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

Average Error: 10.8 → 0.1

Time: 11.0s

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 r385321 = x;
        double r385322 = y;
        double r385323 = z;
        double r385324 = r385322 - r385323;
        double r385325 = 1.0;
        double r385326 = r385324 + r385325;
        double r385327 = r385321 * r385326;
        double r385328 = r385327 / r385323;
        return r385328;
}

↓

double f(double x, double y, double z) {
        double r385329 = z;
        double r385330 = -198391.30145553235;
        bool r385331 = r385329 <= r385330;
        double r385332 = 2.6146347937634723e-28;
        bool r385333 = r385329 <= r385332;
        double r385334 = !r385333;
        bool r385335 = r385331 || r385334;
        double r385336 = x;
        double r385337 = y;
        double r385338 = r385337 - r385329;
        double r385339 = 1.0;
        double r385340 = r385338 + r385339;
        double r385341 = r385340 / r385329;
        double r385342 = r385336 * r385341;
        double r385343 = r385336 * r385340;
        double r385344 = r385343 / r385329;
        double r385345 = r385335 ? r385342 : r385344;
        return r385345;
}

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 +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 -2.7148310671343599e-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))