Average Error: 26.0 → 25.9
Time: 26.5s
Precision: 64
\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\frac{\frac{c \cdot a + d \cdot b}{\sqrt{d \cdot d + c \cdot c}}}{\sqrt{d \cdot d + c \cdot c}}\]
\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}
↓
\frac{\frac{c \cdot a + d \cdot b}{\sqrt{d \cdot d + c \cdot c}}}{\sqrt{d \cdot d + c \cdot c}}
double f(double a, double b, double c, double d) {
        double r6795598 = a;
        double r6795599 = c;
        double r6795600 = r6795598 * r6795599;
        double r6795601 = b;
        double r6795602 = d;
        double r6795603 = r6795601 * r6795602;
        double r6795604 = r6795600 + r6795603;
        double r6795605 = r6795599 * r6795599;
        double r6795606 = r6795602 * r6795602;
        double r6795607 = r6795605 + r6795606;
        double r6795608 = r6795604 / r6795607;
        return r6795608;
}
↓
double f(double a, double b, double c, double d) {
        double r6795609 = c;
        double r6795610 = a;
        double r6795611 = r6795609 * r6795610;
        double r6795612 = d;
        double r6795613 = b;
        double r6795614 = r6795612 * r6795613;
        double r6795615 = r6795611 + r6795614;
        double r6795616 = r6795612 * r6795612;
        double r6795617 = r6795609 * r6795609;
        double r6795618 = r6795616 + r6795617;
        double r6795619 = sqrt(r6795618);
        double r6795620 = r6795615 / r6795619;
        double r6795621 = r6795620 / r6795619;
        return r6795621;
}

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

Derivation

  1. Initial program 26.0

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

  3. Applied add-sqr-sqrt26.0

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

  4. Applied associate-/r*25.9

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

  5. Final simplification25.9

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

Reproduce

herbie shell --seed 2019200 
(FPCore (a b c d)
  :name "Complex division, real part"

  :herbie-target
  (if (< (fabs d) (fabs c)) (/ (+ a (* b (/ d c))) (+ c (* d (/ d c)))) (/ (+ b (* a (/ c d))) (+ d (* c (/ c d)))))

  (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))))