Complex division, real part

Average Error: 26.1 → 26.0

Time: 9.0s

Precision: 64

Math

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

↓

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

C

double f(double a, double b, double c, double d) {
        double r3305106 = a;
        double r3305107 = c;
        double r3305108 = r3305106 * r3305107;
        double r3305109 = b;
        double r3305110 = d;
        double r3305111 = r3305109 * r3305110;
        double r3305112 = r3305108 + r3305111;
        double r3305113 = r3305107 * r3305107;
        double r3305114 = r3305110 * r3305110;
        double r3305115 = r3305113 + r3305114;
        double r3305116 = r3305112 / r3305115;
        return r3305116;
}

↓

double f(double a, double b, double c, double d) {
        double r3305117 = b;
        double r3305118 = d;
        double r3305119 = r3305117 * r3305118;
        double r3305120 = a;
        double r3305121 = c;
        double r3305122 = r3305120 * r3305121;
        double r3305123 = r3305119 + r3305122;
        double r3305124 = r3305121 * r3305121;
        double r3305125 = r3305118 * r3305118;
        double r3305126 = r3305124 + r3305125;
        double r3305127 = sqrt(r3305126);
        double r3305128 = r3305123 / r3305127;
        double r3305129 = r3305128 / r3305127;
        return r3305129;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	26.1
Target	0.5
Herbie	26.0

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

Derivation

1. Initial program 26.1

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

3. Applied add-sqr-sqrt 26.1

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

4. Applied associate-/r* 26.0

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

5. Final simplification 26.0

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

Reproduce

herbie shell --seed 2019128 
(FPCore (a b c d)
  :name "Complex division, real part"

  :herbie-target
  (if (< (fabs d) (fabs c)) (/ (+ a (* b (/ d c))) (+ c (* d (/ d c)))) (/ (+ b (* a (/ c d))) (+ d (* c (/ c d)))))

  (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))))