Average Error: 26.5 → 16.2
Time: 4.1s
Precision: 64
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\begin{array}{l} \mathbf{if}\;c \le -4.4263537617510411 \cdot 10^{138} \lor \neg \left(c \le 8.23624815355425478 \cdot 10^{-31}\right):\\ \;\;\;\;\frac{b}{\frac{{d}^{2}}{c} + c} - \frac{a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\frac{{c}^{2} + {d}^{2}}{c}} - \frac{a}{\frac{c \cdot c + d \cdot d}{d}}\\ \end{array}\]
\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}
↓
\begin{array}{l}
\mathbf{if}\;c \le -4.4263537617510411 \cdot 10^{138} \lor \neg \left(c \le 8.23624815355425478 \cdot 10^{-31}\right):\\
\;\;\;\;\frac{b}{\frac{{d}^{2}}{c} + c} - \frac{a \cdot d}{c \cdot c + d \cdot d}\\

\mathbf{else}:\\
\;\;\;\;\frac{b}{\frac{{c}^{2} + {d}^{2}}{c}} - \frac{a}{\frac{c \cdot c + d \cdot d}{d}}\\

\end{array}
double f(double a, double b, double c, double d) {
        double r105432 = b;
        double r105433 = c;
        double r105434 = r105432 * r105433;
        double r105435 = a;
        double r105436 = d;
        double r105437 = r105435 * r105436;
        double r105438 = r105434 - r105437;
        double r105439 = r105433 * r105433;
        double r105440 = r105436 * r105436;
        double r105441 = r105439 + r105440;
        double r105442 = r105438 / r105441;
        return r105442;
}
↓
double f(double a, double b, double c, double d) {
        double r105443 = c;
        double r105444 = -4.426353761751041e+138;
        bool r105445 = r105443 <= r105444;
        double r105446 = 8.236248153554255e-31;
        bool r105447 = r105443 <= r105446;
        double r105448 = !r105447;
        bool r105449 = r105445 || r105448;
        double r105450 = b;
        double r105451 = d;
        double r105452 = 2.0;
        double r105453 = pow(r105451, r105452);
        double r105454 = r105453 / r105443;
        double r105455 = r105454 + r105443;
        double r105456 = r105450 / r105455;
        double r105457 = a;
        double r105458 = r105457 * r105451;
        double r105459 = r105443 * r105443;
        double r105460 = r105451 * r105451;
        double r105461 = r105459 + r105460;
        double r105462 = r105458 / r105461;
        double r105463 = r105456 - r105462;
        double r105464 = pow(r105443, r105452);
        double r105465 = r105464 + r105453;
        double r105466 = r105465 / r105443;
        double r105467 = r105450 / r105466;
        double r105468 = r105461 / r105451;
        double r105469 = r105457 / r105468;
        double r105470 = r105467 - r105469;
        double r105471 = r105449 ? r105463 : r105470;
        return r105471;
}

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

Original26.5
Target0.4
Herbie16.2
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if c < -4.426353761751041e+138 or 8.236248153554255e-31 < c

    1. Initial program 34.9

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

    3. Applied div-sub34.9

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

    5. Applied associate-/l*32.2

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

    6. Simplified32.2

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

    7. Taylor expanded around 0 15.5

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

    if -4.426353761751041e+138 < c < 8.236248153554255e-31

    1. Initial program 19.9

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

    3. Applied div-sub19.9

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

    5. Applied associate-/l*19.3

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

    6. Simplified19.3

      \[\leadsto \frac{b}{\color{blue}{\frac{{c}^{2} + {d}^{2}}{c}}} - \frac{a \cdot d}{c \cdot c + d \cdot d}\]
    7. Using strategy rm

    8. Applied associate-/l*16.7

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

  4. Final simplification16.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;c \le -4.4263537617510411 \cdot 10^{138} \lor \neg \left(c \le 8.23624815355425478 \cdot 10^{-31}\right):\\ \;\;\;\;\frac{b}{\frac{{d}^{2}}{c} + c} - \frac{a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\frac{{c}^{2} + {d}^{2}}{c}} - \frac{a}{\frac{c \cdot c + d \cdot d}{d}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(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))))