Complex division, imag part

Average Error: 25.7 → 25.7

Time: 8.7s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

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 r104043 = b;
        double r104044 = c;
        double r104045 = r104043 * r104044;
        double r104046 = a;
        double r104047 = d;
        double r104048 = r104046 * r104047;
        double r104049 = r104045 - r104048;
        double r104050 = r104044 * r104044;
        double r104051 = r104047 * r104047;
        double r104052 = r104050 + r104051;
        double r104053 = r104049 / r104052;
        return r104053;
}

↓

double f(double a, double b, double c, double d) {
        double r104054 = b;
        double r104055 = c;
        double r104056 = r104054 * r104055;
        double r104057 = a;
        double r104058 = d;
        double r104059 = r104057 * r104058;
        double r104060 = r104056 - r104059;
        double r104061 = r104055 * r104055;
        double r104062 = r104058 * r104058;
        double r104063 = r104061 + r104062;
        double r104064 = sqrt(r104063);
        double r104065 = r104060 / r104064;
        double r104066 = r104065 / r104064;
        return r104066;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	25.7
Target	0.5
Herbie	25.7

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

Derivation

1. Initial program 25.7

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

3. Applied add-sqr-sqrt 25.7

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

4. Applied associate-/r* 25.7

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

5. Final simplification 25.7

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

Reproduce

herbie shell --seed 2020047 
(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))))