Complex division, real part

Average Error: 26.2 → 13.2

Time: 13.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;c \le -6.99838818051005 \cdot 10^{+73}:\\ \;\;\;\;\frac{-a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \le 1.938821746780901 \cdot 10^{+120}:\\ \;\;\;\;\frac{\frac{1}{\frac{\mathsf{hypot}\left(c, d\right)}{\mathsf{fma}\left(a, c, d \cdot b\right)}}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\mathsf{hypot}\left(c, d\right)}\\ \end{array}\]

C

double f(double a, double b, double c, double d) {
        double r1989581 = a;
        double r1989582 = c;
        double r1989583 = r1989581 * r1989582;
        double r1989584 = b;
        double r1989585 = d;
        double r1989586 = r1989584 * r1989585;
        double r1989587 = r1989583 + r1989586;
        double r1989588 = r1989582 * r1989582;
        double r1989589 = r1989585 * r1989585;
        double r1989590 = r1989588 + r1989589;
        double r1989591 = r1989587 / r1989590;
        return r1989591;
}

↓

double f(double a, double b, double c, double d) {
        double r1989592 = c;
        double r1989593 = -6.99838818051005e+73;
        bool r1989594 = r1989592 <= r1989593;
        double r1989595 = a;
        double r1989596 = -r1989595;
        double r1989597 = d;
        double r1989598 = hypot(r1989592, r1989597);
        double r1989599 = r1989596 / r1989598;
        double r1989600 = 1.938821746780901e+120;
        bool r1989601 = r1989592 <= r1989600;
        double r1989602 = 1.0;
        double r1989603 = b;
        double r1989604 = r1989597 * r1989603;
        double r1989605 = fma(r1989595, r1989592, r1989604);
        double r1989606 = r1989598 / r1989605;
        double r1989607 = r1989602 / r1989606;
        double r1989608 = r1989607 / r1989598;
        double r1989609 = r1989595 / r1989598;
        double r1989610 = r1989601 ? r1989608 : r1989609;
        double r1989611 = r1989594 ? r1989599 : r1989610;
        return r1989611;
}

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	13.2

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

Derivation

1. Split input into 3 regimes

2. if c < -6.99838818051005e+73

   1. Initial program 37.1

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

   3. Applied add-sqr-sqrt 37.1

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

   4. Applied associate-/r* 37.0

      \[\leadsto \color{blue}{\frac{\frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}}\]
   5. Using strategy rm

   6. Applied hypot-def 37.0

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

   7. Taylor expanded around -inf 17.1

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

   8. Simplified 17.1

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

   if -6.99838818051005e+73 < c < 1.938821746780901e+120

   1. Initial program 18.7

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

   3. Applied add-sqr-sqrt 18.7

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

   4. Applied associate-/r* 18.6

      \[\leadsto \color{blue}{\frac{\frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}}\]
   5. Using strategy rm

   6. Applied hypot-def 18.6

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

   8. Applied clear-num 18.7

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

   9. Simplified 11.6

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

   if 1.938821746780901e+120 < c

   1. Initial program 41.2

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

   3. Applied add-sqr-sqrt 41.2

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

   4. Applied associate-/r* 41.1

      \[\leadsto \color{blue}{\frac{\frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}}\]
   5. Using strategy rm

   6. Applied hypot-def 41.1

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

   7. Taylor expanded around inf 14.6

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

4. Final simplification 13.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \le -6.99838818051005 \cdot 10^{+73}:\\ \;\;\;\;\frac{-a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \le 1.938821746780901 \cdot 10^{+120}:\\ \;\;\;\;\frac{\frac{1}{\frac{\mathsf{hypot}\left(c, d\right)}{\mathsf{fma}\left(a, c, d \cdot b\right)}}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\mathsf{hypot}\left(c, d\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 +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))))