Average Error: 26.1 → 26.0
Time: 32.3s
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 r25830464 = a;
        double r25830465 = c;
        double r25830466 = r25830464 * r25830465;
        double r25830467 = b;
        double r25830468 = d;
        double r25830469 = r25830467 * r25830468;
        double r25830470 = r25830466 + r25830469;
        double r25830471 = r25830465 * r25830465;
        double r25830472 = r25830468 * r25830468;
        double r25830473 = r25830471 + r25830472;
        double r25830474 = r25830470 / r25830473;
        return r25830474;
}
↓
double f(double a, double b, double c, double d) {
        double r25830475 = b;
        double r25830476 = d;
        double r25830477 = r25830475 * r25830476;
        double r25830478 = a;
        double r25830479 = c;
        double r25830480 = r25830478 * r25830479;
        double r25830481 = r25830477 + r25830480;
        double r25830482 = r25830479 * r25830479;
        double r25830483 = r25830476 * r25830476;
        double r25830484 = r25830482 + r25830483;
        double r25830485 = sqrt(r25830484);
        double r25830486 = r25830481 / r25830485;
        double r25830487 = r25830486 / r25830485;
        return r25830487;
}

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

Original26.1
Target0.5
Herbie26.0
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Initial program 26.1

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

  3. Applied add-sqr-sqrt26.1

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

  4. Applied associate-/r*26.0

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

  5. Final simplification26.0

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

Reproduce

herbie shell --seed 2019128 
(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))))