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

Average Error: 10.2 → 0.2

Time: 22.5s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z) {
        double r436160 = x;
        double r436161 = y;
        double r436162 = z;
        double r436163 = r436161 - r436162;
        double r436164 = 1.0;
        double r436165 = r436163 + r436164;
        double r436166 = r436160 * r436165;
        double r436167 = r436166 / r436162;
        return r436167;
}

↓

double f(double x, double y, double z) {
        double r436168 = x;
        double r436169 = -2.0208923455408954e-80;
        bool r436170 = r436168 <= r436169;
        double r436171 = 4.5945027814798725e-108;
        bool r436172 = r436168 <= r436171;
        double r436173 = !r436172;
        bool r436174 = r436170 || r436173;
        double r436175 = z;
        double r436176 = r436168 / r436175;
        double r436177 = y;
        double r436178 = r436176 * r436177;
        double r436179 = 1.0;
        double r436180 = r436179 * r436176;
        double r436181 = r436180 - r436168;
        double r436182 = r436178 + r436181;
        double r436183 = r436177 - r436175;
        double r436184 = r436183 + r436179;
        double r436185 = r436168 * r436184;
        double r436186 = r436185 / r436175;
        double r436187 = r436174 ? r436182 : r436186;
        return r436187;
}

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

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

   5. Applied distribute-lft-in 0.2

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

   6. Applied associate--l+ 0.2

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

   7. Simplified 0.2

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

   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

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

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(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))