Average Error: 31.7 → 14.8
Time: 47.0s
Precision: 64
\[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.302473734361689583818414967767065647429 \cdot 10^{64}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le 4.427972959141087508327226963770083374106 \cdot 10^{-125}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \le 4.117086201251834897728735635968684829902 \cdot 10^{60}:\\ \;\;\;\;\log \left(e^{\frac{x \cdot x - y \cdot \left(4 \cdot y\right)}{x \cdot x + y \cdot \left(4 \cdot y\right)}}\right)\\ \mathbf{elif}\;y \le 5.419438360639095008822556977586250635758 \cdot 10^{72}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \le 1.027562094282635282760369761847330220115 \cdot 10^{103}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x \cdot x}{\mathsf{fma}\left(4 \cdot y, y, x \cdot x\right)} - \frac{4 \cdot y}{\mathsf{fma}\left(4 \cdot y, y, x \cdot x\right)} \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array}\]
\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}\;y \le -1.302473734361689583818414967767065647429 \cdot 10^{64}:\\
\;\;\;\;-1\\

\mathbf{elif}\;y \le 4.427972959141087508327226963770083374106 \cdot 10^{-125}:\\
\;\;\;\;1\\

\mathbf{elif}\;y \le 4.117086201251834897728735635968684829902 \cdot 10^{60}:\\
\;\;\;\;\log \left(e^{\frac{x \cdot x - y \cdot \left(4 \cdot y\right)}{x \cdot x + y \cdot \left(4 \cdot y\right)}}\right)\\

\mathbf{elif}\;y \le 5.419438360639095008822556977586250635758 \cdot 10^{72}:\\
\;\;\;\;1\\

\mathbf{elif}\;y \le 1.027562094282635282760369761847330220115 \cdot 10^{103}:\\
\;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x \cdot x}{\mathsf{fma}\left(4 \cdot y, y, x \cdot x\right)} - \frac{4 \cdot y}{\mathsf{fma}\left(4 \cdot y, y, x \cdot x\right)} \cdot y\right)\right)\\

\mathbf{else}:\\
\;\;\;\;-1\\

\end{array}
double f(double x, double y) {
        double r32653569 = x;
        double r32653570 = r32653569 * r32653569;
        double r32653571 = y;
        double r32653572 = 4.0;
        double r32653573 = r32653571 * r32653572;
        double r32653574 = r32653573 * r32653571;
        double r32653575 = r32653570 - r32653574;
        double r32653576 = r32653570 + r32653574;
        double r32653577 = r32653575 / r32653576;
        return r32653577;
}
↓
double f(double x, double y) {
        double r32653578 = y;
        double r32653579 = -1.3024737343616896e+64;
        bool r32653580 = r32653578 <= r32653579;
        double r32653581 = -1.0;
        double r32653582 = 4.4279729591410875e-125;
        bool r32653583 = r32653578 <= r32653582;
        double r32653584 = 1.0;
        double r32653585 = 4.117086201251835e+60;
        bool r32653586 = r32653578 <= r32653585;
        double r32653587 = x;
        double r32653588 = r32653587 * r32653587;
        double r32653589 = 4.0;
        double r32653590 = r32653589 * r32653578;
        double r32653591 = r32653578 * r32653590;
        double r32653592 = r32653588 - r32653591;
        double r32653593 = r32653588 + r32653591;
        double r32653594 = r32653592 / r32653593;
        double r32653595 = exp(r32653594);
        double r32653596 = log(r32653595);
        double r32653597 = 5.419438360639095e+72;
        bool r32653598 = r32653578 <= r32653597;
        double r32653599 = 1.0275620942826353e+103;
        bool r32653600 = r32653578 <= r32653599;
        double r32653601 = fma(r32653590, r32653578, r32653588);
        double r32653602 = r32653588 / r32653601;
        double r32653603 = r32653590 / r32653601;
        double r32653604 = r32653603 * r32653578;
        double r32653605 = r32653602 - r32653604;
        double r32653606 = expm1(r32653605);
        double r32653607 = log1p(r32653606);
        double r32653608 = r32653600 ? r32653607 : r32653581;
        double r32653609 = r32653598 ? r32653584 : r32653608;
        double r32653610 = r32653586 ? r32653596 : r32653609;
        double r32653611 = r32653583 ? r32653584 : r32653610;
        double r32653612 = r32653580 ? r32653581 : r32653611;
        return r32653612;
}

Error

Bits error versus x

Bits error versus y

Target

Original31.7
Target31.4
Herbie14.8
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \lt 0.9743233849626781184483093056769575923681:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot y\right) \cdot 4} - \frac{\left(y \cdot y\right) \cdot 4}{x \cdot x + \left(y \cdot y\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x}{\sqrt{x \cdot x + \left(y \cdot y\right) \cdot 4}}\right)}^{2} - \frac{\left(y \cdot y\right) \cdot 4}{x \cdot x + \left(y \cdot y\right) \cdot 4}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if y < -1.3024737343616896e+64 or 1.0275620942826353e+103 < y

    1. Initial program 48.8

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

    2. Taylor expanded around 0 12.0

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

    if -1.3024737343616896e+64 < y < 4.4279729591410875e-125 or 4.117086201251835e+60 < y < 5.419438360639095e+72

    1. Initial program 24.5

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

    2. Taylor expanded around inf 16.4

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

    if 4.4279729591410875e-125 < y < 4.117086201251835e+60

    1. Initial program 16.0

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

    3. Applied add-log-exp16.0

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

    if 5.419438360639095e+72 < y < 1.0275620942826353e+103

    1. Initial program 18.2

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

    3. Applied add-log-exp18.2

      \[\leadsto \color{blue}{\log \left(e^{\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}}\right)}\]
    4. Using strategy rm

    5. Applied log1p-expm1-u18.2

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(e^{\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}}\right)\right)\right)}\]

    6. Simplified18.3

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\frac{x \cdot x}{\mathsf{fma}\left(y \cdot 4, y, x \cdot x\right)} - y \cdot \frac{y \cdot 4}{\mathsf{fma}\left(y \cdot 4, y, x \cdot x\right)}\right)}\right)\]
  3. Recombined 4 regimes into one program.

  4. Final simplification14.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.302473734361689583818414967767065647429 \cdot 10^{64}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le 4.427972959141087508327226963770083374106 \cdot 10^{-125}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \le 4.117086201251834897728735635968684829902 \cdot 10^{60}:\\ \;\;\;\;\log \left(e^{\frac{x \cdot x - y \cdot \left(4 \cdot y\right)}{x \cdot x + y \cdot \left(4 \cdot y\right)}}\right)\\ \mathbf{elif}\;y \le 5.419438360639095008822556977586250635758 \cdot 10^{72}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \le 1.027562094282635282760369761847330220115 \cdot 10^{103}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x \cdot x}{\mathsf{fma}\left(4 \cdot y, y, x \cdot x\right)} - \frac{4 \cdot y}{\mathsf{fma}\left(4 \cdot y, y, x \cdot x\right)} \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (x y)
  :name "Diagrams.TwoD.Arc:arcBetween from diagrams-lib-1.3.0.3"

  :herbie-target
  (if (< (/ (- (* x x) (* (* y 4.0) y)) (+ (* x x) (* (* y 4.0) y))) 0.9743233849626781) (- (/ (* x x) (+ (* x x) (* (* y y) 4.0))) (/ (* (* y y) 4.0) (+ (* x x) (* (* y y) 4.0)))) (- (pow (/ x (sqrt (+ (* x x) (* (* y y) 4.0)))) 2.0) (/ (* (* y y) 4.0) (+ (* x x) (* (* y y) 4.0)))))

  (/ (- (* x x) (* (* y 4.0) y)) (+ (* x x) (* (* y 4.0) y))))