Complex division, real part

Average Error: 25.9 → 13.6

Time: 4.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;c \le -7.72381449609585197339080432829350912947 \cdot 10^{66}:\\ \;\;\;\;\frac{-1 \cdot a}{\mathsf{hypot}\left(c, d\right) \cdot 1}\\ \mathbf{elif}\;c \le 2.28477586604815435652148617808763730008 \cdot 10^{70}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right) \cdot 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\mathsf{hypot}\left(c, d\right) \cdot 1}\\ \end{array}\]

C

double f(double a, double b, double c, double d) {
        double r105613 = a;
        double r105614 = c;
        double r105615 = r105613 * r105614;
        double r105616 = b;
        double r105617 = d;
        double r105618 = r105616 * r105617;
        double r105619 = r105615 + r105618;
        double r105620 = r105614 * r105614;
        double r105621 = r105617 * r105617;
        double r105622 = r105620 + r105621;
        double r105623 = r105619 / r105622;
        return r105623;
}

↓

double f(double a, double b, double c, double d) {
        double r105624 = c;
        double r105625 = -7.723814496095852e+66;
        bool r105626 = r105624 <= r105625;
        double r105627 = -1.0;
        double r105628 = a;
        double r105629 = r105627 * r105628;
        double r105630 = d;
        double r105631 = hypot(r105624, r105630);
        double r105632 = 1.0;
        double r105633 = r105631 * r105632;
        double r105634 = r105629 / r105633;
        double r105635 = 2.2847758660481544e+70;
        bool r105636 = r105624 <= r105635;
        double r105637 = b;
        double r105638 = r105637 * r105630;
        double r105639 = fma(r105628, r105624, r105638);
        double r105640 = r105639 / r105631;
        double r105641 = r105640 / r105633;
        double r105642 = r105628 / r105633;
        double r105643 = r105636 ? r105641 : r105642;
        double r105644 = r105626 ? r105634 : r105643;
        return r105644;
}

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	13.6

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

Derivation

1. Split input into 3 regimes

2. if c < -7.723814496095852e+66

   1. Initial program 36.2

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

   3. Applied add-sqr-sqrt 36.2

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

   4. Applied *-un-lft-identity 36.2

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

   5. Applied times-frac 36.2

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

   6. Simplified 36.2

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

   7. Simplified 24.1

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

   9. Applied associate-*r/ 24.1

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

   10. Simplified 24.0

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

   11. Taylor expanded around -inf 18.2

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

   if -7.723814496095852e+66 < c < 2.2847758660481544e+70

   1. Initial program 18.0

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

   3. Applied add-sqr-sqrt 18.0

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

   4. Applied *-un-lft-identity 18.0

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

   5. Applied times-frac 18.0

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

   6. Simplified 18.0

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

   7. Simplified 10.6

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

   9. Applied associate-*r/ 10.5

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

   10. Simplified 10.5

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

   if 2.2847758660481544e+70 < c

   1. Initial program 37.9

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

   3. Applied add-sqr-sqrt 37.9

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

   4. Applied *-un-lft-identity 37.9

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

   5. Applied times-frac 37.9

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

   6. Simplified 37.9

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

   7. Simplified 25.3

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

   9. Applied associate-*r/ 25.3

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

   10. Simplified 25.2

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

   11. Taylor expanded around inf 18.1

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

4. Final simplification 13.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \le -7.72381449609585197339080432829350912947 \cdot 10^{66}:\\ \;\;\;\;\frac{-1 \cdot a}{\mathsf{hypot}\left(c, d\right) \cdot 1}\\ \mathbf{elif}\;c \le 2.28477586604815435652148617808763730008 \cdot 10^{70}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right) \cdot 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\mathsf{hypot}\left(c, d\right) \cdot 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 +o rules:numerics
(FPCore (a b c d)
  :name "Complex division, real part"
  :precision binary64

  :herbie-target
  (if (< (fabs d) (fabs c)) (/ (+ a (* b (/ d c))) (+ c (* d (/ d c)))) (/ (+ b (* a (/ c d))) (+ d (* c (/ c d)))))

  (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))))