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

Average Error: 10.4 → 1.9

Time: 3.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le 1.512135366082805601804874055418816683336 \cdot 10^{-70}:\\ \;\;\;\;\left(\frac{1}{\frac{z}{x \cdot y}} + 1 \cdot \frac{x}{z}\right) - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(1 + y\right) - x\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r601507 = x;
        double r601508 = y;
        double r601509 = z;
        double r601510 = r601508 - r601509;
        double r601511 = 1.0;
        double r601512 = r601510 + r601511;
        double r601513 = r601507 * r601512;
        double r601514 = r601513 / r601509;
        return r601514;
}

↓

double f(double x, double y, double z) {
        double r601515 = x;
        double r601516 = 1.5121353660828056e-70;
        bool r601517 = r601515 <= r601516;
        double r601518 = 1.0;
        double r601519 = z;
        double r601520 = y;
        double r601521 = r601515 * r601520;
        double r601522 = r601519 / r601521;
        double r601523 = r601518 / r601522;
        double r601524 = 1.0;
        double r601525 = r601515 / r601519;
        double r601526 = r601524 * r601525;
        double r601527 = r601523 + r601526;
        double r601528 = r601527 - r601515;
        double r601529 = r601524 + r601520;
        double r601530 = r601525 * r601529;
        double r601531 = r601530 - r601515;
        double r601532 = r601517 ? r601528 : r601531;
        return r601532;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	10.4
Target	0.4
Herbie	1.9

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

Derivation

1. Split input into 2 regimes

2. if x < 1.5121353660828056e-70

   1. Initial program 7.1

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

   3. Applied associate-/l* 4.4

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

   4. Taylor expanded around 0 2.4

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

   6. Applied clear-num 2.5

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

   if 1.5121353660828056e-70 < x

   1. Initial program 19.5

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

   3. Applied associate-/l* 0.4

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

   4. Taylor expanded around 0 6.2

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

   5. Taylor expanded around 0 6.2

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

   6. Simplified 0.1

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

4. Final simplification 1.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 1.512135366082805601804874055418816683336 \cdot 10^{-70}:\\ \;\;\;\;\left(\frac{1}{\frac{z}{x \cdot y}} + 1 \cdot \frac{x}{z}\right) - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(1 + y\right) - x\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 
(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))