Complex division, real part

Average Error: 25.9 → 14.0

Time: 11.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;c \le -1.236056920632649636312988321949767458809 \cdot 10^{252}:\\ \;\;\;\;-\frac{a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \le -1.972049955288829430907028189059339655129 \cdot 10^{188}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{1}{\frac{\mathsf{hypot}\left(c, d\right)}{\mathsf{fma}\left(a, c, b \cdot d\right)}}\\ \mathbf{elif}\;c \le -9.931482377190183189903426181814662287963 \cdot 10^{103}:\\ \;\;\;\;-\frac{a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \le 7.28777456571337501443708414977136635674 \cdot 10^{101}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\mathsf{hypot}\left(c, d\right)}\\ \end{array}\]

C

double f(double a, double b, double c, double d) {
        double r78430 = a;
        double r78431 = c;
        double r78432 = r78430 * r78431;
        double r78433 = b;
        double r78434 = d;
        double r78435 = r78433 * r78434;
        double r78436 = r78432 + r78435;
        double r78437 = r78431 * r78431;
        double r78438 = r78434 * r78434;
        double r78439 = r78437 + r78438;
        double r78440 = r78436 / r78439;
        return r78440;
}

↓

double f(double a, double b, double c, double d) {
        double r78441 = c;
        double r78442 = -1.2360569206326496e+252;
        bool r78443 = r78441 <= r78442;
        double r78444 = a;
        double r78445 = d;
        double r78446 = hypot(r78441, r78445);
        double r78447 = r78444 / r78446;
        double r78448 = -r78447;
        double r78449 = -1.9720499552888294e+188;
        bool r78450 = r78441 <= r78449;
        double r78451 = 1.0;
        double r78452 = r78451 / r78446;
        double r78453 = b;
        double r78454 = r78453 * r78445;
        double r78455 = fma(r78444, r78441, r78454);
        double r78456 = r78446 / r78455;
        double r78457 = r78451 / r78456;
        double r78458 = r78452 * r78457;
        double r78459 = -9.931482377190183e+103;
        bool r78460 = r78441 <= r78459;
        double r78461 = 7.287774565713375e+101;
        bool r78462 = r78441 <= r78461;
        double r78463 = r78455 / r78446;
        double r78464 = r78463 / r78446;
        double r78465 = r78462 ? r78464 : r78447;
        double r78466 = r78460 ? r78448 : r78465;
        double r78467 = r78450 ? r78458 : r78466;
        double r78468 = r78443 ? r78448 : r78467;
        return r78468;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	25.9
Target	0.4
Herbie	14.0

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

Derivation

1. Split input into 4 regimes

2. if c < -1.2360569206326496e+252 or -1.9720499552888294e+188 < c < -9.931482377190183e+103

   1. Initial program 36.0

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

   3. Applied add-sqr-sqrt 36.0

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

   4. Applied *-un-lft-identity 36.0

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

   5. Applied times-frac 36.0

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

   6. Simplified 36.0

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

   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 16.2

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

   12. Simplified 16.2

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

   if -1.2360569206326496e+252 < c < -1.9720499552888294e+188

   1. Initial program 45.3

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

   3. Applied add-sqr-sqrt 45.3

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

   4. Applied *-un-lft-identity 45.3

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

   5. Applied times-frac 45.3

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

   6. Simplified 45.3

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

   7. Simplified 27.9

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

   9. Applied clear-num 28.0

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

   if -9.931482377190183e+103 < c < 7.287774565713375e+101

   1. Initial program 18.5

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

   3. Applied add-sqr-sqrt 18.5

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

   4. Applied *-un-lft-identity 18.5

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

   5. Applied times-frac 18.5

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

   6. Simplified 18.5

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

   7. Simplified 11.6

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

   9. Applied associate-*r/ 11.5

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

   10. Simplified 11.4

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

   if 7.287774565713375e+101 < c

   1. Initial program 40.0

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

   3. Applied add-sqr-sqrt 40.0

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

   4. Applied *-un-lft-identity 40.0

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

   5. Applied times-frac 40.0

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

   6. Simplified 40.0

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

   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 17.8

       \[\leadsto \frac{\color{blue}{a}}{\mathsf{hypot}\left(c, d\right)}\]
3. Recombined 4 regimes into one program.

4. Final simplification 14.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \le -1.236056920632649636312988321949767458809 \cdot 10^{252}:\\ \;\;\;\;-\frac{a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \le -1.972049955288829430907028189059339655129 \cdot 10^{188}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{1}{\frac{\mathsf{hypot}\left(c, d\right)}{\mathsf{fma}\left(a, c, b \cdot d\right)}}\\ \mathbf{elif}\;c \le -9.931482377190183189903426181814662287963 \cdot 10^{103}:\\ \;\;\;\;-\frac{a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \le 7.28777456571337501443708414977136635674 \cdot 10^{101}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\mathsf{hypot}\left(c, d\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (a b c d)
  :name "Complex division, real part"

  :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))))