Average Error: 26.5 → 25.3
Time: 3.7s
Precision: 64
\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} = -\infty:\\ \;\;\;\;\frac{a}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{elif}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \le 5.93953377056499191 \cdot 10^{289}:\\ \;\;\;\;\frac{\frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot a}{\sqrt{c \cdot c + d \cdot d}}\\ \end{array}\]
\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} = -\infty:\\
\;\;\;\;\frac{a}{\sqrt{c \cdot c + d \cdot d}}\\

\mathbf{elif}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \le 5.93953377056499191 \cdot 10^{289}:\\
\;\;\;\;\frac{\frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\

\mathbf{else}:\\
\;\;\;\;\frac{-1 \cdot a}{\sqrt{c \cdot c + d \cdot d}}\\

\end{array}
double f(double a, double b, double c, double d) {
        double r83761 = a;
        double r83762 = c;
        double r83763 = r83761 * r83762;
        double r83764 = b;
        double r83765 = d;
        double r83766 = r83764 * r83765;
        double r83767 = r83763 + r83766;
        double r83768 = r83762 * r83762;
        double r83769 = r83765 * r83765;
        double r83770 = r83768 + r83769;
        double r83771 = r83767 / r83770;
        return r83771;
}
↓
double f(double a, double b, double c, double d) {
        double r83772 = a;
        double r83773 = c;
        double r83774 = r83772 * r83773;
        double r83775 = b;
        double r83776 = d;
        double r83777 = r83775 * r83776;
        double r83778 = r83774 + r83777;
        double r83779 = r83773 * r83773;
        double r83780 = r83776 * r83776;
        double r83781 = r83779 + r83780;
        double r83782 = r83778 / r83781;
        double r83783 = -inf.0;
        bool r83784 = r83782 <= r83783;
        double r83785 = sqrt(r83781);
        double r83786 = r83772 / r83785;
        double r83787 = 5.939533770564992e+289;
        bool r83788 = r83782 <= r83787;
        double r83789 = r83778 / r83785;
        double r83790 = r83789 / r83785;
        double r83791 = -1.0;
        double r83792 = r83791 * r83772;
        double r83793 = r83792 / r83785;
        double r83794 = r83788 ? r83790 : r83793;
        double r83795 = r83784 ? r83786 : r83794;
        return r83795;
}

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

Original26.5
Target0.5
Herbie25.3
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))) < -inf.0

    1. Initial program 64.0

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

    3. Applied add-sqr-sqrt64.0

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

    4. Applied associate-/r*64.0

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

    5. Taylor expanded around inf 50.1

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

    if -inf.0 < (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))) < 5.939533770564992e+289

    1. Initial program 12.3

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

    3. Applied add-sqr-sqrt12.3

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

    4. Applied associate-/r*12.2

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

    if 5.939533770564992e+289 < (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))

    1. Initial program 62.8

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

    3. Applied add-sqr-sqrt62.8

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

    4. Applied associate-/r*62.8

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

    5. Taylor expanded around -inf 60.0

      \[\leadsto \frac{\color{blue}{-1 \cdot a}}{\sqrt{c \cdot c + d \cdot d}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification25.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} = -\infty:\\ \;\;\;\;\frac{a}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{elif}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \le 5.93953377056499191 \cdot 10^{289}:\\ \;\;\;\;\frac{\frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot a}{\sqrt{c \cdot c + d \cdot d}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020021 
(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))))