Average Error: 25.9 → 25.7
Time: 13.4s
Precision: 64
\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{b \cdot d + a \cdot c}{c \cdot c + d \cdot d} \le 5.836032336515321 \cdot 10^{+282}:\\ \;\;\;\;\frac{\frac{b \cdot d + a \cdot c}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\sqrt{c \cdot c + d \cdot d}}\\ \end{array}\]
\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}
↓
\begin{array}{l}
\mathbf{if}\;\frac{b \cdot d + a \cdot c}{c \cdot c + d \cdot d} \le 5.836032336515321 \cdot 10^{+282}:\\
\;\;\;\;\frac{\frac{b \cdot d + a \cdot c}{\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 r3626208 = a;
        double r3626209 = c;
        double r3626210 = r3626208 * r3626209;
        double r3626211 = b;
        double r3626212 = d;
        double r3626213 = r3626211 * r3626212;
        double r3626214 = r3626210 + r3626213;
        double r3626215 = r3626209 * r3626209;
        double r3626216 = r3626212 * r3626212;
        double r3626217 = r3626215 + r3626216;
        double r3626218 = r3626214 / r3626217;
        return r3626218;
}
↓
double f(double a, double b, double c, double d) {
        double r3626219 = b;
        double r3626220 = d;
        double r3626221 = r3626219 * r3626220;
        double r3626222 = a;
        double r3626223 = c;
        double r3626224 = r3626222 * r3626223;
        double r3626225 = r3626221 + r3626224;
        double r3626226 = r3626223 * r3626223;
        double r3626227 = r3626220 * r3626220;
        double r3626228 = r3626226 + r3626227;
        double r3626229 = r3626225 / r3626228;
        double r3626230 = 5.836032336515321e+282;
        bool r3626231 = r3626229 <= r3626230;
        double r3626232 = sqrt(r3626228);
        double r3626233 = r3626225 / r3626232;
        double r3626234 = r3626233 / r3626232;
        double r3626235 = r3626219 / r3626232;
        double r3626236 = r3626231 ? r3626234 : r3626235;
        return r3626236;
}

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

Original25.9
Target0.5
Herbie25.7
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))) < 5.836032336515321e+282

    1. Initial program 14.1

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
    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

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

    if 5.836032336515321e+282 < (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))

    1. Initial program 61.0

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

    3. Applied add-sqr-sqrt61.0

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

    4. Applied associate-/r*61.0

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

    5. Taylor expanded around 0 60.2

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

  4. Final simplification25.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{b \cdot d + a \cdot c}{c \cdot c + d \cdot d} \le 5.836032336515321 \cdot 10^{+282}:\\ \;\;\;\;\frac{\frac{b \cdot d + a \cdot c}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\sqrt{c \cdot c + d \cdot d}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019149 
(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))))