Average Error: 26.6 → 22.7
Time: 12.4s
Precision: 64
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\frac{\frac{b}{\frac{\sqrt{d \cdot d + c \cdot c}}{c}} - \frac{d}{\sqrt{d \cdot d + c \cdot c}} \cdot a}{\sqrt{c \cdot c + d \cdot d}}\]
\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}
↓
\frac{\frac{b}{\frac{\sqrt{d \cdot d + c \cdot c}}{c}} - \frac{d}{\sqrt{d \cdot d + c \cdot c}} \cdot a}{\sqrt{c \cdot c + d \cdot d}}
double f(double a, double b, double c, double d) {
        double r110754 = b;
        double r110755 = c;
        double r110756 = r110754 * r110755;
        double r110757 = a;
        double r110758 = d;
        double r110759 = r110757 * r110758;
        double r110760 = r110756 - r110759;
        double r110761 = r110755 * r110755;
        double r110762 = r110758 * r110758;
        double r110763 = r110761 + r110762;
        double r110764 = r110760 / r110763;
        return r110764;
}
↓
double f(double a, double b, double c, double d) {
        double r110765 = b;
        double r110766 = d;
        double r110767 = r110766 * r110766;
        double r110768 = c;
        double r110769 = r110768 * r110768;
        double r110770 = r110767 + r110769;
        double r110771 = sqrt(r110770);
        double r110772 = r110771 / r110768;
        double r110773 = r110765 / r110772;
        double r110774 = r110766 / r110771;
        double r110775 = a;
        double r110776 = r110774 * r110775;
        double r110777 = r110773 - r110776;
        double r110778 = r110769 + r110767;
        double r110779 = sqrt(r110778);
        double r110780 = r110777 / r110779;
        return r110780;
}

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

Original26.6
Target0.5
Herbie22.7
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Initial program 26.6

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

  3. Applied add-sqr-sqrt26.6

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

  4. Applied associate-/r*26.5

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

  6. Applied div-sub26.5

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

  7. Simplified26.5

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

  8. Simplified25.1

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

  10. Applied associate-/l*23.1

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

  12. Applied associate-/r/22.7

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

  13. Final simplification22.7

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

Reproduce

herbie shell --seed 2019347 
(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))))