Average Error: 26.1 → 26.0
Time: 14.5s
Precision: 64
\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\frac{\frac{a \cdot c + d \cdot b}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\]
\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}
↓
\frac{\frac{a \cdot c + d \cdot b}{\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 r114155 = a;
        double r114156 = c;
        double r114157 = r114155 * r114156;
        double r114158 = b;
        double r114159 = d;
        double r114160 = r114158 * r114159;
        double r114161 = r114157 + r114160;
        double r114162 = r114156 * r114156;
        double r114163 = r114159 * r114159;
        double r114164 = r114162 + r114163;
        double r114165 = r114161 / r114164;
        return r114165;
}
↓
double f(double a, double b, double c, double d) {
        double r114166 = a;
        double r114167 = c;
        double r114168 = r114166 * r114167;
        double r114169 = d;
        double r114170 = b;
        double r114171 = r114169 * r114170;
        double r114172 = r114168 + r114171;
        double r114173 = r114167 * r114167;
        double r114174 = r114169 * r114169;
        double r114175 = r114173 + r114174;
        double r114176 = sqrt(r114175);
        double r114177 = r114172 / r114176;
        double r114178 = r114177 / r114176;
        return r114178;
}

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. Simplified26.0

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

  6. Final simplification26.0

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

Reproduce

herbie shell --seed 2019179 
(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))))