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

Average Error: 10.2 → 0.3

Time: 16.1s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.2174262221561359 \cdot 10^{-94} \lor \neg \left(x \le 1.12601737674282078 \cdot 10^{-171}\right):\\ \;\;\;\;\frac{x}{z} \cdot \left(1 + y\right) - x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r422809 = x;
        double r422810 = y;
        double r422811 = z;
        double r422812 = r422810 - r422811;
        double r422813 = 1.0;
        double r422814 = r422812 + r422813;
        double r422815 = r422809 * r422814;
        double r422816 = r422815 / r422811;
        return r422816;
}

↓

double f(double x, double y, double z) {
        double r422817 = x;
        double r422818 = -1.217426222156136e-94;
        bool r422819 = r422817 <= r422818;
        double r422820 = 1.1260173767428208e-171;
        bool r422821 = r422817 <= r422820;
        double r422822 = !r422821;
        bool r422823 = r422819 || r422822;
        double r422824 = z;
        double r422825 = r422817 / r422824;
        double r422826 = 1.0;
        double r422827 = y;
        double r422828 = r422826 + r422827;
        double r422829 = r422825 * r422828;
        double r422830 = r422829 - r422817;
        double r422831 = r422827 - r422824;
        double r422832 = r422831 + r422826;
        double r422833 = r422817 * r422832;
        double r422834 = 1.0;
        double r422835 = r422834 / r422824;
        double r422836 = r422833 * r422835;
        double r422837 = r422823 ? r422830 : r422836;
        return r422837;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	10.2
Target	0.4
Herbie	0.3

\[\begin{array}{l} \mathbf{if}\;x \lt -2.7148310671343599 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x \lt 3.87410881643954616 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if x < -1.217426222156136e-94 or 1.1260173767428208e-171 < x

   1. Initial program 15.5

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

   2. Taylor expanded around 0 5.2

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

   3. Simplified 0.3

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

   if -1.217426222156136e-94 < x < 1.1260173767428208e-171

   1. Initial program 0.2

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

   3. Applied div-inv 0.3

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.2174262221561359 \cdot 10^{-94} \lor \neg \left(x \le 1.12601737674282078 \cdot 10^{-171}\right):\\ \;\;\;\;\frac{x}{z} \cdot \left(1 + y\right) - x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020043 +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))