Average Error: 26.0 → 25.2
Time: 12.2s
Precision: 64
\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \le 8.757218044757217587065948698275685768631 \cdot 10^{298}:\\ \;\;\;\;\frac{\left(a \cdot c + b \cdot d\right) \cdot \frac{1}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\sqrt{c \cdot c + d \cdot d}}\\ \end{array}\]
\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 8.757218044757217587065948698275685768631 \cdot 10^{298}:\\
\;\;\;\;\frac{\left(a \cdot c + b \cdot d\right) \cdot \frac{1}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\

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

\end{array}
double f(double a, double b, double c, double d) {
        double r70792 = a;
        double r70793 = c;
        double r70794 = r70792 * r70793;
        double r70795 = b;
        double r70796 = d;
        double r70797 = r70795 * r70796;
        double r70798 = r70794 + r70797;
        double r70799 = r70793 * r70793;
        double r70800 = r70796 * r70796;
        double r70801 = r70799 + r70800;
        double r70802 = r70798 / r70801;
        return r70802;
}
↓
double f(double a, double b, double c, double d) {
        double r70803 = a;
        double r70804 = c;
        double r70805 = r70803 * r70804;
        double r70806 = b;
        double r70807 = d;
        double r70808 = r70806 * r70807;
        double r70809 = r70805 + r70808;
        double r70810 = r70804 * r70804;
        double r70811 = r70807 * r70807;
        double r70812 = r70810 + r70811;
        double r70813 = r70809 / r70812;
        double r70814 = 8.757218044757218e+298;
        bool r70815 = r70813 <= r70814;
        double r70816 = 1.0;
        double r70817 = sqrt(r70812);
        double r70818 = r70816 / r70817;
        double r70819 = r70809 * r70818;
        double r70820 = r70819 / r70817;
        double r70821 = r70806 / r70817;
        double r70822 = r70815 ? r70820 : r70821;
        return r70822;
}

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

Original26.0
Target0.5
Herbie25.2
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))) < 8.757218044757218e+298

    1. Initial program 14.0

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

    3. Applied add-sqr-sqrt14.1

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

    4. Applied associate-/r*14.0

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

    6. Applied div-inv14.0

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

    if 8.757218044757218e+298 < (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))

    1. Initial program 63.6

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

    3. Applied add-sqr-sqrt63.6

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

    4. Applied associate-/r*63.6

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

    5. Taylor expanded around 0 60.6

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

  4. Final simplification25.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \le 8.757218044757217587065948698275685768631 \cdot 10^{298}:\\ \;\;\;\;\frac{\left(a \cdot c + b \cdot d\right) \cdot \frac{1}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\sqrt{c \cdot c + d \cdot d}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(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))))