Average Error: 26.4 → 26.4
Time: 3.5s
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 r112527 = a;
        double r112528 = c;
        double r112529 = r112527 * r112528;
        double r112530 = b;
        double r112531 = d;
        double r112532 = r112530 * r112531;
        double r112533 = r112529 + r112532;
        double r112534 = r112528 * r112528;
        double r112535 = r112531 * r112531;
        double r112536 = r112534 + r112535;
        double r112537 = r112533 / r112536;
        return r112537;
}
↓
double f(double a, double b, double c, double d) {
        double r112538 = a;
        double r112539 = c;
        double r112540 = r112538 * r112539;
        double r112541 = b;
        double r112542 = d;
        double r112543 = r112541 * r112542;
        double r112544 = r112540 + r112543;
        double r112545 = r112539 * r112539;
        double r112546 = r112542 * r112542;
        double r112547 = r112545 + r112546;
        double r112548 = sqrt(r112547);
        double r112549 = r112544 / r112548;
        double r112550 = r112549 / r112548;
        return r112550;
}

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
Target0.5
Herbie26.4
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

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*26.4

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

  5. Final simplification26.4

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

Reproduce

herbie shell --seed 2020065 
(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))))