Complex division, imag part

Average Error: 25.4 → 25.3

Time: 10.3s

Precision: 64

Math

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

↓

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

C

double f(double a, double b, double c, double d) {
        double r4804951 = b;
        double r4804952 = c;
        double r4804953 = r4804951 * r4804952;
        double r4804954 = a;
        double r4804955 = d;
        double r4804956 = r4804954 * r4804955;
        double r4804957 = r4804953 - r4804956;
        double r4804958 = r4804952 * r4804952;
        double r4804959 = r4804955 * r4804955;
        double r4804960 = r4804958 + r4804959;
        double r4804961 = r4804957 / r4804960;
        return r4804961;
}

↓

double f(double a, double b, double c, double d) {
        double r4804962 = b;
        double r4804963 = c;
        double r4804964 = r4804962 * r4804963;
        double r4804965 = a;
        double r4804966 = d;
        double r4804967 = r4804965 * r4804966;
        double r4804968 = r4804964 - r4804967;
        double r4804969 = r4804963 * r4804963;
        double r4804970 = r4804966 * r4804966;
        double r4804971 = r4804969 + r4804970;
        double r4804972 = sqrt(r4804971);
        double r4804973 = r4804968 / r4804972;
        double r4804974 = r4804973 / r4804972;
        return r4804974;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	25.4
Target	0.4
Herbie	25.3

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

Derivation

1. Initial program 25.4

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

3. Applied add-sqr-sqrt 25.4

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

4. Applied associate-/r* 25.3

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

5. Final simplification 25.3

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

Reproduce

herbie shell --seed 2019152 
(FPCore (a b c d)
  :name "Complex division, imag part"

  :herbie-target
  (if (< (fabs d) (fabs c)) (/ (- b (* a (/ d c))) (+ c (* d (/ d c)))) (/ (+ (- a) (* b (/ c d))) (+ d (* c (/ c d)))))

  (/ (- (* b c) (* a d)) (+ (* c c) (* d d))))