Complex division, imag part

Average Error: 26.0 → 25.9

Time: 9.2s

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 r254121 = b;
        double r254122 = c;
        double r254123 = r254121 * r254122;
        double r254124 = a;
        double r254125 = d;
        double r254126 = r254124 * r254125;
        double r254127 = r254123 - r254126;
        double r254128 = r254122 * r254122;
        double r254129 = r254125 * r254125;
        double r254130 = r254128 + r254129;
        double r254131 = r254127 / r254130;
        return r254131;
}

↓

double f(double a, double b, double c, double d) {
        double r254132 = b;
        double r254133 = c;
        double r254134 = r254132 * r254133;
        double r254135 = a;
        double r254136 = d;
        double r254137 = r254135 * r254136;
        double r254138 = r254134 - r254137;
        double r254139 = r254133 * r254133;
        double r254140 = r254136 * r254136;
        double r254141 = r254139 + r254140;
        double r254142 = sqrt(r254141);
        double r254143 = r254138 / r254142;
        double r254144 = r254143 / r254142;
        return r254144;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	26.0
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 26.0

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

3. Applied add-sqr-sqrt 26.0

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

4. Applied associate-/r* 25.9

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

5. Final simplification 25.9

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

Reproduce

herbie shell --seed 2019350 
(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))))