Average Error: 26.0 → 26.0
Time: 3.2s
Precision: binary64
\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\frac{\frac{1}{\frac{\sqrt{c \cdot c + d \cdot d}}{c \cdot a + d \cdot b}}}{\sqrt{c \cdot c + d \cdot d}}\]
\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}
↓
\frac{\frac{1}{\frac{\sqrt{c \cdot c + d \cdot d}}{c \cdot a + d \cdot b}}}{\sqrt{c \cdot c + d \cdot d}}
(FPCore (a b c d)
 :precision binary64
 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))))
↓
(FPCore (a b c d)
 :precision binary64
 (/
  (/ 1.0 (/ (sqrt (+ (* c c) (* d d))) (+ (* c a) (* d b))))
  (sqrt (+ (* c c) (* d d)))))
double code(double a, double b, double c, double d) {
	return ((a * c) + (b * d)) / ((c * c) + (d * d));
}
↓
double code(double a, double b, double c, double d) {
	return (1.0 / (sqrt((c * c) + (d * d)) / ((c * a) + (d * b)))) / sqrt((c * c) + (d * d));
}

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
\[\begin{array}{l} \mathbf{if}\;\left|d\right| < \left|c\right|:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

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-sqrt_binary6426.0

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

  4. Applied associate-/r*_binary6426.0

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

  6. Applied clear-num_binary6426.0

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

  7. Simplified26.0

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

  8. Final simplification26.0

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

Reproduce

herbie shell --seed 2020224 
(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))))