Average Error: 25.8 → 25.8
Time: 10.4s
Precision: 64
\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\frac{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \left(b \cdot d + a \cdot c\right)}{\sqrt{c \cdot c + d \cdot d}}\]
\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}
↓
\frac{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \left(b \cdot d + a \cdot c\right)}{\sqrt{c \cdot c + d \cdot d}}
double f(double a, double b, double c, double d) {
        double r2817366 = a;
        double r2817367 = c;
        double r2817368 = r2817366 * r2817367;
        double r2817369 = b;
        double r2817370 = d;
        double r2817371 = r2817369 * r2817370;
        double r2817372 = r2817368 + r2817371;
        double r2817373 = r2817367 * r2817367;
        double r2817374 = r2817370 * r2817370;
        double r2817375 = r2817373 + r2817374;
        double r2817376 = r2817372 / r2817375;
        return r2817376;
}
↓
double f(double a, double b, double c, double d) {
        double r2817377 = 1.0;
        double r2817378 = c;
        double r2817379 = r2817378 * r2817378;
        double r2817380 = d;
        double r2817381 = r2817380 * r2817380;
        double r2817382 = r2817379 + r2817381;
        double r2817383 = sqrt(r2817382);
        double r2817384 = r2817377 / r2817383;
        double r2817385 = b;
        double r2817386 = r2817385 * r2817380;
        double r2817387 = a;
        double r2817388 = r2817387 * r2817378;
        double r2817389 = r2817386 + r2817388;
        double r2817390 = r2817384 * r2817389;
        double r2817391 = r2817390 / r2817383;
        return r2817391;
}

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

Original25.8
Target0.5
Herbie25.8
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Initial program 25.8

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

  3. Applied add-sqr-sqrt25.8

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

  4. Applied associate-/r*25.8

    \[\leadsto \color{blue}{\frac{\frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}}\]
  5. Using strategy rm

  6. Applied div-inv25.8

    \[\leadsto \frac{\color{blue}{\left(a \cdot c + b \cdot d\right) \cdot \frac{1}{\sqrt{c \cdot c + d \cdot d}}}}{\sqrt{c \cdot c + d \cdot d}}\]

  7. Final simplification25.8

    \[\leadsto \frac{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \left(b \cdot d + a \cdot c\right)}{\sqrt{c \cdot c + d \cdot d}}\]

Reproduce

herbie shell --seed 2019139 
(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))))