Average Error: 26.1 → 3.3
Time: 13.4s
Precision: 64
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\frac{b}{c + \frac{d}{\frac{c}{d}}} - \frac{a}{d + c \cdot \frac{c}{d}}\]
\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}
↓
\frac{b}{c + \frac{d}{\frac{c}{d}}} - \frac{a}{d + c \cdot \frac{c}{d}}
double f(double a, double b, double c, double d) {
        double r83630 = b;
        double r83631 = c;
        double r83632 = r83630 * r83631;
        double r83633 = a;
        double r83634 = d;
        double r83635 = r83633 * r83634;
        double r83636 = r83632 - r83635;
        double r83637 = r83631 * r83631;
        double r83638 = r83634 * r83634;
        double r83639 = r83637 + r83638;
        double r83640 = r83636 / r83639;
        return r83640;
}
↓
double f(double a, double b, double c, double d) {
        double r83641 = b;
        double r83642 = c;
        double r83643 = d;
        double r83644 = r83642 / r83643;
        double r83645 = r83643 / r83644;
        double r83646 = r83642 + r83645;
        double r83647 = r83641 / r83646;
        double r83648 = a;
        double r83649 = r83642 * r83644;
        double r83650 = r83643 + r83649;
        double r83651 = r83648 / r83650;
        double r83652 = r83647 - r83651;
        return r83652;
}

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
Herbie3.3
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Initial program 26.1

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

  3. Applied div-sub26.1

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

  4. Simplified24.4

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

  5. Simplified22.9

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

  6. Taylor expanded around 0 15.6

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

  7. Simplified14.1

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

  8. Taylor expanded around 0 5.5

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

  9. Simplified3.3

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

  11. Applied associate-/r/3.3

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

  12. Final simplification3.3

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

Reproduce

herbie shell --seed 2019179 
(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))))