Average Error: 26.7 → 26.5
Time: 8.3s
Precision: 64
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\begin{array}{l} \mathbf{if}\;d \le 3.67504458165974214 \cdot 10^{89}:\\ \;\;\;\;\frac{\frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{\sqrt{c \cdot c + d \cdot d}}\\ \end{array}\]
\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}
↓
\begin{array}{l}
\mathbf{if}\;d \le 3.67504458165974214 \cdot 10^{89}:\\
\;\;\;\;\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 r122101 = b;
        double r122102 = c;
        double r122103 = r122101 * r122102;
        double r122104 = a;
        double r122105 = d;
        double r122106 = r122104 * r122105;
        double r122107 = r122103 - r122106;
        double r122108 = r122102 * r122102;
        double r122109 = r122105 * r122105;
        double r122110 = r122108 + r122109;
        double r122111 = r122107 / r122110;
        return r122111;
}
↓
double f(double a, double b, double c, double d) {
        double r122112 = d;
        double r122113 = 3.675044581659742e+89;
        bool r122114 = r122112 <= r122113;
        double r122115 = b;
        double r122116 = c;
        double r122117 = r122115 * r122116;
        double r122118 = a;
        double r122119 = r122118 * r122112;
        double r122120 = r122117 - r122119;
        double r122121 = r122116 * r122116;
        double r122122 = r122112 * r122112;
        double r122123 = r122121 + r122122;
        double r122124 = sqrt(r122123);
        double r122125 = r122120 / r122124;
        double r122126 = r122125 / r122124;
        double r122127 = -r122118;
        double r122128 = r122127 / r122124;
        double r122129 = r122114 ? r122126 : r122128;
        return r122129;
}

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

Original26.7
Target0.5
Herbie26.5
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if d < 3.675044581659742e+89

    1. Initial program 23.7

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

    3. Applied add-sqr-sqrt23.7

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

    4. Applied associate-/r*23.6

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

    if 3.675044581659742e+89 < d

    1. Initial program 38.6

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

    3. Applied add-sqr-sqrt38.7

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

    4. Applied associate-/r*38.6

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

    5. Taylor expanded around 0 38.0

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

    6. Simplified38.0

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

  4. Final simplification26.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;d \le 3.67504458165974214 \cdot 10^{89}:\\ \;\;\;\;\frac{\frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{\sqrt{c \cdot c + d \cdot d}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(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))))