Average Error: 26.5 → 8.6
Time: 4.1s
Precision: 64
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\begin{array}{l} \mathbf{if}\;d \le -1.351214721549483218261281106569258966909 \cdot 10^{154} \lor \neg \left(d \le 8.000358861136643239312158095843514305232 \cdot 10^{153}\right):\\ \;\;\;\;\frac{b}{\frac{c \cdot c + d \cdot d}{c}} - \frac{a}{d + \frac{{c}^{2}}{d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\frac{{d}^{2}}{c} + c} - \frac{a}{\frac{c \cdot c + d \cdot d}{d}}\\ \end{array}\]
\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}
↓
\begin{array}{l}
\mathbf{if}\;d \le -1.351214721549483218261281106569258966909 \cdot 10^{154} \lor \neg \left(d \le 8.000358861136643239312158095843514305232 \cdot 10^{153}\right):\\
\;\;\;\;\frac{b}{\frac{c \cdot c + d \cdot d}{c}} - \frac{a}{d + \frac{{c}^{2}}{d}}\\

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

\end{array}
double f(double a, double b, double c, double d) {
        double r140301 = b;
        double r140302 = c;
        double r140303 = r140301 * r140302;
        double r140304 = a;
        double r140305 = d;
        double r140306 = r140304 * r140305;
        double r140307 = r140303 - r140306;
        double r140308 = r140302 * r140302;
        double r140309 = r140305 * r140305;
        double r140310 = r140308 + r140309;
        double r140311 = r140307 / r140310;
        return r140311;
}
↓
double f(double a, double b, double c, double d) {
        double r140312 = d;
        double r140313 = -1.3512147215494832e+154;
        bool r140314 = r140312 <= r140313;
        double r140315 = 8.000358861136643e+153;
        bool r140316 = r140312 <= r140315;
        double r140317 = !r140316;
        bool r140318 = r140314 || r140317;
        double r140319 = b;
        double r140320 = c;
        double r140321 = r140320 * r140320;
        double r140322 = r140312 * r140312;
        double r140323 = r140321 + r140322;
        double r140324 = r140323 / r140320;
        double r140325 = r140319 / r140324;
        double r140326 = a;
        double r140327 = 2.0;
        double r140328 = pow(r140320, r140327);
        double r140329 = r140328 / r140312;
        double r140330 = r140312 + r140329;
        double r140331 = r140326 / r140330;
        double r140332 = r140325 - r140331;
        double r140333 = pow(r140312, r140327);
        double r140334 = r140333 / r140320;
        double r140335 = r140334 + r140320;
        double r140336 = r140319 / r140335;
        double r140337 = r140323 / r140312;
        double r140338 = r140326 / r140337;
        double r140339 = r140336 - r140338;
        double r140340 = r140318 ? r140332 : r140339;
        return r140340;
}

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
Target0.5
Herbie8.6
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if d < -1.3512147215494832e+154 or 8.000358861136643e+153 < d

    1. Initial program 46.6

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

    3. Applied add-sqr-sqrt46.6

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

    5. Applied div-sub46.6

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

    6. Simplified46.4

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

    7. Simplified45.3

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

    8. Taylor expanded around 0 16.8

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

    if -1.3512147215494832e+154 < d < 8.000358861136643e+153

    1. Initial program 18.8

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

    3. Applied add-sqr-sqrt18.8

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

    5. Applied div-sub18.8

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

    6. Simplified17.0

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

    7. Simplified15.2

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

    8. Taylor expanded around 0 5.4

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

  4. Final simplification8.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;d \le -1.351214721549483218261281106569258966909 \cdot 10^{154} \lor \neg \left(d \le 8.000358861136643239312158095843514305232 \cdot 10^{153}\right):\\ \;\;\;\;\frac{b}{\frac{c \cdot c + d \cdot d}{c}} - \frac{a}{d + \frac{{c}^{2}}{d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\frac{{d}^{2}}{c} + c} - \frac{a}{\frac{c \cdot c + d \cdot d}{d}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 
(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))))