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

Average Error: 9.8 → 0.1

Time: 8.9s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -2371258724462086320000:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \mathbf{elif}\;z \le 7.031010429090227 \cdot 10^{-38}:\\ \;\;\;\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\left(y - z\right) + 1}{z}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r602616 = x;
        double r602617 = y;
        double r602618 = z;
        double r602619 = r602617 - r602618;
        double r602620 = 1.0;
        double r602621 = r602619 + r602620;
        double r602622 = r602616 * r602621;
        double r602623 = r602622 / r602618;
        return r602623;
}

↓

double f(double x, double y, double z) {
        double r602624 = z;
        double r602625 = -2.3712587244620863e+21;
        bool r602626 = r602624 <= r602625;
        double r602627 = x;
        double r602628 = y;
        double r602629 = r602628 - r602624;
        double r602630 = 1.0;
        double r602631 = r602629 + r602630;
        double r602632 = r602624 / r602631;
        double r602633 = r602627 / r602632;
        double r602634 = 7.031010429090227e-38;
        bool r602635 = r602624 <= r602634;
        double r602636 = r602627 * r602631;
        double r602637 = r602636 / r602624;
        double r602638 = r602631 / r602624;
        double r602639 = r602627 * r602638;
        double r602640 = r602635 ? r602637 : r602639;
        double r602641 = r602626 ? r602633 : r602640;
        return r602641;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	9.8
Target	0.4
Herbie	0.1

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

Derivation

1. Split input into 3 regimes

2. if z < -2.3712587244620863e+21

   1. Initial program 17.3

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

   3. Applied associate-/l* 0.1

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

   if -2.3712587244620863e+21 < z < 7.031010429090227e-38

   1. Initial program 0.2

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

   if 7.031010429090227e-38 < z

   1. Initial program 14.8

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

   3. Applied *-un-lft-identity 14.8

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

   4. Applied times-frac 0.2

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

   5. Simplified 0.2

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2371258724462086320000:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \mathbf{elif}\;z \le 7.031010429090227 \cdot 10^{-38}:\\ \;\;\;\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\left(y - z\right) + 1}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 +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))