Complex division, imag part

Average Error: 25.7 → 25.7

Time: 8.4s

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 r115700 = b;
        double r115701 = c;
        double r115702 = r115700 * r115701;
        double r115703 = a;
        double r115704 = d;
        double r115705 = r115703 * r115704;
        double r115706 = r115702 - r115705;
        double r115707 = r115701 * r115701;
        double r115708 = r115704 * r115704;
        double r115709 = r115707 + r115708;
        double r115710 = r115706 / r115709;
        return r115710;
}

↓

double f(double a, double b, double c, double d) {
        double r115711 = b;
        double r115712 = c;
        double r115713 = r115711 * r115712;
        double r115714 = a;
        double r115715 = d;
        double r115716 = r115714 * r115715;
        double r115717 = r115713 - r115716;
        double r115718 = r115712 * r115712;
        double r115719 = r115715 * r115715;
        double r115720 = r115718 + r115719;
        double r115721 = sqrt(r115720);
        double r115722 = r115717 / r115721;
        double r115723 = r115722 / r115721;
        return r115723;
}

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))))