Average Error: 25.3 → 25.2
Time: 18.4s
Precision: 64
\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\frac{\frac{b \cdot d + a \cdot c}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\]
\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}
↓
\frac{\frac{b \cdot d + a \cdot c}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}
double f(double a, double b, double c, double d) {
        double r10364845 = a;
        double r10364846 = c;
        double r10364847 = r10364845 * r10364846;
        double r10364848 = b;
        double r10364849 = d;
        double r10364850 = r10364848 * r10364849;
        double r10364851 = r10364847 + r10364850;
        double r10364852 = r10364846 * r10364846;
        double r10364853 = r10364849 * r10364849;
        double r10364854 = r10364852 + r10364853;
        double r10364855 = r10364851 / r10364854;
        return r10364855;
}
↓
double f(double a, double b, double c, double d) {
        double r10364856 = b;
        double r10364857 = d;
        double r10364858 = r10364856 * r10364857;
        double r10364859 = a;
        double r10364860 = c;
        double r10364861 = r10364859 * r10364860;
        double r10364862 = r10364858 + r10364861;
        double r10364863 = r10364860 * r10364860;
        double r10364864 = r10364857 * r10364857;
        double r10364865 = r10364863 + r10364864;
        double r10364866 = sqrt(r10364865);
        double r10364867 = r10364862 / r10364866;
        double r10364868 = r10364867 / r10364866;
        return r10364868;
}

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

Original25.3
Target0.4
Herbie25.2
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Initial program 25.3

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

  3. Applied add-sqr-sqrt25.3

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

  4. Applied associate-/r*25.2

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

  5. Final simplification25.2

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

Reproduce

herbie shell --seed 2019158 
(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))))