Complex division, imag part

Average Error: 25.8 → 25.2

Time: 10.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \le 3.0586257786179306 \cdot 10^{292}:\\ \;\;\;\;\frac{\frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\sqrt{c \cdot c + d \cdot d}}\\ \end{array}\]

C

double f(double a, double b, double c, double d) {
        double r86848 = b;
        double r86849 = c;
        double r86850 = r86848 * r86849;
        double r86851 = a;
        double r86852 = d;
        double r86853 = r86851 * r86852;
        double r86854 = r86850 - r86853;
        double r86855 = r86849 * r86849;
        double r86856 = r86852 * r86852;
        double r86857 = r86855 + r86856;
        double r86858 = r86854 / r86857;
        return r86858;
}

↓

double f(double a, double b, double c, double d) {
        double r86859 = b;
        double r86860 = c;
        double r86861 = r86859 * r86860;
        double r86862 = a;
        double r86863 = d;
        double r86864 = r86862 * r86863;
        double r86865 = r86861 - r86864;
        double r86866 = r86860 * r86860;
        double r86867 = r86863 * r86863;
        double r86868 = r86866 + r86867;
        double r86869 = r86865 / r86868;
        double r86870 = 3.0586257786179306e+292;
        bool r86871 = r86869 <= r86870;
        double r86872 = sqrt(r86868);
        double r86873 = r86865 / r86872;
        double r86874 = r86873 / r86872;
        double r86875 = r86859 / r86872;
        double r86876 = r86871 ? r86874 : r86875;
        return r86876;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	25.8
Target	0.5
Herbie	25.2

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

Derivation

1. Split input into 2 regimes

2. if (/ (- (* b c) (* a d)) (+ (* c c) (* d d))) < 3.0586257786179306e+292

   1. Initial program 13.9

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

   3. Applied add-sqr-sqrt 13.9

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

   4. Applied associate-/r* 13.8

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

   if 3.0586257786179306e+292 < (/ (- (* b c) (* a d)) (+ (* c c) (* d d)))

   1. Initial program 63.0

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

   3. Applied add-sqr-sqrt 63.0

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

   4. Applied associate-/r* 63.0

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

   5. Taylor expanded around inf 60.6

      \[\leadsto \frac{\color{blue}{b}}{\sqrt{c \cdot c + d \cdot d}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 25.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \le 3.0586257786179306 \cdot 10^{292}:\\ \;\;\;\;\frac{\frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\sqrt{c \cdot c + d \cdot d}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019198 
(FPCore (a b c d)
  :name "Complex division, imag part"

  :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))))