Complex division, real part

Average Error: 26.3 → 13.1

Time: 11.4s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;c \le -6.75150761769019798 \cdot 10^{142}:\\ \;\;\;\;\frac{-a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \le 1.8955542797321017 \cdot 10^{121}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\mathsf{hypot}\left(c, d\right)}\\ \end{array}\]

C

double f(double a, double b, double c, double d) {
        double r121028 = a;
        double r121029 = c;
        double r121030 = r121028 * r121029;
        double r121031 = b;
        double r121032 = d;
        double r121033 = r121031 * r121032;
        double r121034 = r121030 + r121033;
        double r121035 = r121029 * r121029;
        double r121036 = r121032 * r121032;
        double r121037 = r121035 + r121036;
        double r121038 = r121034 / r121037;
        return r121038;
}

↓

double f(double a, double b, double c, double d) {
        double r121039 = c;
        double r121040 = -6.751507617690198e+142;
        bool r121041 = r121039 <= r121040;
        double r121042 = a;
        double r121043 = -r121042;
        double r121044 = d;
        double r121045 = hypot(r121039, r121044);
        double r121046 = r121043 / r121045;
        double r121047 = 1.8955542797321017e+121;
        bool r121048 = r121039 <= r121047;
        double r121049 = 1.0;
        double r121050 = r121049 / r121045;
        double r121051 = b;
        double r121052 = r121051 * r121044;
        double r121053 = fma(r121042, r121039, r121052);
        double r121054 = r121053 / r121045;
        double r121055 = r121050 * r121054;
        double r121056 = r121042 / r121045;
        double r121057 = r121048 ? r121055 : r121056;
        double r121058 = r121041 ? r121046 : r121057;
        return r121058;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	26.3
Target	0.4
Herbie	13.1

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

Derivation

1. Split input into 3 regimes

2. if c < -6.751507617690198e+142

   1. Initial program 43.0

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

   2. Simplified 43.0

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

   4. Applied add-sqr-sqrt 43.0

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

   5. Applied *-un-lft-identity 43.0

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

   6. Applied times-frac 43.0

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

   7. Simplified 43.0

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

   8. Simplified 28.4

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

   10. Applied *-un-lft-identity 28.4

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

   11. Applied associate-*l* 28.4

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

   12. Simplified 28.3

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

   13. Taylor expanded around -inf 12.8

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

   14. Simplified 12.8

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

   if -6.751507617690198e+142 < c < 1.8955542797321017e+121

   1. Initial program 19.3

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

   2. Simplified 19.3

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

   4. Applied add-sqr-sqrt 19.3

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

   5. Applied *-un-lft-identity 19.3

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

   6. Applied times-frac 19.3

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

   7. Simplified 19.3

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

   8. Simplified 12.7

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

   10. Applied *-un-lft-identity 12.7

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

   11. Applied associate-*l* 12.7

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

   12. Simplified 12.6

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

   14. Applied add-sqr-sqrt 12.9

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

   15. Applied add-sqr-sqrt 13.0

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

   16. Applied *-un-lft-identity 13.0

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

   17. Applied times-frac 13.1

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

   18. Applied times-frac 13.1

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

   19. Simplified 12.9

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

   20. Simplified 12.7

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

   if 1.8955542797321017e+121 < c

   1. Initial program 41.9

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

   2. Simplified 41.9

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

   4. Applied add-sqr-sqrt 41.9

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

   5. Applied *-un-lft-identity 41.9

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

   6. Applied times-frac 41.9

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

   7. Simplified 41.9

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

   8. Simplified 27.9

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

   10. Applied *-un-lft-identity 27.9

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

   11. Applied associate-*l* 27.9

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

   12. Simplified 27.8

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

   13. Taylor expanded around inf 14.7

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

4. Final simplification 13.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \le -6.75150761769019798 \cdot 10^{142}:\\ \;\;\;\;\frac{-a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \le 1.8955542797321017 \cdot 10^{121}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\mathsf{hypot}\left(c, d\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +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))))