Average Error: 25.2 → 24.8
Time: 12.3s
Precision: 64
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \le 1.1631964723081969 \cdot 10^{+277}:\\ \;\;\;\;\frac{\frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{\sqrt{c \cdot c + d \cdot d}}\\ \end{array}\]
\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}
↓
\begin{array}{l}
\mathbf{if}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \le 1.1631964723081969 \cdot 10^{+277}:\\
\;\;\;\;\frac{\frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\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 r2793720 = b;
        double r2793721 = c;
        double r2793722 = r2793720 * r2793721;
        double r2793723 = a;
        double r2793724 = d;
        double r2793725 = r2793723 * r2793724;
        double r2793726 = r2793722 - r2793725;
        double r2793727 = r2793721 * r2793721;
        double r2793728 = r2793724 * r2793724;
        double r2793729 = r2793727 + r2793728;
        double r2793730 = r2793726 / r2793729;
        return r2793730;
}
↓
double f(double a, double b, double c, double d) {
        double r2793731 = b;
        double r2793732 = c;
        double r2793733 = r2793731 * r2793732;
        double r2793734 = a;
        double r2793735 = d;
        double r2793736 = r2793734 * r2793735;
        double r2793737 = r2793733 - r2793736;
        double r2793738 = r2793732 * r2793732;
        double r2793739 = r2793735 * r2793735;
        double r2793740 = r2793738 + r2793739;
        double r2793741 = r2793737 / r2793740;
        double r2793742 = 1.1631964723081969e+277;
        bool r2793743 = r2793741 <= r2793742;
        double r2793744 = sqrt(r2793740);
        double r2793745 = r2793737 / r2793744;
        double r2793746 = r2793745 / r2793744;
        double r2793747 = -r2793734;
        double r2793748 = r2793747 / r2793744;
        double r2793749 = r2793743 ? r2793746 : r2793748;
        return r2793749;
}

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

Original25.2
Target0.5
Herbie24.8
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (- (* b c) (* a d)) (+ (* c c) (* d d))) < 1.1631964723081969e+277

    1. Initial program 13.6

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

    3. Applied add-sqr-sqrt13.6

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

    4. Applied associate-/r*13.5

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

    if 1.1631964723081969e+277 < (/ (- (* b c) (* a d)) (+ (* c c) (* d d)))

    1. Initial program 61.2

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

    3. Applied add-sqr-sqrt61.2

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

    4. Applied associate-/r*61.2

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

    5. Taylor expanded around 0 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 simplification24.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \le 1.1631964723081969 \cdot 10^{+277}:\\ \;\;\;\;\frac{\frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{\sqrt{c \cdot c + d \cdot d}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019129 
(FPCore (a b c d)
  :name "Complex division, imag part"

  :herbie-target
  (if (< (fabs d) (fabs c)) (/ (- b (* a (/ d c))) (+ c (* d (/ d c)))) (/ (+ (- a) (* b (/ c d))) (+ d (* c (/ c d)))))

  (/ (- (* b c) (* a d)) (+ (* c c) (* d d))))