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

Average Error: 10.8 → 0.1

Time: 8.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -198391.3014555323461536318063735961914062 \lor \neg \left(z \le 2.614634793763472256491750599341237196353 \cdot 10^{-28}\right):\\ \;\;\;\;x \cdot \frac{\left(y - z\right) + 1}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r457778 = x;
        double r457779 = y;
        double r457780 = z;
        double r457781 = r457779 - r457780;
        double r457782 = 1.0;
        double r457783 = r457781 + r457782;
        double r457784 = r457778 * r457783;
        double r457785 = r457784 / r457780;
        return r457785;
}

↓

double f(double x, double y, double z) {
        double r457786 = z;
        double r457787 = -198391.30145553235;
        bool r457788 = r457786 <= r457787;
        double r457789 = 2.6146347937634723e-28;
        bool r457790 = r457786 <= r457789;
        double r457791 = !r457790;
        bool r457792 = r457788 || r457791;
        double r457793 = x;
        double r457794 = y;
        double r457795 = r457794 - r457786;
        double r457796 = 1.0;
        double r457797 = r457795 + r457796;
        double r457798 = r457797 / r457786;
        double r457799 = r457793 * r457798;
        double r457800 = r457793 * r457797;
        double r457801 = r457800 / r457786;
        double r457802 = r457792 ? r457799 : r457801;
        return r457802;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	10.8
Target	0.5
Herbie	0.1

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

Derivation

1. Split input into 2 regimes

2. if z < -198391.30145553235 or 2.6146347937634723e-28 < z

   1. Initial program 17.4

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

   3. Applied *-un-lft-identity 17.4

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

   4. Applied times-frac 0.1

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

   5. Simplified 0.1

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

   if -198391.30145553235 < z < 2.6146347937634723e-28

   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 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -198391.3014555323461536318063735961914062 \lor \neg \left(z \le 2.614634793763472256491750599341237196353 \cdot 10^{-28}\right):\\ \;\;\;\;x \cdot \frac{\left(y - z\right) + 1}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019208 
(FPCore (x y z)
  :name "Diagrams.TwoD.Segment.Bernstein:evaluateBernstein from diagrams-lib-1.3.0.3"
  :precision binary64

  :herbie-target
  (if (< x -2.7148310671343599e-162) (- (* (+ 1 y) (/ x z)) x) (if (< x 3.87410881643954616e-197) (* (* x (+ (- y z) 1)) (/ 1 z)) (- (* (+ 1 y) (/ x z)) x)))

  (/ (* x (+ (- y z) 1)) z))