Average Error: 26.0 → 25.9
Time: 9.5s
Precision: 64
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\frac{\frac{1}{\frac{\sqrt{c \cdot c + d \cdot d}}{b \cdot c - d \cdot a}}}{\sqrt{c \cdot c + d \cdot d}}\]
\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}
↓
\frac{\frac{1}{\frac{\sqrt{c \cdot c + d \cdot d}}{b \cdot c - d \cdot a}}}{\sqrt{c \cdot c + d \cdot d}}
double f(double a, double b, double c, double d) {
        double r111229 = b;
        double r111230 = c;
        double r111231 = r111229 * r111230;
        double r111232 = a;
        double r111233 = d;
        double r111234 = r111232 * r111233;
        double r111235 = r111231 - r111234;
        double r111236 = r111230 * r111230;
        double r111237 = r111233 * r111233;
        double r111238 = r111236 + r111237;
        double r111239 = r111235 / r111238;
        return r111239;
}
↓
double f(double a, double b, double c, double d) {
        double r111240 = 1.0;
        double r111241 = c;
        double r111242 = r111241 * r111241;
        double r111243 = d;
        double r111244 = r111243 * r111243;
        double r111245 = r111242 + r111244;
        double r111246 = sqrt(r111245);
        double r111247 = b;
        double r111248 = r111247 * r111241;
        double r111249 = a;
        double r111250 = r111243 * r111249;
        double r111251 = r111248 - r111250;
        double r111252 = r111246 / r111251;
        double r111253 = r111240 / r111252;
        double r111254 = r111253 / r111246;
        return r111254;
}

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

Original26.0
Target0.5
Herbie25.9
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Initial program 26.0

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

  3. Applied add-sqr-sqrt26.0

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

  4. Applied associate-/r*25.9

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

  6. Applied clear-num25.9

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

  7. Simplified25.9

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

  8. Final simplification25.9

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

Reproduce

herbie shell --seed 2020045 
(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))))