Average Error: 25.9 → 25.9
Time: 3.5s
Precision: 64
\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\frac{\left(a \cdot c + b \cdot d\right) \cdot \frac{1}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\]
\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}
↓
\frac{\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}}
double f(double a, double b, double c, double d) {
        double r85375 = a;
        double r85376 = c;
        double r85377 = r85375 * r85376;
        double r85378 = b;
        double r85379 = d;
        double r85380 = r85378 * r85379;
        double r85381 = r85377 + r85380;
        double r85382 = r85376 * r85376;
        double r85383 = r85379 * r85379;
        double r85384 = r85382 + r85383;
        double r85385 = r85381 / r85384;
        return r85385;
}
↓
double f(double a, double b, double c, double d) {
        double r85386 = a;
        double r85387 = c;
        double r85388 = r85386 * r85387;
        double r85389 = b;
        double r85390 = d;
        double r85391 = r85389 * r85390;
        double r85392 = r85388 + r85391;
        double r85393 = 1.0;
        double r85394 = r85387 * r85387;
        double r85395 = r85390 * r85390;
        double r85396 = r85394 + r85395;
        double r85397 = sqrt(r85396);
        double r85398 = r85393 / r85397;
        double r85399 = r85392 * r85398;
        double r85400 = r85399 / r85397;
        return r85400;
}

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

Original25.9
Target0.5
Herbie25.9
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Initial program 25.9

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

  3. Applied add-sqr-sqrt25.9

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

  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-inv25.9

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

  7. Final simplification25.9

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

Reproduce

herbie shell --seed 2019353 
(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))))