Average Error: 32.3 → 13.6
Time: 4.6s
Precision: 64
\[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -7.71073899284975746 \cdot 10^{142}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -1.38415960656676321 \cdot 10^{-97}:\\ \;\;\;\;\frac{x \cdot x}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)} - \frac{\left(y \cdot 4\right) \cdot y}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}\\ \mathbf{elif}\;y \le 1.1196094733529732 \cdot 10^{-55}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \le 28161126.972837694:\\ \;\;\;\;\frac{x \cdot x}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)} - \frac{\left(y \cdot 4\right) \cdot y}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}\\ \mathbf{elif}\;y \le 4.01162503157803517 \cdot 10^{38}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \le 2.16201455188241352 \cdot 10^{107}:\\ \;\;\;\;\log \left(e^{\frac{\mathsf{fma}\left(x, x, -\left(y \cdot 4\right) \cdot y\right)}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}}\right)\\ \mathbf{elif}\;y \le 4.36744780112662469 \cdot 10^{128}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array}\]
\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 -7.71073899284975746 \cdot 10^{142}:\\
\;\;\;\;-1\\

\mathbf{elif}\;y \le -1.38415960656676321 \cdot 10^{-97}:\\
\;\;\;\;\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}\;y \le 1.1196094733529732 \cdot 10^{-55}:\\
\;\;\;\;1\\

\mathbf{elif}\;y \le 28161126.972837694:\\
\;\;\;\;\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}\;y \le 4.01162503157803517 \cdot 10^{38}:\\
\;\;\;\;1\\

\mathbf{elif}\;y \le 2.16201455188241352 \cdot 10^{107}:\\
\;\;\;\;\log \left(e^{\frac{\mathsf{fma}\left(x, x, -\left(y \cdot 4\right) \cdot y\right)}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}}\right)\\

\mathbf{elif}\;y \le 4.36744780112662469 \cdot 10^{128}:\\
\;\;\;\;1\\

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

\end{array}
double f(double x, double y) {
        double r764028 = x;
        double r764029 = r764028 * r764028;
        double r764030 = y;
        double r764031 = 4.0;
        double r764032 = r764030 * r764031;
        double r764033 = r764032 * r764030;
        double r764034 = r764029 - r764033;
        double r764035 = r764029 + r764033;
        double r764036 = r764034 / r764035;
        return r764036;
}
↓
double f(double x, double y) {
        double r764037 = y;
        double r764038 = -7.710738992849757e+142;
        bool r764039 = r764037 <= r764038;
        double r764040 = -1.0;
        double r764041 = -1.3841596065667632e-97;
        bool r764042 = r764037 <= r764041;
        double r764043 = x;
        double r764044 = r764043 * r764043;
        double r764045 = 4.0;
        double r764046 = r764037 * r764045;
        double r764047 = r764046 * r764037;
        double r764048 = fma(r764043, r764043, r764047);
        double r764049 = r764044 / r764048;
        double r764050 = r764047 / r764048;
        double r764051 = r764049 - r764050;
        double r764052 = 1.1196094733529732e-55;
        bool r764053 = r764037 <= r764052;
        double r764054 = 1.0;
        double r764055 = 28161126.972837694;
        bool r764056 = r764037 <= r764055;
        double r764057 = 4.011625031578035e+38;
        bool r764058 = r764037 <= r764057;
        double r764059 = 2.1620145518824135e+107;
        bool r764060 = r764037 <= r764059;
        double r764061 = -r764047;
        double r764062 = fma(r764043, r764043, r764061);
        double r764063 = r764062 / r764048;
        double r764064 = exp(r764063);
        double r764065 = log(r764064);
        double r764066 = 4.367447801126625e+128;
        bool r764067 = r764037 <= r764066;
        double r764068 = r764067 ? r764054 : r764040;
        double r764069 = r764060 ? r764065 : r764068;
        double r764070 = r764058 ? r764054 : r764069;
        double r764071 = r764056 ? r764051 : r764070;
        double r764072 = r764053 ? r764054 : r764071;
        double r764073 = r764042 ? r764051 : r764072;
        double r764074 = r764039 ? r764040 : r764073;
        return r764074;
}

Error

Bits error versus x

Bits error versus y

Target

Original32.3
Target32.0
Herbie13.6
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \lt 0.974323384962678118:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot y\right) \cdot 4} - \frac{\left(y \cdot y\right) \cdot 4}{x \cdot x + \left(y \cdot y\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x}{\sqrt{x \cdot x + \left(y \cdot y\right) \cdot 4}}\right)}^{2} - \frac{\left(y \cdot y\right) \cdot 4}{x \cdot x + \left(y \cdot y\right) \cdot 4}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if y < -7.710738992849757e+142 or 4.367447801126625e+128 < y

    1. Initial program 59.1

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

    2. Simplified59.1

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

    3. Taylor expanded around 0 8.6

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

    if -7.710738992849757e+142 < y < -1.3841596065667632e-97 or 1.1196094733529732e-55 < y < 28161126.972837694

    1. Initial program 17.4

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

    2. Simplified17.4

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

    4. Applied div-sub17.4

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

    if -1.3841596065667632e-97 < y < 1.1196094733529732e-55 or 28161126.972837694 < y < 4.011625031578035e+38 or 2.1620145518824135e+107 < y < 4.367447801126625e+128

    1. Initial program 25.0

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

    2. Simplified25.0

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

    3. Taylor expanded around inf 14.8

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

    if 4.011625031578035e+38 < y < 2.1620145518824135e+107

    1. Initial program 13.9

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

    2. Simplified13.9

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

    4. Applied fma-neg13.9

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

    6. Applied add-log-exp13.9

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

  4. Final simplification13.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -7.71073899284975746 \cdot 10^{142}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -1.38415960656676321 \cdot 10^{-97}:\\ \;\;\;\;\frac{x \cdot x}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)} - \frac{\left(y \cdot 4\right) \cdot y}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}\\ \mathbf{elif}\;y \le 1.1196094733529732 \cdot 10^{-55}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \le 28161126.972837694:\\ \;\;\;\;\frac{x \cdot x}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)} - \frac{\left(y \cdot 4\right) \cdot y}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}\\ \mathbf{elif}\;y \le 4.01162503157803517 \cdot 10^{38}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \le 2.16201455188241352 \cdot 10^{107}:\\ \;\;\;\;\log \left(e^{\frac{\mathsf{fma}\left(x, x, -\left(y \cdot 4\right) \cdot y\right)}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}}\right)\\ \mathbf{elif}\;y \le 4.36744780112662469 \cdot 10^{128}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +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))))