Average Error: 26.4 → 14.6
Time: 12.5s
Precision: 64
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\frac{b}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{c}{\sqrt{c \cdot c + d \cdot d}} - \frac{a}{d + \frac{c}{\frac{d}{c}}}\]
\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}
↓
\frac{b}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{c}{\sqrt{c \cdot c + d \cdot d}} - \frac{a}{d + \frac{c}{\frac{d}{c}}}
double f(double a, double b, double c, double d) {
        double r73008 = b;
        double r73009 = c;
        double r73010 = r73008 * r73009;
        double r73011 = a;
        double r73012 = d;
        double r73013 = r73011 * r73012;
        double r73014 = r73010 - r73013;
        double r73015 = r73009 * r73009;
        double r73016 = r73012 * r73012;
        double r73017 = r73015 + r73016;
        double r73018 = r73014 / r73017;
        return r73018;
}
↓
double f(double a, double b, double c, double d) {
        double r73019 = b;
        double r73020 = c;
        double r73021 = r73020 * r73020;
        double r73022 = d;
        double r73023 = r73022 * r73022;
        double r73024 = r73021 + r73023;
        double r73025 = sqrt(r73024);
        double r73026 = r73019 / r73025;
        double r73027 = r73020 / r73025;
        double r73028 = r73026 * r73027;
        double r73029 = a;
        double r73030 = r73022 / r73020;
        double r73031 = r73020 / r73030;
        double r73032 = r73022 + r73031;
        double r73033 = r73029 / r73032;
        double r73034 = r73028 - r73033;
        return r73034;
}

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

Original26.4
Target0.5
Herbie14.6
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Initial program 26.4

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

  3. Applied div-sub26.4

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

  4. Simplified25.2

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

  5. Taylor expanded around 0 18.0

    \[\leadsto \frac{b \cdot c}{c \cdot c + d \cdot d} - \frac{a}{\color{blue}{\frac{{c}^{2}}{d} + d}}\]

  6. Simplified16.5

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

  8. Applied add-sqr-sqrt16.5

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

  9. Applied times-frac14.6

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

  10. Final simplification14.6

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

Reproduce

herbie shell --seed 2019174 
(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))))