Average Error: 26.1 → 26.0
Time: 3.8s
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 r98513 = a;
        double r98514 = c;
        double r98515 = r98513 * r98514;
        double r98516 = b;
        double r98517 = d;
        double r98518 = r98516 * r98517;
        double r98519 = r98515 + r98518;
        double r98520 = r98514 * r98514;
        double r98521 = r98517 * r98517;
        double r98522 = r98520 + r98521;
        double r98523 = r98519 / r98522;
        return r98523;
}
↓
double f(double a, double b, double c, double d) {
        double r98524 = a;
        double r98525 = c;
        double r98526 = r98524 * r98525;
        double r98527 = b;
        double r98528 = d;
        double r98529 = r98527 * r98528;
        double r98530 = r98526 + r98529;
        double r98531 = r98525 * r98525;
        double r98532 = r98528 * r98528;
        double r98533 = r98531 + r98532;
        double r98534 = sqrt(r98533);
        double r98535 = r98530 / r98534;
        double r98536 = r98535 / r98534;
        return r98536;
}

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
Target0.4
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

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

Reproduce

herbie shell --seed 2020100 
(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))))