Diagrams.TwoD.Arc:arcBetween from diagrams-lib-1.3.0.3

Average Error: 32.2 → 12.6

Time: 8.0s

Precision: 64

Math

\[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.517398075841731978978804721327597516736 \cdot 10^{145}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \le -6.621300319852041212687540142873364646976 \cdot 10^{-102}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, -y \cdot \left(y \cdot 4\right)\right)}{\mathsf{fma}\left(x, x, {y}^{2} \cdot 4\right)}\\ \mathbf{elif}\;x \le 1.295898151448028641074794033905884537542 \cdot 10^{-152}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \le 6.052303689076668472383783695774322400393 \cdot 10^{95}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, -y \cdot \left(y \cdot 4\right)\right)}{\mathsf{fma}\left(x, x, {y}^{2} \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

C

double f(double x, double y) {
        double r883498 = x;
        double r883499 = r883498 * r883498;
        double r883500 = y;
        double r883501 = 4.0;
        double r883502 = r883500 * r883501;
        double r883503 = r883502 * r883500;
        double r883504 = r883499 - r883503;
        double r883505 = r883499 + r883503;
        double r883506 = r883504 / r883505;
        return r883506;
}

↓

double f(double x, double y) {
        double r883507 = x;
        double r883508 = -1.517398075841732e+145;
        bool r883509 = r883507 <= r883508;
        double r883510 = 1.0;
        double r883511 = -6.621300319852041e-102;
        bool r883512 = r883507 <= r883511;
        double r883513 = y;
        double r883514 = 4.0;
        double r883515 = r883513 * r883514;
        double r883516 = r883513 * r883515;
        double r883517 = -r883516;
        double r883518 = fma(r883507, r883507, r883517);
        double r883519 = 2.0;
        double r883520 = pow(r883513, r883519);
        double r883521 = r883520 * r883514;
        double r883522 = fma(r883507, r883507, r883521);
        double r883523 = r883518 / r883522;
        double r883524 = 1.2958981514480286e-152;
        bool r883525 = r883507 <= r883524;
        double r883526 = -1.0;
        double r883527 = 6.052303689076668e+95;
        bool r883528 = r883507 <= r883527;
        double r883529 = r883528 ? r883523 : r883510;
        double r883530 = r883525 ? r883526 : r883529;
        double r883531 = r883512 ? r883523 : r883530;
        double r883532 = r883509 ? r883510 : r883531;
        return r883532;
}

Error

Bits error versus x

Bits error versus y

Target

Original	32.2
Target	31.9
Herbie	12.6

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \lt 0.9743233849626781184483093056769575923681:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot y\right) \cdot 4} - \frac{\left(y \cdot y\right) \cdot 4}{x \cdot x + \left(y \cdot y\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x}{\sqrt{x \cdot x + \left(y \cdot y\right) \cdot 4}}\right)}^{2} - \frac{\left(y \cdot y\right) \cdot 4}{x \cdot x + \left(y \cdot y\right) \cdot 4}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if x < -1.517398075841732e+145 or 6.052303689076668e+95 < x

   1. Initial program 55.2

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\]

   2. Simplified 55.2

      \[\leadsto \color{blue}{\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}}\]
   3. Using strategy rm

   4. Applied *-un-lft-identity 55.2

      \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{1 \cdot \mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}}\]

   5. Applied *-un-lft-identity 55.2

      \[\leadsto \frac{\color{blue}{1 \cdot \left(x \cdot x - \left(y \cdot 4\right) \cdot y\right)}}{1 \cdot \mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}\]

   6. Applied times-frac 55.2

      \[\leadsto \color{blue}{\frac{1}{1} \cdot \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}}\]

   7. Simplified 55.2

      \[\leadsto \color{blue}{1} \cdot \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}\]

   8. Simplified 55.2

      \[\leadsto 1 \cdot \color{blue}{\frac{\mathsf{fma}\left(x, x, -y \cdot \left(y \cdot 4\right)\right)}{\mathsf{fma}\left(x, x, {y}^{2} \cdot 4\right)}}\]
   9. Using strategy rm

   10. Applied add-cbrt-cube 63.7

       \[\leadsto 1 \cdot \frac{\mathsf{fma}\left(x, x, -y \cdot \left(y \cdot 4\right)\right)}{\color{blue}{\sqrt[3]{\left(\mathsf{fma}\left(x, x, {y}^{2} \cdot 4\right) \cdot \mathsf{fma}\left(x, x, {y}^{2} \cdot 4\right)\right) \cdot \mathsf{fma}\left(x, x, {y}^{2} \cdot 4\right)}}}\]

   11. Applied add-cbrt-cube 64.0

       \[\leadsto 1 \cdot \frac{\color{blue}{\sqrt[3]{\left(\mathsf{fma}\left(x, x, -y \cdot \left(y \cdot 4\right)\right) \cdot \mathsf{fma}\left(x, x, -y \cdot \left(y \cdot 4\right)\right)\right) \cdot \mathsf{fma}\left(x, x, -y \cdot \left(y \cdot 4\right)\right)}}}{\sqrt[3]{\left(\mathsf{fma}\left(x, x, {y}^{2} \cdot 4\right) \cdot \mathsf{fma}\left(x, x, {y}^{2} \cdot 4\right)\right) \cdot \mathsf{fma}\left(x, x, {y}^{2} \cdot 4\right)}}\]

   12. Applied cbrt-undiv 64.0

       \[\leadsto 1 \cdot \color{blue}{\sqrt[3]{\frac{\left(\mathsf{fma}\left(x, x, -y \cdot \left(y \cdot 4\right)\right) \cdot \mathsf{fma}\left(x, x, -y \cdot \left(y \cdot 4\right)\right)\right) \cdot \mathsf{fma}\left(x, x, -y \cdot \left(y \cdot 4\right)\right)}{\left(\mathsf{fma}\left(x, x, {y}^{2} \cdot 4\right) \cdot \mathsf{fma}\left(x, x, {y}^{2} \cdot 4\right)\right) \cdot \mathsf{fma}\left(x, x, {y}^{2} \cdot 4\right)}}}\]

   13. Simplified 55.2

       \[\leadsto 1 \cdot \sqrt[3]{\color{blue}{{\left(\frac{\mathsf{fma}\left(x, x, -y \cdot \left(y \cdot 4\right)\right)}{\mathsf{fma}\left(x, x, {y}^{2} \cdot 4\right)}\right)}^{3}}}\]

   14. Taylor expanded around inf 10.1

       \[\leadsto 1 \cdot \color{blue}{1}\]

   if -1.517398075841732e+145 < x < -6.621300319852041e-102 or 1.2958981514480286e-152 < x < 6.052303689076668e+95

   1. Initial program 17.1

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\]

   2. Simplified 17.1

      \[\leadsto \color{blue}{\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}}\]
   3. Using strategy rm

   4. Applied *-un-lft-identity 17.1

      \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{1 \cdot \mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}}\]

   5. Applied *-un-lft-identity 17.1

      \[\leadsto \frac{\color{blue}{1 \cdot \left(x \cdot x - \left(y \cdot 4\right) \cdot y\right)}}{1 \cdot \mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}\]

   6. Applied times-frac 17.1

      \[\leadsto \color{blue}{\frac{1}{1} \cdot \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}}\]

   7. Simplified 17.1

      \[\leadsto \color{blue}{1} \cdot \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}\]

   8. Simplified 17.1

      \[\leadsto 1 \cdot \color{blue}{\frac{\mathsf{fma}\left(x, x, -y \cdot \left(y \cdot 4\right)\right)}{\mathsf{fma}\left(x, x, {y}^{2} \cdot 4\right)}}\]

   if -6.621300319852041e-102 < x < 1.2958981514480286e-152

   1. Initial program 28.6

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\]

   2. Simplified 28.6

      \[\leadsto \color{blue}{\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}}\]

   3. Taylor expanded around 0 9.1

      \[\leadsto \color{blue}{-1}\]
3. Recombined 3 regimes into one program.

4. Final simplification 12.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.517398075841731978978804721327597516736 \cdot 10^{145}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \le -6.621300319852041212687540142873364646976 \cdot 10^{-102}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, -y \cdot \left(y \cdot 4\right)\right)}{\mathsf{fma}\left(x, x, {y}^{2} \cdot 4\right)}\\ \mathbf{elif}\;x \le 1.295898151448028641074794033905884537542 \cdot 10^{-152}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \le 6.052303689076668472383783695774322400393 \cdot 10^{95}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, -y \cdot \left(y \cdot 4\right)\right)}{\mathsf{fma}\left(x, x, {y}^{2} \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Reproduce

herbie shell --seed 2019305 +o rules:numerics
(FPCore (x y)
  :name "Diagrams.TwoD.Arc:arcBetween from diagrams-lib-1.3.0.3"
  :precision binary64

  :herbie-target
  (if (< (/ (- (* x x) (* (* y 4) y)) (+ (* x x) (* (* y 4) y))) 0.974323384962678118) (- (/ (* x x) (+ (* x x) (* (* y y) 4))) (/ (* (* y y) 4) (+ (* x x) (* (* y y) 4)))) (- (pow (/ x (sqrt (+ (* x x) (* (* y y) 4)))) 2) (/ (* (* y y) 4) (+ (* x x) (* (* y y) 4)))))

  (/ (- (* x x) (* (* y 4) y)) (+ (* x x) (* (* y 4) y))))