Complex division, real part

Average Error: 26.2 → 13.4

Time: 17.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;c \le -6.019013903669939680914943016175030615548 \cdot 10^{211}:\\ \;\;\;\;\frac{-a}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{elif}\;c \le 8.140603411282212805110303745543003453499 \cdot 10^{83}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{hypot}\left(d, c\right)}}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\mathsf{hypot}\left(d, c\right)}\\ \end{array}\]

C

double f(double a, double b, double c, double d) {
        double r77936 = a;
        double r77937 = c;
        double r77938 = r77936 * r77937;
        double r77939 = b;
        double r77940 = d;
        double r77941 = r77939 * r77940;
        double r77942 = r77938 + r77941;
        double r77943 = r77937 * r77937;
        double r77944 = r77940 * r77940;
        double r77945 = r77943 + r77944;
        double r77946 = r77942 / r77945;
        return r77946;
}

↓

double f(double a, double b, double c, double d) {
        double r77947 = c;
        double r77948 = -6.01901390366994e+211;
        bool r77949 = r77947 <= r77948;
        double r77950 = a;
        double r77951 = -r77950;
        double r77952 = d;
        double r77953 = hypot(r77952, r77947);
        double r77954 = r77951 / r77953;
        double r77955 = 8.140603411282213e+83;
        bool r77956 = r77947 <= r77955;
        double r77957 = b;
        double r77958 = r77952 * r77957;
        double r77959 = fma(r77950, r77947, r77958);
        double r77960 = r77959 / r77953;
        double r77961 = r77960 / r77953;
        double r77962 = r77950 / r77953;
        double r77963 = r77956 ? r77961 : r77962;
        double r77964 = r77949 ? r77954 : r77963;
        return r77964;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	26.2
Target	0.4
Herbie	13.4

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

Derivation

1. Split input into 3 regimes

2. if c < -6.01901390366994e+211

   1. Initial program 42.4

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

   2. Simplified 42.4

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

   4. Applied add-sqr-sqrt 42.4

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

   5. Applied *-un-lft-identity 42.4

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

   6. Applied times-frac 42.4

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

   7. Simplified 42.4

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

   8. Simplified 31.7

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

   10. Applied *-un-lft-identity 31.7

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

   11. Applied associate-*l* 31.7

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

   12. Simplified 31.7

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

   13. Taylor expanded around -inf 10.9

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

   14. Simplified 10.9

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

   if -6.01901390366994e+211 < c < 8.140603411282213e+83

   1. Initial program 21.0

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

   2. Simplified 21.0

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

   4. Applied add-sqr-sqrt 21.0

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

   5. Applied *-un-lft-identity 21.0

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

   6. Applied times-frac 21.0

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

   7. Simplified 21.0

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

   8. Simplified 12.7

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

   10. Applied *-un-lft-identity 12.7

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

   11. Applied associate-*l* 12.7

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

   12. Simplified 12.6

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

   if 8.140603411282213e+83 < c

   1. Initial program 39.9

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

   2. Simplified 39.9

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

   4. Applied add-sqr-sqrt 39.9

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

   5. Applied *-un-lft-identity 39.9

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

   6. Applied times-frac 39.9

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

   7. Simplified 39.9

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

   8. Simplified 27.4

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

   10. Applied *-un-lft-identity 27.4

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

   11. Applied associate-*l* 27.4

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

   12. Simplified 27.3

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

   13. Taylor expanded around inf 17.5

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

4. Final simplification 13.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \le -6.019013903669939680914943016175030615548 \cdot 10^{211}:\\ \;\;\;\;\frac{-a}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{elif}\;c \le 8.140603411282212805110303745543003453499 \cdot 10^{83}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{hypot}\left(d, c\right)}}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\mathsf{hypot}\left(d, c\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +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))))