Average Error: 26.3 → 26.3
Time: 8.4s
Precision: 64
\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\begin{array}{l} \mathbf{if}\;c \le 1.44736579362970321 \cdot 10^{65}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\sqrt{c \cdot c + d \cdot d}}\\ \end{array}\]
\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}
↓
\begin{array}{l}
\mathbf{if}\;c \le 1.44736579362970321 \cdot 10^{65}:\\
\;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\

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

\end{array}
double f(double a, double b, double c, double d) {
        double r97845 = a;
        double r97846 = c;
        double r97847 = r97845 * r97846;
        double r97848 = b;
        double r97849 = d;
        double r97850 = r97848 * r97849;
        double r97851 = r97847 + r97850;
        double r97852 = r97846 * r97846;
        double r97853 = r97849 * r97849;
        double r97854 = r97852 + r97853;
        double r97855 = r97851 / r97854;
        return r97855;
}
↓
double f(double a, double b, double c, double d) {
        double r97856 = c;
        double r97857 = 1.4473657936297032e+65;
        bool r97858 = r97856 <= r97857;
        double r97859 = a;
        double r97860 = r97859 * r97856;
        double r97861 = b;
        double r97862 = d;
        double r97863 = r97861 * r97862;
        double r97864 = r97860 + r97863;
        double r97865 = r97856 * r97856;
        double r97866 = r97862 * r97862;
        double r97867 = r97865 + r97866;
        double r97868 = r97864 / r97867;
        double r97869 = sqrt(r97867);
        double r97870 = r97859 / r97869;
        double r97871 = r97858 ? r97868 : r97870;
        return r97871;
}

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
Target0.4
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 c < 1.4473657936297032e+65

    1. Initial program 23.4

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

    if 1.4473657936297032e+65 < c

    1. Initial program 37.2

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

    3. Applied add-sqr-sqrt37.2

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

    4. Applied associate-/r*37.2

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

    5. Taylor expanded around inf 37.2

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

  4. Final simplification26.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;c \le 1.44736579362970321 \cdot 10^{65}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\sqrt{c \cdot c + d \cdot d}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(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))))