Average Error: 25.7 → 25.7
Time: 13.9s
Precision: 64
\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b \cdot d + a \cdot c}{\sqrt{c \cdot c + d \cdot d}}\]
\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}
↓
\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b \cdot d + a \cdot c}{\sqrt{c \cdot c + d \cdot d}}
double f(double a, double b, double c, double d) {
        double r3218193 = a;
        double r3218194 = c;
        double r3218195 = r3218193 * r3218194;
        double r3218196 = b;
        double r3218197 = d;
        double r3218198 = r3218196 * r3218197;
        double r3218199 = r3218195 + r3218198;
        double r3218200 = r3218194 * r3218194;
        double r3218201 = r3218197 * r3218197;
        double r3218202 = r3218200 + r3218201;
        double r3218203 = r3218199 / r3218202;
        return r3218203;
}
↓
double f(double a, double b, double c, double d) {
        double r3218204 = 1.0;
        double r3218205 = c;
        double r3218206 = r3218205 * r3218205;
        double r3218207 = d;
        double r3218208 = r3218207 * r3218207;
        double r3218209 = r3218206 + r3218208;
        double r3218210 = sqrt(r3218209);
        double r3218211 = r3218204 / r3218210;
        double r3218212 = b;
        double r3218213 = r3218212 * r3218207;
        double r3218214 = a;
        double r3218215 = r3218214 * r3218205;
        double r3218216 = r3218213 + r3218215;
        double r3218217 = r3218216 / r3218210;
        double r3218218 = r3218211 * r3218217;
        return r3218218;
}

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

Original25.7
Target0.5
Herbie25.7
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Initial program 25.7

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

  3. Applied add-sqr-sqrt25.7

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

  4. Applied *-un-lft-identity25.7

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

  5. Applied times-frac25.7

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

  6. Final simplification25.7

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

Reproduce

herbie shell --seed 2019143 
(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))))