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

Average Error: 31.7 → 12.4

Time: 13.4s

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 r566963 = x;
        double r566964 = r566963 * r566963;
        double r566965 = y;
        double r566966 = 4.0;
        double r566967 = r566965 * r566966;
        double r566968 = r566967 * r566965;
        double r566969 = r566964 - r566968;
        double r566970 = r566964 + r566968;
        double r566971 = r566969 / r566970;
        return r566971;
}

↓

double f(double x, double y) {
        double r566972 = y;
        double r566973 = 4.0;
        double r566974 = r566972 * r566973;
        double r566975 = r566974 * r566972;
        double r566976 = 1.8839870956276886e-179;
        bool r566977 = r566975 <= r566976;
        double r566978 = 1.0;
        double r566979 = 2.5951328951665388e+25;
        bool r566980 = r566975 <= r566979;
        double r566981 = x;
        double r566982 = r566981 * r566981;
        double r566983 = fma(r566981, r566981, r566975);
        double r566984 = r566982 / r566983;
        double r566985 = r566975 / r566983;
        double r566986 = r566984 - r566985;
        double r566987 = cbrt(r566986);
        double r566988 = r566987 * r566987;
        double r566989 = r566988 * r566987;
        double r566990 = 5.630971785438847e+28;
        bool r566991 = r566975 <= r566990;
        double r566992 = 7.251271519567313e+285;
        bool r566993 = r566975 <= r566992;
        double r566994 = 3.0;
        double r566995 = pow(r566985, r566994);
        double r566996 = cbrt(r566995);
        double r566997 = r566984 - r566996;
        double r566998 = -1.0;
        double r566999 = r566993 ? r566997 : r566998;
        double r567000 = r566991 ? r566978 : r566999;
        double r567001 = r566980 ? r566989 : r567000;
        double r567002 = r566977 ? r566978 : r567001;
        return r567002;
}

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))))