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

Average Error: 10.4 → 0.2

Time: 3.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -4537181300022055465139962904576:\\ \;\;\;\;x \cdot \frac{\left(y - z\right) + 1}{z}\\ \mathbf{elif}\;z \le 10852378853767397241265820316179722403840:\\ \;\;\;\;\frac{x \cdot \left(y - z\right) + x \cdot 1}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r697781 = x;
        double r697782 = y;
        double r697783 = z;
        double r697784 = r697782 - r697783;
        double r697785 = 1.0;
        double r697786 = r697784 + r697785;
        double r697787 = r697781 * r697786;
        double r697788 = r697787 / r697783;
        return r697788;
}

↓

double f(double x, double y, double z) {
        double r697789 = z;
        double r697790 = -4.5371813000220555e+30;
        bool r697791 = r697789 <= r697790;
        double r697792 = x;
        double r697793 = y;
        double r697794 = r697793 - r697789;
        double r697795 = 1.0;
        double r697796 = r697794 + r697795;
        double r697797 = r697796 / r697789;
        double r697798 = r697792 * r697797;
        double r697799 = 1.0852378853767397e+40;
        bool r697800 = r697789 <= r697799;
        double r697801 = r697792 * r697794;
        double r697802 = r697792 * r697795;
        double r697803 = r697801 + r697802;
        double r697804 = r697803 / r697789;
        double r697805 = r697789 / r697796;
        double r697806 = r697792 / r697805;
        double r697807 = r697800 ? r697804 : r697806;
        double r697808 = r697791 ? r697798 : r697807;
        return r697808;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	10.4
Target	0.4
Herbie	0.2

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

Derivation

1. Split input into 3 regimes

2. if z < -4.5371813000220555e+30

   1. Initial program 18.8

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

   3. Applied *-un-lft-identity 18.8

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

   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 -4.5371813000220555e+30 < z < 1.0852378853767397e+40

   1. Initial program 0.3

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

   3. Applied distribute-lft-in 0.3

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

   if 1.0852378853767397e+40 < z

   1. Initial program 19.2

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

   3. Applied associate-/l* 0.1

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -4537181300022055465139962904576:\\ \;\;\;\;x \cdot \frac{\left(y - z\right) + 1}{z}\\ \mathbf{elif}\;z \le 10852378853767397241265820316179722403840:\\ \;\;\;\;\frac{x \cdot \left(y - z\right) + x \cdot 1}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 
(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))