Complex division, imag part

Average Error: 25.9 → 25.9

Time: 14.5s

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 r70321 = b;
        double r70322 = c;
        double r70323 = r70321 * r70322;
        double r70324 = a;
        double r70325 = d;
        double r70326 = r70324 * r70325;
        double r70327 = r70323 - r70326;
        double r70328 = r70322 * r70322;
        double r70329 = r70325 * r70325;
        double r70330 = r70328 + r70329;
        double r70331 = r70327 / r70330;
        return r70331;
}

↓

double f(double a, double b, double c, double d) {
        double r70332 = b;
        double r70333 = c;
        double r70334 = r70332 * r70333;
        double r70335 = a;
        double r70336 = d;
        double r70337 = r70335 * r70336;
        double r70338 = r70334 - r70337;
        double r70339 = 1.0;
        double r70340 = r70333 * r70333;
        double r70341 = r70336 * r70336;
        double r70342 = r70340 + r70341;
        double r70343 = sqrt(r70342);
        double r70344 = r70339 / r70343;
        double r70345 = r70338 * r70344;
        double r70346 = r70345 / r70343;
        return r70346;
}

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))))