Average Error: 26.3 → 25.8
Time: 4.5s
Precision: 64
\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \le 3.318211213550137477706442015578382060034 \cdot 10^{283}:\\ \;\;\;\;\frac{\frac{1}{\frac{\sqrt{c \cdot c + d \cdot d}}{a \cdot c + b \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot a}{\sqrt{c \cdot c + d \cdot d}}\\ \end{array}\]
\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 3.318211213550137477706442015578382060034 \cdot 10^{283}:\\
\;\;\;\;\frac{\frac{1}{\frac{\sqrt{c \cdot c + d \cdot d}}{a \cdot c + b \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\

\mathbf{else}:\\
\;\;\;\;\frac{-1 \cdot a}{\sqrt{c \cdot c + d \cdot d}}\\

\end{array}
double f(double a, double b, double c, double d) {
        double r95754 = a;
        double r95755 = c;
        double r95756 = r95754 * r95755;
        double r95757 = b;
        double r95758 = d;
        double r95759 = r95757 * r95758;
        double r95760 = r95756 + r95759;
        double r95761 = r95755 * r95755;
        double r95762 = r95758 * r95758;
        double r95763 = r95761 + r95762;
        double r95764 = r95760 / r95763;
        return r95764;
}
↓
double f(double a, double b, double c, double d) {
        double r95765 = a;
        double r95766 = c;
        double r95767 = r95765 * r95766;
        double r95768 = b;
        double r95769 = d;
        double r95770 = r95768 * r95769;
        double r95771 = r95767 + r95770;
        double r95772 = r95766 * r95766;
        double r95773 = r95769 * r95769;
        double r95774 = r95772 + r95773;
        double r95775 = r95771 / r95774;
        double r95776 = 3.3182112135501375e+283;
        bool r95777 = r95775 <= r95776;
        double r95778 = 1.0;
        double r95779 = sqrt(r95774);
        double r95780 = r95779 / r95771;
        double r95781 = r95778 / r95780;
        double r95782 = r95781 / r95779;
        double r95783 = -1.0;
        double r95784 = r95783 * r95765;
        double r95785 = r95784 / r95779;
        double r95786 = r95777 ? r95782 : r95785;
        return r95786;
}

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

Original26.3
Target0.5
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))) < 3.3182112135501375e+283

    1. Initial program 14.3

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

    3. Applied add-sqr-sqrt14.3

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

    4. Applied associate-/r*14.3

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

    6. Applied clear-num14.3

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

    if 3.3182112135501375e+283 < (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))

    1. Initial program 62.6

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

    3. Applied add-sqr-sqrt62.6

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

    4. Applied associate-/r*62.6

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

    5. Taylor expanded around -inf 60.3

      \[\leadsto \frac{\color{blue}{-1 \cdot a}}{\sqrt{c \cdot c + d \cdot d}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification25.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \le 3.318211213550137477706442015578382060034 \cdot 10^{283}:\\ \;\;\;\;\frac{\frac{1}{\frac{\sqrt{c \cdot c + d \cdot d}}{a \cdot c + b \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot a}{\sqrt{c \cdot c + d \cdot d}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 
(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))))