Complex division, imag part

Average Error: 26.4 → 12.8

Time: 13.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;d \le -2.556682598034695703330250466492269315027 \cdot 10^{120}:\\ \;\;\;\;\frac{a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \le 5.654413025831701701732390598974065197851 \cdot 10^{178}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-d, a, c \cdot b\right)}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{\mathsf{hypot}\left(c, d\right)}\\ \end{array}\]

C

double f(double a, double b, double c, double d) {
        double r78737 = b;
        double r78738 = c;
        double r78739 = r78737 * r78738;
        double r78740 = a;
        double r78741 = d;
        double r78742 = r78740 * r78741;
        double r78743 = r78739 - r78742;
        double r78744 = r78738 * r78738;
        double r78745 = r78741 * r78741;
        double r78746 = r78744 + r78745;
        double r78747 = r78743 / r78746;
        return r78747;
}

↓

double f(double a, double b, double c, double d) {
        double r78748 = d;
        double r78749 = -2.5566825980346957e+120;
        bool r78750 = r78748 <= r78749;
        double r78751 = a;
        double r78752 = c;
        double r78753 = hypot(r78752, r78748);
        double r78754 = r78751 / r78753;
        double r78755 = 5.654413025831702e+178;
        bool r78756 = r78748 <= r78755;
        double r78757 = -r78748;
        double r78758 = b;
        double r78759 = r78752 * r78758;
        double r78760 = fma(r78757, r78751, r78759);
        double r78761 = r78760 / r78753;
        double r78762 = r78761 / r78753;
        double r78763 = -r78751;
        double r78764 = r78763 / r78753;
        double r78765 = r78756 ? r78762 : r78764;
        double r78766 = r78750 ? r78754 : r78765;
        return r78766;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	26.4
Target	0.5
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 d < -2.5566825980346957e+120

   1. Initial program 40.8

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

   3. Applied add-sqr-sqrt 40.8

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

   4. Applied *-un-lft-identity 40.8

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

   5. Applied times-frac 40.8

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

   6. Simplified 40.8

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

   7. Simplified 27.0

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

   9. Applied *-un-lft-identity 27.0

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

   10. Applied associate-*l* 27.0

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

   11. Simplified 27.0

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

   13. Applied *-un-lft-identity 27.0

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

   14. Taylor expanded around -inf 15.4

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

   if -2.5566825980346957e+120 < d < 5.654413025831702e+178

   1. Initial program 20.1

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

   3. Applied add-sqr-sqrt 20.1

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

   4. Applied *-un-lft-identity 20.1

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

   5. Applied times-frac 20.1

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

   6. Simplified 20.1

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

   7. Simplified 12.3

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

   9. Applied *-un-lft-identity 12.3

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

   10. Applied associate-*l* 12.3

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

   11. Simplified 12.1

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

   13. Applied *-un-lft-identity 12.1

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

   if 5.654413025831702e+178 < d

   1. Initial program 45.3

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

   3. Applied add-sqr-sqrt 45.3

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

   4. Applied *-un-lft-identity 45.3

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

   5. Applied times-frac 45.3

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

   6. Simplified 45.3

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

   7. Simplified 32.6

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

   9. Applied *-un-lft-identity 32.6

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

   10. Applied associate-*l* 32.6

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

   11. Simplified 32.6

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

   13. Applied *-un-lft-identity 32.6

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

   14. Taylor expanded around inf 13.3

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

   15. Simplified 13.3

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

4. Final simplification 12.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \le -2.556682598034695703330250466492269315027 \cdot 10^{120}:\\ \;\;\;\;\frac{a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \le 5.654413025831701701732390598974065197851 \cdot 10^{178}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-d, a, c \cdot b\right)}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{\mathsf{hypot}\left(c, d\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(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))))