Average Error: 26.4 → 26.3
Time: 3.9s
Precision: 64
\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\frac{\frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\]
\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}
↓
\frac{\frac{a \cdot c + b \cdot d}{\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 r139570 = a;
        double r139571 = c;
        double r139572 = r139570 * r139571;
        double r139573 = b;
        double r139574 = d;
        double r139575 = r139573 * r139574;
        double r139576 = r139572 + r139575;
        double r139577 = r139571 * r139571;
        double r139578 = r139574 * r139574;
        double r139579 = r139577 + r139578;
        double r139580 = r139576 / r139579;
        return r139580;
}
↓
double f(double a, double b, double c, double d) {
        double r139581 = a;
        double r139582 = c;
        double r139583 = r139581 * r139582;
        double r139584 = b;
        double r139585 = d;
        double r139586 = r139584 * r139585;
        double r139587 = r139583 + r139586;
        double r139588 = r139582 * r139582;
        double r139589 = r139585 * r139585;
        double r139590 = r139588 + r139589;
        double r139591 = sqrt(r139590);
        double r139592 = r139587 / r139591;
        double r139593 = r139592 / r139591;
        return r139593;
}

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

Original26.4
Target0.4
Herbie26.3
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Initial program 26.4

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

  3. Applied add-sqr-sqrt26.4

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

  4. Applied associate-/r*26.3

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

  5. Final simplification26.3

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

Reproduce

herbie shell --seed 2020062 
(FPCore (a b c d)
  :name "Complex division, real part"
  :precision binary64

  :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))))