Average Error: 26.0 → 26.0
Time: 10.5s
Precision: 64
\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}
↓
\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}
double f(double a, double b, double c, double d) {
        double r82014 = a;
        double r82015 = c;
        double r82016 = r82014 * r82015;
        double r82017 = b;
        double r82018 = d;
        double r82019 = r82017 * r82018;
        double r82020 = r82016 + r82019;
        double r82021 = r82015 * r82015;
        double r82022 = r82018 * r82018;
        double r82023 = r82021 + r82022;
        double r82024 = r82020 / r82023;
        return r82024;
}
↓
double f(double a, double b, double c, double d) {
        double r82025 = a;
        double r82026 = c;
        double r82027 = r82025 * r82026;
        double r82028 = b;
        double r82029 = d;
        double r82030 = r82028 * r82029;
        double r82031 = r82027 + r82030;
        double r82032 = r82026 * r82026;
        double r82033 = r82029 * r82029;
        double r82034 = r82032 + r82033;
        double r82035 = r82031 / r82034;
        return r82035;
}

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

Original26.0
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 26.0

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

  3. Applied add-sqr-sqrt26.0

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

  4. Applied associate-/r*25.9

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

  6. Applied clear-num26.0

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

  7. Final simplification26.0

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

Reproduce

herbie shell --seed 2019291 
(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))))