 + r426634;
        double r426642 = r426625 * r426641;
        double r426643 = r426642 / r426632;
        double r426644 = r426631 ? r426639 : r426643;
        return r426644;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	10.2
Target	0.5
Herbie	0.2

\[\begin{array}{l} \mathbf{if}\;x \lt -2.714831067134359919650240696134672137284 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x \lt 3.874108816439546156869494499878029491333 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if x < -2.0208923455408954e-80 or 4.5945027814798725e-108 < x

   1. Initial program 17.7

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

   2. Taylor expanded around 0 5.9

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

   3. Simplified 0.2

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

   5. Applied distribute-lft-in 0.2

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

   6. Applied associate--l+ 0.2

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

   if -2.0208923455408954e-80 < x < 4.5945027814798725e-108

   1. Initial program 0.1

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.020892345540895412577476308946463194127 \cdot 10^{-80} \lor \neg \left(x \le 4.594502781479872539407817327237920661379 \cdot 10^{-108}\right):\\ \;\;\;\;\frac{x}{z} \cdot 1 + \left(\frac{x}{z} \cdot y - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(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))