Average Error: 26.1 → 24.5
Time: 3.4s
Precision: 64
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\frac{\frac{b \cdot c}{\sqrt{c \cdot c + d \cdot d}} - \frac{a}{\frac{\sqrt{c \cdot c + d \cdot d}}{d}}}{\sqrt{c \cdot c + d \cdot d}}\]
\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}
↓
\frac{\frac{b \cdot c}{\sqrt{c \cdot c + d \cdot d}} - \frac{a}{\frac{\sqrt{c \cdot c + d \cdot d}}{d}}}{\sqrt{c \cdot c + d \cdot d}}
double f(double a, double b, double c, double d) {
        double r151767 = b;
        double r151768 = c;
        double r151769 = r151767 * r151768;
        double r151770 = a;
        double r151771 = d;
        double r151772 = r151770 * r151771;
        double r151773 = r151769 - r151772;
        double r151774 = r151768 * r151768;
        double r151775 = r151771 * r151771;
        double r151776 = r151774 + r151775;
        double r151777 = r151773 / r151776;
        return r151777;
}
↓
double f(double a, double b, double c, double d) {
        double r151778 = b;
        double r151779 = c;
        double r151780 = r151778 * r151779;
        double r151781 = r151779 * r151779;
        double r151782 = d;
        double r151783 = r151782 * r151782;
        double r151784 = r151781 + r151783;
        double r151785 = sqrt(r151784);
        double r151786 = r151780 / r151785;
        double r151787 = a;
        double r151788 = r151785 / r151782;
        double r151789 = r151787 / r151788;
        double r151790 = r151786 - r151789;
        double r151791 = r151790 / r151785;
        return r151791;
}

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
Target0.5
Herbie24.5
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Initial program 26.1

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

  3. Applied add-sqr-sqrt26.1

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

  4. Applied associate-/r*26.1

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

  6. Applied div-sub26.1

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

  8. Applied associate-/l*24.5

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

  9. Final simplification24.5

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

Reproduce

herbie shell --seed 2020062 
(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))))