Complex division, imag part

Average Error: 27.0 → 24.1

Time: 3.7s

Precision: binary64

Math

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

↓

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

C

double code(double a, double b, double c, double d) {
	return (((double) (((double) (b * c)) - ((double) (a * d)))) / ((double) (((double) (c * c)) + ((double) (d * d)))));
}

↓

double code(double a, double b, double c, double d) {
	return ((double) ((1.0 / ((double) sqrt(((double) (((double) (c * c)) + ((double) (d * d))))))) * ((double) (((double) (c * (b / ((double) sqrt(((double) (((double) (c * c)) + ((double) (d * d))))))))) - ((double) (d * (a / ((double) sqrt(((double) (((double) (c * c)) + ((double) (d * d)))))))))))));
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	27.0
Target	0.4
Herbie	24.1

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

Derivation

1. Initial program 27.0

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

3. Applied add-sqr-sqrt 27.0

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

4. Applied *-un-lft-identity 27.0

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

5. Applied times-frac 27.0

   \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]
6. Using strategy rm

7. Applied div-sub 27.0

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

8. Simplified 25.5

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

9. Simplified 24.1

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

10. Final simplification 24.1

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

Reproduce

herbie shell --seed 2020196 
(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))))