Complex division, imag part

Average Error: 25.7 → 12.8

Time: 13.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;c \le -1.565167565781187104932955349587597625361 \cdot 10^{162}:\\ \;\;\;\;\frac{-b}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \le 6.152282888480768325547605387077337858415 \cdot 10^{76}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(c, b, \left(-a\right) \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\mathsf{hypot}\left(c, d\right)}\\ \end{array}\]

C

double f(double a, double b, double c, double d) {
        double r83295 = b;
        double r83296 = c;
        double r83297 = r83295 * r83296;
        double r83298 = a;
        double r83299 = d;
        double r83300 = r83298 * r83299;
        double r83301 = r83297 - r83300;
        double r83302 = r83296 * r83296;
        double r83303 = r83299 * r83299;
        double r83304 = r83302 + r83303;
        double r83305 = r83301 / r83304;
        return r83305;
}

↓

double f(double a, double b, double c, double d) {
        double r83306 = c;
        double r83307 = -1.5651675657811871e+162;
        bool r83308 = r83306 <= r83307;
        double r83309 = b;
        double r83310 = -r83309;
        double r83311 = d;
        double r83312 = hypot(r83306, r83311);
        double r83313 = r83310 / r83312;
        double r83314 = 6.152282888480768e+76;
        bool r83315 = r83306 <= r83314;
        double r83316 = a;
        double r83317 = -r83316;
        double r83318 = r83317 * r83311;
        double r83319 = fma(r83306, r83309, r83318);
        double r83320 = r83319 / r83312;
        double r83321 = r83320 / r83312;
        double r83322 = r83309 / r83312;
        double r83323 = r83315 ? r83321 : r83322;
        double r83324 = r83308 ? r83313 : r83323;
        return r83324;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	25.7
Target	0.3
Herbie	12.8

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

Derivation

1. Split input into 3 regimes

2. if c < -1.5651675657811871e+162

   1. Initial program 45.9

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

   3. Applied add-sqr-sqrt 45.9

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

   4. Applied *-un-lft-identity 45.9

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

   5. Applied times-frac 45.9

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

   6. Simplified 45.9

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

   7. Simplified 30.7

      \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \color{blue}{\frac{\mathsf{fma}\left(b, c, -d \cdot a\right)}{\mathsf{hypot}\left(c, d\right)}}\]
   8. Using strategy rm

   9. Applied associate-*r/ 30.7

      \[\leadsto \color{blue}{\frac{\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \mathsf{fma}\left(b, c, -d \cdot a\right)}{\mathsf{hypot}\left(c, d\right)}}\]

   10. Simplified 30.7

       \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(c, b, \left(-a\right) \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}}}{\mathsf{hypot}\left(c, d\right)}\]

   11. Taylor expanded around -inf 13.3

       \[\leadsto \frac{\color{blue}{-1 \cdot b}}{\mathsf{hypot}\left(c, d\right)}\]

   12. Simplified 13.3

       \[\leadsto \frac{\color{blue}{-b}}{\mathsf{hypot}\left(c, d\right)}\]

   if -1.5651675657811871e+162 < c < 6.152282888480768e+76

   1. Initial program 18.7

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

   3. Applied add-sqr-sqrt 18.7

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

   4. Applied *-un-lft-identity 18.7

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

   5. Applied times-frac 18.7

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

   6. Simplified 18.7

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

   7. Simplified 11.4

      \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \color{blue}{\frac{\mathsf{fma}\left(b, c, -d \cdot a\right)}{\mathsf{hypot}\left(c, d\right)}}\]
   8. Using strategy rm

   9. Applied associate-*r/ 11.4

      \[\leadsto \color{blue}{\frac{\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \mathsf{fma}\left(b, c, -d \cdot a\right)}{\mathsf{hypot}\left(c, d\right)}}\]

   10. Simplified 11.3

       \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(c, b, \left(-a\right) \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}}}{\mathsf{hypot}\left(c, d\right)}\]

   if 6.152282888480768e+76 < c

   1. Initial program 37.3

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

   3. Applied add-sqr-sqrt 37.3

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

   4. Applied *-un-lft-identity 37.3

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

   5. Applied times-frac 37.3

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

   6. Simplified 37.3

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

   7. Simplified 26.1

      \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \color{blue}{\frac{\mathsf{fma}\left(b, c, -d \cdot a\right)}{\mathsf{hypot}\left(c, d\right)}}\]
   8. Using strategy rm

   9. Applied associate-*r/ 26.1

      \[\leadsto \color{blue}{\frac{\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \mathsf{fma}\left(b, c, -d \cdot a\right)}{\mathsf{hypot}\left(c, d\right)}}\]

   10. Simplified 26.1

       \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(c, b, \left(-a\right) \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}}}{\mathsf{hypot}\left(c, d\right)}\]

   11. Taylor expanded around inf 17.8

       \[\leadsto \frac{\color{blue}{b}}{\mathsf{hypot}\left(c, d\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 12.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \le -1.565167565781187104932955349587597625361 \cdot 10^{162}:\\ \;\;\;\;\frac{-b}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \le 6.152282888480768325547605387077337858415 \cdot 10^{76}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(c, b, \left(-a\right) \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\mathsf{hypot}\left(c, d\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019209 +o rules:numerics
(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))))