Average Error: 25.8 → 26.3
Time: 13.8s
Precision: 64
\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\begin{array}{l} \mathbf{if}\;d \le 28148546215012511376658377374508777472:\\ \;\;\;\;\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b \cdot d + c \cdot a}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\sqrt{c \cdot c + d \cdot d}}\\ \end{array}\]
\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}
↓
\begin{array}{l}
\mathbf{if}\;d \le 28148546215012511376658377374508777472:\\
\;\;\;\;\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b \cdot d + c \cdot a}{\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 r7499820 = a;
        double r7499821 = c;
        double r7499822 = r7499820 * r7499821;
        double r7499823 = b;
        double r7499824 = d;
        double r7499825 = r7499823 * r7499824;
        double r7499826 = r7499822 + r7499825;
        double r7499827 = r7499821 * r7499821;
        double r7499828 = r7499824 * r7499824;
        double r7499829 = r7499827 + r7499828;
        double r7499830 = r7499826 / r7499829;
        return r7499830;
}
↓
double f(double a, double b, double c, double d) {
        double r7499831 = d;
        double r7499832 = 2.814854621501251e+37;
        bool r7499833 = r7499831 <= r7499832;
        double r7499834 = 1.0;
        double r7499835 = c;
        double r7499836 = r7499835 * r7499835;
        double r7499837 = r7499831 * r7499831;
        double r7499838 = r7499836 + r7499837;
        double r7499839 = sqrt(r7499838);
        double r7499840 = r7499834 / r7499839;
        double r7499841 = b;
        double r7499842 = r7499841 * r7499831;
        double r7499843 = a;
        double r7499844 = r7499835 * r7499843;
        double r7499845 = r7499842 + r7499844;
        double r7499846 = r7499845 / r7499839;
        double r7499847 = r7499840 * r7499846;
        double r7499848 = r7499841 / r7499839;
        double r7499849 = r7499833 ? r7499847 : r7499848;
        return r7499849;
}

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
Herbie26.3
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if d < 2.814854621501251e+37

    1. Initial program 22.8

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

    3. Applied add-sqr-sqrt22.8

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

    4. Applied *-un-lft-identity22.8

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

    5. Applied times-frac22.8

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

    if 2.814854621501251e+37 < d

    1. Initial program 35.3

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

    3. Applied add-sqr-sqrt35.3

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

    4. Applied associate-/r*35.3

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

    5. Taylor expanded around 0 37.0

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

  4. Final simplification26.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;d \le 28148546215012511376658377374508777472:\\ \;\;\;\;\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b \cdot d + c \cdot a}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\sqrt{c \cdot c + d \cdot d}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(FPCore (a b c d)
  :name "Complex division, real part"

  :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))))