Average Error: 25.8 → 26.1
Time: 13.0s
Precision: 64
\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\begin{array}{l} \mathbf{if}\;c \le -4.876668595961939 \cdot 10^{+48}:\\ \;\;\;\;\frac{-a}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b \cdot d + a \cdot c}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \end{array}\]
\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}
↓
\begin{array}{l}
\mathbf{if}\;c \le -4.876668595961939 \cdot 10^{+48}:\\
\;\;\;\;\frac{-a}{\sqrt{c \cdot c + d \cdot d}}\\

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

\end{array}
double f(double a, double b, double c, double d) {
        double r3488155 = a;
        double r3488156 = c;
        double r3488157 = r3488155 * r3488156;
        double r3488158 = b;
        double r3488159 = d;
        double r3488160 = r3488158 * r3488159;
        double r3488161 = r3488157 + r3488160;
        double r3488162 = r3488156 * r3488156;
        double r3488163 = r3488159 * r3488159;
        double r3488164 = r3488162 + r3488163;
        double r3488165 = r3488161 / r3488164;
        return r3488165;
}
↓
double f(double a, double b, double c, double d) {
        double r3488166 = c;
        double r3488167 = -4.876668595961939e+48;
        bool r3488168 = r3488166 <= r3488167;
        double r3488169 = a;
        double r3488170 = -r3488169;
        double r3488171 = r3488166 * r3488166;
        double r3488172 = d;
        double r3488173 = r3488172 * r3488172;
        double r3488174 = r3488171 + r3488173;
        double r3488175 = sqrt(r3488174);
        double r3488176 = r3488170 / r3488175;
        double r3488177 = b;
        double r3488178 = r3488177 * r3488172;
        double r3488179 = r3488169 * r3488166;
        double r3488180 = r3488178 + r3488179;
        double r3488181 = r3488180 / r3488175;
        double r3488182 = r3488181 / r3488175;
        double r3488183 = r3488168 ? r3488176 : r3488182;
        return r3488183;
}

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
Herbie26.1
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if c < -4.876668595961939e+48

    1. Initial program 35.1

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

    3. Applied add-sqr-sqrt35.1

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

    4. Applied associate-/r*35.1

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

    5. Taylor expanded around -inf 36.8

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

    6. Simplified36.8

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

    if -4.876668595961939e+48 < c

    1. Initial program 23.1

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

    3. Applied add-sqr-sqrt23.1

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

    4. Applied associate-/r*23.0

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

  4. Final simplification26.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;c \le -4.876668595961939 \cdot 10^{+48}:\\ \;\;\;\;\frac{-a}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b \cdot d + a \cdot c}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019138 
(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))))