Average Error: 31.5 → 12.7
Time: 7.7s
Precision: 64
\[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(y \cdot 4\right) \cdot y \le 3.414623317947593663745953537661023590748 \cdot 10^{-192}:\\ \;\;\;\;1\\ \mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 3.810003643719863336668138253262668476651 \cdot 10^{67}:\\ \;\;\;\;\frac{x \cdot x}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)} - \frac{\left(y \cdot 4\right) \cdot y}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}\\ \mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 1.756338886358467686795570653182400240677 \cdot 10^{115}:\\ \;\;\;\;1\\ \mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 9.415768721852470917873482124314739876564 \cdot 10^{215}:\\ \;\;\;\;\frac{x \cdot x}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)} - \frac{\left(y \cdot 4\right) \cdot y}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array}\]
\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 3.414623317947593663745953537661023590748 \cdot 10^{-192}:\\
\;\;\;\;1\\

\mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 3.810003643719863336668138253262668476651 \cdot 10^{67}:\\
\;\;\;\;\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 1.756338886358467686795570653182400240677 \cdot 10^{115}:\\
\;\;\;\;1\\

\mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 9.415768721852470917873482124314739876564 \cdot 10^{215}:\\
\;\;\;\;\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{else}:\\
\;\;\;\;-1\\

\end{array}
double f(double x, double y) {
        double r520075 = x;
        double r520076 = r520075 * r520075;
        double r520077 = y;
        double r520078 = 4.0;
        double r520079 = r520077 * r520078;
        double r520080 = r520079 * r520077;
        double r520081 = r520076 - r520080;
        double r520082 = r520076 + r520080;
        double r520083 = r520081 / r520082;
        return r520083;
}
↓
double f(double x, double y) {
        double r520084 = y;
        double r520085 = 4.0;
        double r520086 = r520084 * r520085;
        double r520087 = r520086 * r520084;
        double r520088 = 3.4146233179475937e-192;
        bool r520089 = r520087 <= r520088;
        double r520090 = 1.0;
        double r520091 = 3.8100036437198633e+67;
        bool r520092 = r520087 <= r520091;
        double r520093 = x;
        double r520094 = r520093 * r520093;
        double r520095 = fma(r520093, r520093, r520087);
        double r520096 = r520094 / r520095;
        double r520097 = r520087 / r520095;
        double r520098 = r520096 - r520097;
        double r520099 = 1.7563388863584677e+115;
        bool r520100 = r520087 <= r520099;
        double r520101 = 9.415768721852471e+215;
        bool r520102 = r520087 <= r520101;
        double r520103 = -1.0;
        double r520104 = r520102 ? r520098 : r520103;
        double r520105 = r520100 ? r520090 : r520104;
        double r520106 = r520092 ? r520098 : r520105;
        double r520107 = r520089 ? r520090 : r520106;
        return r520107;
}

Error

Bits error versus x

Bits error versus y

Target

Original31.5
Target31.2
Herbie12.7
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \lt 0.9743233849626781184483093056769575923681:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot y\right) \cdot 4} - \frac{\left(y \cdot y\right) \cdot 4}{x \cdot x + \left(y \cdot y\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x}{\sqrt{x \cdot x + \left(y \cdot y\right) \cdot 4}}\right)}^{2} - \frac{\left(y \cdot y\right) \cdot 4}{x \cdot x + \left(y \cdot y\right) \cdot 4}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* (* y 4.0) y) < 3.4146233179475937e-192 or 3.8100036437198633e+67 < (* (* y 4.0) y) < 1.7563388863584677e+115

    1. Initial program 26.4

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

    2. Taylor expanded around inf 13.0

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

    if 3.4146233179475937e-192 < (* (* y 4.0) y) < 3.8100036437198633e+67 or 1.7563388863584677e+115 < (* (* y 4.0) y) < 9.415768721852471e+215

    1. Initial program 15.0

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\]
    2. Using strategy rm

    3. Applied div-sub15.0

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

    4. Simplified15.0

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

    5. Simplified15.0

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

    if 9.415768721852471e+215 < (* (* y 4.0) y)

    1. Initial program 52.5

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

    2. Taylor expanded around 0 10.3

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

  4. Final simplification12.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot 4\right) \cdot y \le 3.414623317947593663745953537661023590748 \cdot 10^{-192}:\\ \;\;\;\;1\\ \mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 3.810003643719863336668138253262668476651 \cdot 10^{67}:\\ \;\;\;\;\frac{x \cdot x}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)} - \frac{\left(y \cdot 4\right) \cdot y}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}\\ \mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 1.756338886358467686795570653182400240677 \cdot 10^{115}:\\ \;\;\;\;1\\ \mathbf{elif}\;\left(y \cdot 4\right) \cdot y \le 9.415768721852470917873482124314739876564 \cdot 10^{215}:\\ \;\;\;\;\frac{x \cdot x}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)} - \frac{\left(y \cdot 4\right) \cdot y}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +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))))