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

Average Error: 10.2 → 0.2

Time: 22.5s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z) {
        double r454105 = x;
        double r454106 = y;
        double r454107 = z;
        double r454108 = r454106 - r454107;
        double r454109 = 1.0;
        double r454110 = r454108 + r454109;
        double r454111 = r454105 * r454110;
        double r454112 = r454111 / r454107;
        return r454112;
}

↓

double f(double x, double y, double z) {
        double r454113 = x;
        double r454114 = -2.0208923455408954e-80;
        bool r454115 = r454113 <= r454114;
        double r454116 = 4.5945027814798725e-108;
        bool r454117 = r454113 <= r454116;
        double r454118 = !r454117;
        bool r454119 = r454115 || r454118;
        double r454120 = 1.0;
        double r454121 = z;
        double r454122 = r454113 / r454121;
        double r454123 = r454120 * r454122;
        double r454124 = y;
        double r454125 = r454122 * r454124;
        double r454126 = r454125 - r454113;
        double r454127 = r454123 + r454126;
        double r454128 = r454124 - r454121;
        double r454129 = r454128 + r454120;
        double r454130 = r454113 * r454129;
        double r454131 = r454130 / r454121;
        double r454132 = r454119 ? r454127 : r454131;
        return r454132;
}

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-rgt-in 0.2

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

   6. Applied associate--l+ 0.2

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

   7. Simplified 0.2

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

   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

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

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))