Average Error: 25.8 → 25.8
Time: 10.9s
Precision: 64
\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \le 3.7234357932591811 \cdot 10^{232}:\\ \;\;\;\;\frac{\frac{1}{\frac{\sqrt{c \cdot c + d \cdot d}}{a \cdot c + d \cdot b}}}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{\sqrt{c \cdot c + d \cdot d}}\\ \end{array}\]
\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}
↓
\begin{array}{l}
\mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \le 3.7234357932591811 \cdot 10^{232}:\\
\;\;\;\;\frac{\frac{1}{\frac{\sqrt{c \cdot c + d \cdot d}}{a \cdot c + d \cdot b}}}{\sqrt{c \cdot c + d \cdot d}}\\

\mathbf{else}:\\
\;\;\;\;\frac{-a}{\sqrt{c \cdot c + d \cdot d}}\\

\end{array}
double f(double a, double b, double c, double d) {
        double r51751 = a;
        double r51752 = c;
        double r51753 = r51751 * r51752;
        double r51754 = b;
        double r51755 = d;
        double r51756 = r51754 * r51755;
        double r51757 = r51753 + r51756;
        double r51758 = r51752 * r51752;
        double r51759 = r51755 * r51755;
        double r51760 = r51758 + r51759;
        double r51761 = r51757 / r51760;
        return r51761;
}
↓
double f(double a, double b, double c, double d) {
        double r51762 = a;
        double r51763 = c;
        double r51764 = r51762 * r51763;
        double r51765 = b;
        double r51766 = d;
        double r51767 = r51765 * r51766;
        double r51768 = r51764 + r51767;
        double r51769 = r51763 * r51763;
        double r51770 = r51766 * r51766;
        double r51771 = r51769 + r51770;
        double r51772 = r51768 / r51771;
        double r51773 = 3.723435793259181e+232;
        bool r51774 = r51772 <= r51773;
        double r51775 = 1.0;
        double r51776 = sqrt(r51771);
        double r51777 = r51766 * r51765;
        double r51778 = r51764 + r51777;
        double r51779 = r51776 / r51778;
        double r51780 = r51775 / r51779;
        double r51781 = r51780 / r51776;
        double r51782 = -r51762;
        double r51783 = r51782 / r51776;
        double r51784 = r51774 ? r51781 : r51783;
        return r51784;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original25.8
Target0.4
Herbie25.8
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))) < 3.723435793259181e+232

    1. Initial program 13.9

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt13.9

      \[\leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]

    4. Applied associate-/r*13.8

      \[\leadsto \color{blue}{\frac{\frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}}\]
    5. Using strategy rm

    6. Applied clear-num13.8

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{\sqrt{c \cdot c + d \cdot d}}{a \cdot c + b \cdot d}}}}{\sqrt{c \cdot c + d \cdot d}}\]

    7. Simplified13.8

      \[\leadsto \frac{\frac{1}{\color{blue}{\frac{\sqrt{c \cdot c + d \cdot d}}{a \cdot c + d \cdot b}}}}{\sqrt{c \cdot c + d \cdot d}}\]

    if 3.723435793259181e+232 < (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))

    1. Initial program 60.0

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt60.0

      \[\leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]

    4. Applied associate-/r*60.0

      \[\leadsto \color{blue}{\frac{\frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}}\]

    5. Taylor expanded around -inf 59.9

      \[\leadsto \frac{\color{blue}{-1 \cdot a}}{\sqrt{c \cdot c + d \cdot d}}\]

    6. Simplified59.9

      \[\leadsto \frac{\color{blue}{-a}}{\sqrt{c \cdot c + d \cdot d}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification25.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \le 3.7234357932591811 \cdot 10^{232}:\\ \;\;\;\;\frac{\frac{1}{\frac{\sqrt{c \cdot c + d \cdot d}}{a \cdot c + d \cdot b}}}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{\sqrt{c \cdot c + d \cdot d}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(FPCore (a b c d)
  :name "Complex division, real part"
  :precision binary64

  :herbie-target
  (if (< (fabs d) (fabs c)) (/ (+ a (* b (/ d c))) (+ c (* d (/ d c)))) (/ (+ b (* a (/ c d))) (+ d (* c (/ c d)))))

  (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))))