Complex division, imag part

Average Error: 26.2 → 26.2

Time: 3.4s

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 r102953 = b;
        double r102954 = c;
        double r102955 = r102953 * r102954;
        double r102956 = a;
        double r102957 = d;
        double r102958 = r102956 * r102957;
        double r102959 = r102955 - r102958;
        double r102960 = r102954 * r102954;
        double r102961 = r102957 * r102957;
        double r102962 = r102960 + r102961;
        double r102963 = r102959 / r102962;
        return r102963;
}

↓

double f(double a, double b, double c, double d) {
        double r102964 = b;
        double r102965 = c;
        double r102966 = r102964 * r102965;
        double r102967 = a;
        double r102968 = d;
        double r102969 = r102967 * r102968;
        double r102970 = r102966 - r102969;
        double r102971 = 1.0;
        double r102972 = r102965 * r102965;
        double r102973 = r102968 * r102968;
        double r102974 = r102972 + r102973;
        double r102975 = sqrt(r102974);
        double r102976 = r102971 / r102975;
        double r102977 = r102970 * r102976;
        double r102978 = r102977 / r102975;
        return r102978;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	26.2
Target	0.5
Herbie	26.2

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

Derivation

1. Initial program 26.2

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

3. Applied add-sqr-sqrt 26.2

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

4. Applied associate-/r* 26.1

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

6. Applied div-inv 26.2

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

7. Final simplification 26.2

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

Reproduce

herbie shell --seed 2020024 
(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))))