Complex division, imag part

Average Error: 25.9 → 25.9

Time: 12.7s

Precision: 64

Math

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

↓

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

C

double f(double a, double b, double c, double d) {
        double r78058 = b;
        double r78059 = c;
        double r78060 = r78058 * r78059;
        double r78061 = a;
        double r78062 = d;
        double r78063 = r78061 * r78062;
        double r78064 = r78060 - r78063;
        double r78065 = r78059 * r78059;
        double r78066 = r78062 * r78062;
        double r78067 = r78065 + r78066;
        double r78068 = r78064 / r78067;
        return r78068;
}

↓

double f(double a, double b, double c, double d) {
        double r78069 = b;
        double r78070 = c;
        double r78071 = r78069 * r78070;
        double r78072 = a;
        double r78073 = d;
        double r78074 = r78072 * r78073;
        double r78075 = r78071 - r78074;
        double r78076 = 1.0;
        double r78077 = r78070 * r78070;
        double r78078 = r78073 * r78073;
        double r78079 = r78077 + r78078;
        double r78080 = sqrt(r78079);
        double r78081 = r78076 / r78080;
        double r78082 = r78075 * r78081;
        double r78083 = r78082 / r78080;
        return r78083;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	25.9
Target	0.4
Herbie	25.9

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

Derivation

1. Initial program 25.9

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

3. Applied add-sqr-sqrt 25.9

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

4. Applied associate-/r* 25.8

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

6. Applied div-inv 25.9

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

7. Final simplification 25.9

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

Reproduce

herbie shell --seed 2019323 
(FPCore (a b c d)
  :name "Complex division, imag part"
  :precision binary64

  :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))))