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

Average Error: 10.5 → 0.7

Time: 3.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \le -4.54809909365855772791856768185597692522 \cdot 10^{-9} \lor \neg \left(\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \le 1.231384797345861064411732315459653816138 \cdot 10^{161}\right):\\ \;\;\;\;\frac{x}{z} \cdot \left(\left(y - z\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r863632 = x;
        double r863633 = y;
        double r863634 = z;
        double r863635 = r863633 - r863634;
        double r863636 = 1.0;
        double r863637 = r863635 + r863636;
        double r863638 = r863632 * r863637;
        double r863639 = r863638 / r863634;
        return r863639;
}

↓

double f(double x, double y, double z) {
        double r863640 = x;
        double r863641 = y;
        double r863642 = z;
        double r863643 = r863641 - r863642;
        double r863644 = 1.0;
        double r863645 = r863643 + r863644;
        double r863646 = r863640 * r863645;
        double r863647 = r863646 / r863642;
        double r863648 = -4.548099093658558e-09;
        bool r863649 = r863647 <= r863648;
        double r863650 = 1.231384797345861e+161;
        bool r863651 = r863647 <= r863650;
        double r863652 = !r863651;
        bool r863653 = r863649 || r863652;
        double r863654 = r863640 / r863642;
        double r863655 = r863654 * r863645;
        double r863656 = r863642 / r863645;
        double r863657 = r863640 / r863656;
        double r863658 = r863653 ? r863655 : r863657;
        return r863658;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	10.5
Target	0.4
Herbie	0.7

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

Derivation

1. Split input into 2 regimes

2. if (/ (* x (+ (- y z) 1.0)) z) < -4.548099093658558e-09 or 1.231384797345861e+161 < (/ (* x (+ (- y z) 1.0)) z)

   1. Initial program 21.5

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

   3. Applied associate-/l* 5.6

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

   5. Applied associate-/r/ 0.2

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

   if -4.548099093658558e-09 < (/ (* x (+ (- y z) 1.0)) z) < 1.231384797345861e+161

   1. Initial program 0.1

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

   3. Applied associate-/l* 1.2

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \le -4.54809909365855772791856768185597692522 \cdot 10^{-9} \lor \neg \left(\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \le 1.231384797345861064411732315459653816138 \cdot 10^{161}\right):\\ \;\;\;\;\frac{x}{z} \cdot \left(\left(y - z\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 
(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))