Average Error: 26.5 → 26.5
Time: 27.7s
Precision: 64
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\frac{c \cdot b - a \cdot d}{c \cdot c + d \cdot d}\]
\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}
↓
\frac{c \cdot b - a \cdot d}{c \cdot c + d \cdot d}
double f(double a, double b, double c, double d) {
        double r44066413 = b;
        double r44066414 = c;
        double r44066415 = r44066413 * r44066414;
        double r44066416 = a;
        double r44066417 = d;
        double r44066418 = r44066416 * r44066417;
        double r44066419 = r44066415 - r44066418;
        double r44066420 = r44066414 * r44066414;
        double r44066421 = r44066417 * r44066417;
        double r44066422 = r44066420 + r44066421;
        double r44066423 = r44066419 / r44066422;
        return r44066423;
}
↓
double f(double a, double b, double c, double d) {
        double r44066424 = c;
        double r44066425 = b;
        double r44066426 = r44066424 * r44066425;
        double r44066427 = a;
        double r44066428 = d;
        double r44066429 = r44066427 * r44066428;
        double r44066430 = r44066426 - r44066429;
        double r44066431 = r44066424 * r44066424;
        double r44066432 = r44066428 * r44066428;
        double r44066433 = r44066431 + r44066432;
        double r44066434 = r44066430 / r44066433;
        return r44066434;
}

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
Herbie26.5
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Initial program 26.5

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

  3. Applied clear-num26.7

    \[\leadsto \color{blue}{\frac{1}{\frac{c \cdot c + d \cdot d}{b \cdot c - a \cdot d}}}\]
  4. Using strategy rm

  5. Applied *-un-lft-identity26.7

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

  6. Applied *-un-lft-identity26.7

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

  7. Applied times-frac26.7

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

  8. Simplified26.7

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

  9. Simplified26.5

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

  10. Final simplification26.5

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

Reproduce

herbie shell --seed 2019120 
(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))))