Complex division, imag part

Average Error: 26.6 → 13.0

Time: 12.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;c \le -9.626862058130032698667930010363888244365 \cdot 10^{163}:\\ \;\;\;\;\frac{-b}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \le 7.666528689436016414053909814985902231564 \cdot 10^{118}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b, c, -d \cdot a\right)}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\mathsf{hypot}\left(c, d\right)}\\ \end{array}\]

C

double f(double a, double b, double c, double d) {
        double r76817 = b;
        double r76818 = c;
        double r76819 = r76817 * r76818;
        double r76820 = a;
        double r76821 = d;
        double r76822 = r76820 * r76821;
        double r76823 = r76819 - r76822;
        double r76824 = r76818 * r76818;
        double r76825 = r76821 * r76821;
        double r76826 = r76824 + r76825;
        double r76827 = r76823 / r76826;
        return r76827;
}

↓

double f(double a, double b, double c, double d) {
        double r76828 = c;
        double r76829 = -9.626862058130033e+163;
        bool r76830 = r76828 <= r76829;
        double r76831 = b;
        double r76832 = -r76831;
        double r76833 = d;
        double r76834 = hypot(r76828, r76833);
        double r76835 = r76832 / r76834;
        double r76836 = 7.666528689436016e+118;
        bool r76837 = r76828 <= r76836;
        double r76838 = a;
        double r76839 = r76833 * r76838;
        double r76840 = -r76839;
        double r76841 = fma(r76831, r76828, r76840);
        double r76842 = r76841 / r76834;
        double r76843 = r76842 / r76834;
        double r76844 = r76831 / r76834;
        double r76845 = r76837 ? r76843 : r76844;
        double r76846 = r76830 ? r76835 : r76845;
        return r76846;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	26.6
Target	0.5
Herbie	13.0

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

Derivation

1. Split input into 3 regimes

2. if c < -9.626862058130033e+163

   1. Initial program 44.5

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

   3. Applied add-sqr-sqrt 44.5

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

   4. Applied *-un-lft-identity 44.5

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

   5. Applied times-frac 44.5

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

   6. Simplified 44.5

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

   7. Simplified 29.6

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

   9. Applied *-un-lft-identity 29.6

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

   10. Applied add-sqr-sqrt 29.6

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

   11. Applied times-frac 29.6

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

   12. Applied associate-*l* 29.6

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

   13. Simplified 29.5

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

   14. Taylor expanded around -inf 12.3

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

   15. Simplified 12.3

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

   if -9.626862058130033e+163 < c < 7.666528689436016e+118

   1. Initial program 19.9

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

   3. Applied add-sqr-sqrt 19.9

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

   4. Applied *-un-lft-identity 19.9

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

   5. Applied times-frac 19.9

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

   6. Simplified 19.9

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

   7. Simplified 12.6

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

   9. Applied *-un-lft-identity 12.6

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

   10. Applied add-sqr-sqrt 12.6

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

   11. Applied times-frac 12.6

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

   12. Applied associate-*l* 12.6

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

   13. Simplified 12.4

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

   if 7.666528689436016e+118 < c

   1. Initial program 42.3

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

   3. Applied add-sqr-sqrt 42.3

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

   4. Applied *-un-lft-identity 42.3

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

   5. Applied times-frac 42.3

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

   6. Simplified 42.3

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

   7. Simplified 27.9

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

   9. Applied *-un-lft-identity 27.9

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

   10. Applied add-sqr-sqrt 27.9

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

   11. Applied times-frac 27.9

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

   12. Applied associate-*l* 27.9

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

   13. Simplified 27.8

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

   14. Taylor expanded around inf 16.1

       \[\leadsto \frac{\sqrt{1}}{1} \cdot \frac{\color{blue}{b}}{\mathsf{hypot}\left(c, d\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 13.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \le -9.626862058130032698667930010363888244365 \cdot 10^{163}:\\ \;\;\;\;\frac{-b}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \le 7.666528689436016414053909814985902231564 \cdot 10^{118}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b, c, -d \cdot a\right)}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\mathsf{hypot}\left(c, d\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 +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))))