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

Average Error: 31.7 → 12.4

Time: 11.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(y \cdot 4\right) \cdot y \le 1.883987095627688634773256576903935146579 \cdot 10^{-179}:\\ \;\;\;\;1\\ \mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 25951328951665387827625984:\\ \;\;\;\;\left(\sqrt[3]{\frac{x \cdot x}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)} - \frac{\left(y \cdot 4\right) \cdot y}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}} \cdot \sqrt[3]{\frac{x \cdot x}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)} - \frac{\left(y \cdot 4\right) \cdot y}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}}\right) \cdot \sqrt[3]{\frac{x \cdot x}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)} - \frac{\left(y \cdot 4\right) \cdot y}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}}\\ \mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 56309717854388472133950898176:\\ \;\;\;\;1\\ \mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 7.251271519567312889559392701775878259286 \cdot 10^{285}:\\ \;\;\;\;\frac{x \cdot x}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)} - \sqrt[3]{{\left(\frac{\left(y \cdot 4\right) \cdot y}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array}\]

C

double f(double x, double y) {
        double r443274 = x;
        double r443275 = r443274 * r443274;
        double r443276 = y;
        double r443277 = 4.0;
        double r443278 = r443276 * r443277;
        double r443279 = r443278 * r443276;
        double r443280 = r443275 - r443279;
        double r443281 = r443275 + r443279;
        double r443282 = r443280 / r443281;
        return r443282;
}

↓

double f(double x, double y) {
        double r443283 = y;
        double r443284 = 4.0;
        double r443285 = r443283 * r443284;
        double r443286 = r443285 * r443283;
        double r443287 = 1.8839870956276886e-179;
        bool r443288 = r443286 <= r443287;
        double r443289 = 1.0;
        double r443290 = 2.5951328951665388e+25;
        bool r443291 = r443286 <= r443290;
        double r443292 = x;
        double r443293 = r443292 * r443292;
        double r443294 = fma(r443292, r443292, r443286);
        double r443295 = r443293 / r443294;
        double r443296 = r443286 / r443294;
        double r443297 = r443295 - r443296;
        double r443298 = cbrt(r443297);
        double r443299 = r443298 * r443298;
        double r443300 = r443299 * r443298;
        double r443301 = 5.630971785438847e+28;
        bool r443302 = r443286 <= r443301;
        double r443303 = 7.251271519567313e+285;
        bool r443304 = r443286 <= r443303;
        double r443305 = 3.0;
        double r443306 = pow(r443296, r443305);
        double r443307 = cbrt(r443306);
        double r443308 = r443295 - r443307;
        double r443309 = -1.0;
        double r443310 = r443304 ? r443308 : r443309;
        double r443311 = r443302 ? r443289 : r443310;
        double r443312 = r443291 ? r443300 : r443311;
        double r443313 = r443288 ? r443289 : r443312;
        return r443313;
}

Error

Bits error versus x

Bits error versus y

Target

Original	31.7
Target	31.4
Herbie	12.4

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

Derivation

1. Split input into 4 regimes

2. if (* (* y 4.0) y) < 1.8839870956276886e-179 or 2.5951328951665388e+25 < (* (* y 4.0) y) < 5.630971785438847e+28

   1. Initial program 26.3

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

   2. Simplified 26.3

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

   3. Taylor expanded around inf 11.7

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

   if 1.8839870956276886e-179 < (* (* y 4.0) y) < 2.5951328951665388e+25

   1. Initial program 16.4

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

   2. Simplified 16.4

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

   4. Applied div-sub 16.4

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

   6. Applied add-cube-cbrt 16.4

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

   if 5.630971785438847e+28 < (* (* y 4.0) y) < 7.251271519567313e+285

   1. Initial program 15.2

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

   2. Simplified 15.2

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

   4. Applied div-sub 15.2

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

   6. Applied add-cbrt-cube 44.2

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

   7. Applied add-cbrt-cube 45.8

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

   8. Applied add-cbrt-cube 45.8

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

   9. Applied add-cbrt-cube 45.8

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

   10. Applied cbrt-unprod 45.9

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

   11. Applied cbrt-unprod 49.3

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

   12. Applied cbrt-undiv 49.3

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

   13. Simplified 15.2

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

   if 7.251271519567313e+285 < (* (* y 4.0) y)

   1. Initial program 61.2

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

   2. Simplified 61.2

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

   3. Taylor expanded around 0 8.9

      \[\leadsto \color{blue}{-1}\]
3. Recombined 4 regimes into one program.

4. Final simplification 12.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot 4\right) \cdot y \le 1.883987095627688634773256576903935146579 \cdot 10^{-179}:\\ \;\;\;\;1\\ \mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 25951328951665387827625984:\\ \;\;\;\;\left(\sqrt[3]{\frac{x \cdot x}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)} - \frac{\left(y \cdot 4\right) \cdot y}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}} \cdot \sqrt[3]{\frac{x \cdot x}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)} - \frac{\left(y \cdot 4\right) \cdot y}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}}\right) \cdot \sqrt[3]{\frac{x \cdot x}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)} - \frac{\left(y \cdot 4\right) \cdot y}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}}\\ \mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 56309717854388472133950898176:\\ \;\;\;\;1\\ \mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 7.251271519567312889559392701775878259286 \cdot 10^{285}:\\ \;\;\;\;\frac{x \cdot x}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)} - \sqrt[3]{{\left(\frac{\left(y \cdot 4\right) \cdot y}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +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))) 0.9743233849626781) (- (/ (* 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))))