Average Error: 26.1 → 25.3
Time: 5.4s
Precision: 64
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \le 6.918361099356831363837397929747071264075 \cdot 10^{305}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\sqrt{c \cdot c + d \cdot d}}\\ \end{array}\]
\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 6.918361099356831363837397929747071264075 \cdot 10^{305}:\\
\;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\

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

\end{array}
double f(double a, double b, double c, double d) {
        double r80094 = b;
        double r80095 = c;
        double r80096 = r80094 * r80095;
        double r80097 = a;
        double r80098 = d;
        double r80099 = r80097 * r80098;
        double r80100 = r80096 - r80099;
        double r80101 = r80095 * r80095;
        double r80102 = r80098 * r80098;
        double r80103 = r80101 + r80102;
        double r80104 = r80100 / r80103;
        return r80104;
}
↓
double f(double a, double b, double c, double d) {
        double r80105 = b;
        double r80106 = c;
        double r80107 = r80105 * r80106;
        double r80108 = a;
        double r80109 = d;
        double r80110 = r80108 * r80109;
        double r80111 = r80107 - r80110;
        double r80112 = r80106 * r80106;
        double r80113 = r80109 * r80109;
        double r80114 = r80112 + r80113;
        double r80115 = r80111 / r80114;
        double r80116 = 6.918361099356831e+305;
        bool r80117 = r80115 <= r80116;
        double r80118 = sqrt(r80114);
        double r80119 = r80105 / r80118;
        double r80120 = r80117 ? r80115 : r80119;
        return r80120;
}

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

Original26.1
Target0.4
Herbie25.3
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (- (* b c) (* a d)) (+ (* c c) (* d d))) < 6.918361099356831e+305

    1. Initial program 13.9

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

    if 6.918361099356831e+305 < (/ (- (* b c) (* a d)) (+ (* c c) (* d d)))

    1. Initial program 64.0

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

    3. Applied add-sqr-sqrt64.0

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

    4. Applied associate-/r*64.0

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

    5. Taylor expanded around inf 60.6

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

  4. Final simplification25.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \le 6.918361099356831363837397929747071264075 \cdot 10^{305}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\sqrt{c \cdot c + d \cdot d}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(FPCore (a b c d)
  :name "Complex division, imag part"
  :precision binary64

  :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))))