Average Error: 32.3 → 13.6
Time: 5.8s
Precision: 64
\[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -7.71073899284975746 \cdot 10^{142}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -1.38415960656676321 \cdot 10^{-97}:\\ \;\;\;\;\log \left(e^{\frac{\mathsf{fma}\left(x, x, -\left(y \cdot 4\right) \cdot y\right)}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}}\right)\\ \mathbf{elif}\;y \le 1.1196094733529732 \cdot 10^{-55}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \le 28161126.972837694:\\ \;\;\;\;\log \left(e^{\frac{\mathsf{fma}\left(x, x, -\left(y \cdot 4\right) \cdot y\right)}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}}\right)\\ \mathbf{elif}\;y \le 4.01162503157803517 \cdot 10^{38}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \le 2.16201455188241352 \cdot 10^{107}:\\ \;\;\;\;\log \left(e^{\frac{\mathsf{fma}\left(x, x, -\left(y \cdot 4\right) \cdot y\right)}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}}\right)\\ \mathbf{elif}\;y \le 4.36744780112662469 \cdot 10^{128}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array}\]
\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}:\\
\;\;\;\;\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 1.1196094733529732 \cdot 10^{-55}:\\
\;\;\;\;1\\

\mathbf{elif}\;y \le 28161126.972837694:\\
\;\;\;\;\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 4.01162503157803517 \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 r616902 = x;
        double r616903 = r616902 * r616902;
        double r616904 = y;
        double r616905 = 4.0;
        double r616906 = r616904 * r616905;
        double r616907 = r616906 * r616904;
        double r616908 = r616903 - r616907;
        double r616909 = r616903 + r616907;
        double r616910 = r616908 / r616909;
        return r616910;
}
↓
double f(double x, double y) {
        double r616911 = y;
        double r616912 = -7.710738992849757e+142;
        bool r616913 = r616911 <= r616912;
        double r616914 = -1.0;
        double r616915 = -1.3841596065667632e-97;
        bool r616916 = r616911 <= r616915;
        double r616917 = x;
        double r616918 = 4.0;
        double r616919 = r616911 * r616918;
        double r616920 = r616919 * r616911;
        double r616921 = -r616920;
        double r616922 = fma(r616917, r616917, r616921);
        double r616923 = fma(r616917, r616917, r616920);
        double r616924 = r616922 / r616923;
        double r616925 = exp(r616924);
        double r616926 = log(r616925);
        double r616927 = 1.1196094733529732e-55;
        bool r616928 = r616911 <= r616927;
        double r616929 = 1.0;
        double r616930 = 28161126.972837694;
        bool r616931 = r616911 <= r616930;
        double r616932 = 4.011625031578035e+38;
        bool r616933 = r616911 <= r616932;
        double r616934 = 2.1620145518824135e+107;
        bool r616935 = r616911 <= r616934;
        double r616936 = 4.367447801126625e+128;
        bool r616937 = r616911 <= r616936;
        double r616938 = r616937 ? r616929 : r616914;
        double r616939 = r616935 ? r616926 : r616938;
        double r616940 = r616933 ? r616929 : r616939;
        double r616941 = r616931 ? r616926 : r616940;
        double r616942 = r616928 ? r616929 : r616941;
        double r616943 = r616916 ? r616926 : r616942;
        double r616944 = r616913 ? r616914 : r616943;
        return r616944;
}

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 3 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 or 4.011625031578035e+38 < y < 2.1620145518824135e+107

    1. Initial program 16.7

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

    2. Simplified16.7

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

    4. Applied fma-neg16.7

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

    6. Applied add-log-exp16.7

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

    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

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

  4. Final simplification13.6

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

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))))