Average Error: 25.8 → 25.8
Time: 17.7s
Precision: 64
\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \le 3.7234357932591811 \cdot 10^{232}:\\ \;\;\;\;\frac{\frac{1}{\frac{\sqrt{c \cdot c + d \cdot d}}{a \cdot c + d \cdot b}}}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{\sqrt{c \cdot c + d \cdot d}}\\ \end{array}\]
\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}
↓
\begin{array}{l}
\mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \le 3.7234357932591811 \cdot 10^{232}:\\
\;\;\;\;\frac{\frac{1}{\frac{\sqrt{c \cdot c + d \cdot d}}{a \cdot c + d \cdot b}}}{\sqrt{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 r87695 = a;
        double r87696 = c;
        double r87697 = r87695 * r87696;
        double r87698 = b;
        double r87699 = d;
        double r87700 = r87698 * r87699;
        double r87701 = r87697 + r87700;
        double r87702 = r87696 * r87696;
        double r87703 = r87699 * r87699;
        double r87704 = r87702 + r87703;
        double r87705 = r87701 / r87704;
        return r87705;
}
↓
double f(double a, double b, double c, double d) {
        double r87706 = a;
        double r87707 = c;
        double r87708 = r87706 * r87707;
        double r87709 = b;
        double r87710 = d;
        double r87711 = r87709 * r87710;
        double r87712 = r87708 + r87711;
        double r87713 = r87707 * r87707;
        double r87714 = r87710 * r87710;
        double r87715 = r87713 + r87714;
        double r87716 = r87712 / r87715;
        double r87717 = 3.723435793259181e+232;
        bool r87718 = r87716 <= r87717;
        double r87719 = 1.0;
        double r87720 = sqrt(r87715);
        double r87721 = r87710 * r87709;
        double r87722 = r87708 + r87721;
        double r87723 = r87720 / r87722;
        double r87724 = r87719 / r87723;
        double r87725 = r87724 / r87720;
        double r87726 = -r87706;
        double r87727 = r87726 / r87720;
        double r87728 = r87718 ? r87725 : r87727;
        return r87728;
}

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
Herbie25.8
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))) < 3.723435793259181e+232

    1. Initial program 13.9

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

    3. Applied add-sqr-sqrt13.9

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

    4. Applied associate-/r*13.8

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

    6. Applied clear-num13.8

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

    7. Simplified13.8

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

    if 3.723435793259181e+232 < (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))

    1. Initial program 60.0

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

    3. Applied add-sqr-sqrt60.0

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

    4. Applied associate-/r*60.0

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

    5. Taylor expanded around -inf 59.9

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

    6. Simplified59.9

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

  4. Final simplification25.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \le 3.7234357932591811 \cdot 10^{232}:\\ \;\;\;\;\frac{\frac{1}{\frac{\sqrt{c \cdot c + d \cdot d}}{a \cdot c + d \cdot b}}}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{\sqrt{c \cdot c + d \cdot d}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(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))))