Complex division, real part

Average Error: 26.3 → 13.2

Time: 12.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;c \le -6.62587008307858845004116821212687409398 \cdot 10^{87}:\\ \;\;\;\;\frac{-a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \le 1.897415265281475970584947986850512250588 \cdot 10^{140}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c, b \cdot d\right) \cdot \frac{1}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\mathsf{hypot}\left(c, d\right)}\\ \end{array}\]

C

double f(double a, double b, double c, double d) {
        double r91617 = a;
        double r91618 = c;
        double r91619 = r91617 * r91618;
        double r91620 = b;
        double r91621 = d;
        double r91622 = r91620 * r91621;
        double r91623 = r91619 + r91622;
        double r91624 = r91618 * r91618;
        double r91625 = r91621 * r91621;
        double r91626 = r91624 + r91625;
        double r91627 = r91623 / r91626;
        return r91627;
}

↓

double f(double a, double b, double c, double d) {
        double r91628 = c;
        double r91629 = -6.6258700830785885e+87;
        bool r91630 = r91628 <= r91629;
        double r91631 = a;
        double r91632 = -r91631;
        double r91633 = d;
        double r91634 = hypot(r91628, r91633);
        double r91635 = r91632 / r91634;
        double r91636 = 1.897415265281476e+140;
        bool r91637 = r91628 <= r91636;
        double r91638 = b;
        double r91639 = r91638 * r91633;
        double r91640 = fma(r91631, r91628, r91639);
        double r91641 = 1.0;
        double r91642 = r91641 / r91634;
        double r91643 = r91640 * r91642;
        double r91644 = r91643 / r91634;
        double r91645 = r91631 / r91634;
        double r91646 = r91637 ? r91644 : r91645;
        double r91647 = r91630 ? r91635 : r91646;
        return r91647;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	26.3
Target	0.5
Herbie	13.2

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

Derivation

1. Split input into 3 regimes

2. if c < -6.6258700830785885e+87

   1. Initial program 39.2

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

   3. Applied add-sqr-sqrt 39.2

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

   4. Applied *-un-lft-identity 39.2

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

   5. Applied times-frac 39.2

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

   6. Simplified 39.2

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

   7. Simplified 26.2

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

   9. Applied associate-*r/ 26.2

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

   10. Simplified 26.1

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

   11. Taylor expanded around -inf 18.0

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

   12. Simplified 18.0

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

   if -6.6258700830785885e+87 < c < 1.897415265281476e+140

   1. Initial program 18.9

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

   3. Applied add-sqr-sqrt 18.9

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

   4. Applied *-un-lft-identity 18.9

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

   5. Applied times-frac 18.9

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

   6. Simplified 18.9

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

   7. Simplified 11.9

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

   9. Applied associate-*r/ 11.9

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

   10. Simplified 11.8

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

   12. Applied div-inv 11.9

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

   if 1.897415265281476e+140 < c

   1. Initial program 43.4

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

   3. Applied add-sqr-sqrt 43.4

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

   4. Applied *-un-lft-identity 43.4

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

   5. Applied times-frac 43.4

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

   6. Simplified 43.4

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

   7. Simplified 28.0

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

   9. Applied associate-*r/ 28.0

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

   10. Simplified 28.0

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

   11. Taylor expanded around inf 13.3

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

4. Final simplification 13.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \le -6.62587008307858845004116821212687409398 \cdot 10^{87}:\\ \;\;\;\;\frac{-a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \le 1.897415265281475970584947986850512250588 \cdot 10^{140}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c, b \cdot d\right) \cdot \frac{1}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\mathsf{hypot}\left(c, d\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 +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))))