Complex division, imag part

Average Error: 25.9 → 25.2

Time: 8.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \le 1.960885555577793652414476012337475353598 \cdot 10^{292}:\\ \;\;\;\;\frac{\frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\sqrt{c \cdot c + d \cdot d}}\\ \end{array}\]

C

double f(double a, double b, double c, double d) {
        double r76323 = b;
        double r76324 = c;
        double r76325 = r76323 * r76324;
        double r76326 = a;
        double r76327 = d;
        double r76328 = r76326 * r76327;
        double r76329 = r76325 - r76328;
        double r76330 = r76324 * r76324;
        double r76331 = r76327 * r76327;
        double r76332 = r76330 + r76331;
        double r76333 = r76329 / r76332;
        return r76333;
}

↓

double f(double a, double b, double c, double d) {
        double r76334 = b;
        double r76335 = c;
        double r76336 = r76334 * r76335;
        double r76337 = a;
        double r76338 = d;
        double r76339 = r76337 * r76338;
        double r76340 = r76336 - r76339;
        double r76341 = r76335 * r76335;
        double r76342 = r76338 * r76338;
        double r76343 = r76341 + r76342;
        double r76344 = r76340 / r76343;
        double r76345 = 1.9608855555777937e+292;
        bool r76346 = r76344 <= r76345;
        double r76347 = sqrt(r76343);
        double r76348 = r76340 / r76347;
        double r76349 = r76348 / r76347;
        double r76350 = r76334 / r76347;
        double r76351 = r76346 ? r76349 : r76350;
        return r76351;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	25.9
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))) < 1.9608855555777937e+292

   1. Initial program 14.3

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

   3. Applied add-sqr-sqrt 14.3

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

   4. Applied associate-/r* 14.2

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

   if 1.9608855555777937e+292 < (/ (- (* b c) (* a d)) (+ (* c c) (* d d)))

   1. Initial program 62.7

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

   3. Applied add-sqr-sqrt 62.7

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

   4. Applied associate-/r* 62.7

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

   5. Taylor expanded around inf 60.1

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

4. Final simplification 25.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \le 1.960885555577793652414476012337475353598 \cdot 10^{292}:\\ \;\;\;\;\frac{\frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\sqrt{c \cdot c + d \cdot d}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019298 
(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))))