powComplex, real part

Average Error: 33.4 → 9.6

Time: 12.5s

Precision: binary64

Math

\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]

↓

\[\begin{array}{l} \mathbf{if}\;x.im \leq -326275111805596.6:\\ \;\;\;\;e^{\log \left(-x.im\right) \cdot y.re - \left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}} \cdot \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}\right) \cdot \left(y.im \cdot \left(\sqrt[3]{\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \left(\sqrt[3]{\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sqrt[3]{\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}}\right)\right)\right)} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;x.im \leq -1.0023614876036504 \cdot 10^{-65}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;x.im \leq 4.3354624121682 \cdot 10^{-310}:\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{\log \left(-x.im\right) \cdot y.re - \left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}} \cdot \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}\right) \cdot \left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}} \cdot y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log x.im - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array}\]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (*
  (exp
   (-
    (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re)
    (* (atan2 x.im x.re) y.im)))
  (cos
   (+
    (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.im)
    (* (atan2 x.im x.re) y.re)))))

↓

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.im -326275111805596.6)
   (*
    (exp
     (-
      (* (log (- x.im)) y.re)
      (*
       (* (cbrt (atan2 x.im x.re)) (cbrt (atan2 x.im x.re)))
       (*
        y.im
        (*
         (cbrt (cbrt (atan2 x.im x.re)))
         (*
          (cbrt (cbrt (atan2 x.im x.re)))
          (cbrt (cbrt (atan2 x.im x.re)))))))))
    (cos (* y.re (atan2 x.im x.re))))
   (if (<= x.im -1.0023614876036504e-65)
     (exp
      (-
       (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im)))))
       (* (atan2 x.im x.re) y.im)))
     (if (<= x.im 4.3354624121682e-310)
       (*
        (cos (* y.re (atan2 x.im x.re)))
        (exp
         (-
          (* (log (- x.im)) y.re)
          (*
           (* (cbrt (atan2 x.im x.re)) (cbrt (atan2 x.im x.re)))
           (* (cbrt (atan2 x.im x.re)) y.im)))))
       (*
        (cos (* y.re (atan2 x.im x.re)))
        (exp (- (* y.re (log x.im)) (* (atan2 x.im x.re) y.im))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return exp((log(sqrt((x_46_re * x_46_re) + (x_46_im * x_46_im))) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im)) * cos((log(sqrt((x_46_re * x_46_re) + (x_46_im * x_46_im))) * y_46_im) + (atan2(x_46_im, x_46_re) * y_46_re));
}

↓

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_im <= -326275111805596.6) {
		tmp = exp((log(-x_46_im) * y_46_re) - ((cbrt(atan2(x_46_im, x_46_re)) * cbrt(atan2(x_46_im, x_46_re))) * (y_46_im * (cbrt(cbrt(atan2(x_46_im, x_46_re))) * (cbrt(cbrt(atan2(x_46_im, x_46_re))) * cbrt(cbrt(atan2(x_46_im, x_46_re)))))))) * cos(y_46_re * atan2(x_46_im, x_46_re));
	} else if (x_46_im <= -1.0023614876036504e-65) {
		tmp = exp((y_46_re * log(sqrt((x_46_re * x_46_re) + (x_46_im * x_46_im)))) - (atan2(x_46_im, x_46_re) * y_46_im));
	} else if (x_46_im <= 4.3354624121682e-310) {
		tmp = cos(y_46_re * atan2(x_46_im, x_46_re)) * exp((log(-x_46_im) * y_46_re) - ((cbrt(atan2(x_46_im, x_46_re)) * cbrt(atan2(x_46_im, x_46_re))) * (cbrt(atan2(x_46_im, x_46_re)) * y_46_im)));
	} else {
		tmp = cos(y_46_re * atan2(x_46_im, x_46_re)) * exp((y_46_re * log(x_46_im)) - (atan2(x_46_im, x_46_re) * y_46_im));
	}
	return tmp;
}

Error

Bits error versus x.re

Bits error versus x.im

Bits error versus y.re

Bits error versus y.im

Derivation

1. Split input into 4 regimes

2. if x.im < -326275111805596.62

   1. Initial program 41.8

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]

   2. Taylor expanded around 0 23.3

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\]

   3. Taylor expanded around -inf 1.6

      \[\leadsto e^{\log \color{blue}{\left(-1 \cdot x.im\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\]

   4. Simplified 1.6

      \[\leadsto e^{\log \color{blue}{\left(-x.im\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\]
   5. Using strategy rm

   6. Applied add-cube-cbrt_binary64 1.6

      \[\leadsto e^{\log \left(-x.im\right) \cdot y.re - \color{blue}{\left(\left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}} \cdot \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}\right) \cdot \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}\right)} \cdot y.im} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\]

   7. Applied associate-*l*_binary64 1.6

      \[\leadsto e^{\log \left(-x.im\right) \cdot y.re - \color{blue}{\left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}} \cdot \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}\right) \cdot \left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}} \cdot y.im\right)}} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\]

   8. Simplified 1.6

      \[\leadsto e^{\log \left(-x.im\right) \cdot y.re - \left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}} \cdot \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}\right) \cdot \color{blue}{\left(y.im \cdot \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}\right)}} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\]
   9. Using strategy rm

   10. Applied add-cube-cbrt_binary64 1.6

       \[\leadsto e^{\log \left(-x.im\right) \cdot y.re - \left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}} \cdot \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}\right) \cdot \left(y.im \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sqrt[3]{\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}}\right) \cdot \sqrt[3]{\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}}\right)}\right)} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\]

   if -326275111805596.62 < x.im < -1.00236148760365044e-65

   1. Initial program 20.9

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]

   2. Taylor expanded around 0 12.1

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\]

   3. Taylor expanded around 0 12.0

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{1}\]

   if -1.00236148760365044e-65 < x.im < 4.33546241216819e-310

   1. Initial program 28.1

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]

   2. Taylor expanded around 0 17.3

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\]

   3. Taylor expanded around -inf 17.8

      \[\leadsto e^{\log \color{blue}{\left(-1 \cdot x.im\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\]

   4. Simplified 17.8

      \[\leadsto e^{\log \color{blue}{\left(-x.im\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\]
   5. Using strategy rm

   6. Applied add-cube-cbrt_binary64 17.8

      \[\leadsto e^{\log \left(-x.im\right) \cdot y.re - \color{blue}{\left(\left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}} \cdot \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}\right) \cdot \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}\right)} \cdot y.im} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\]

   7. Applied associate-*l*_binary64 17.8

      \[\leadsto e^{\log \left(-x.im\right) \cdot y.re - \color{blue}{\left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}} \cdot \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}\right) \cdot \left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}} \cdot y.im\right)}} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\]

   if 4.33546241216819e-310 < x.im

   1. Initial program 33.2

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]

   2. Taylor expanded around 0 20.0

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\]

   3. Taylor expanded around 0 9.6

      \[\leadsto e^{\log \color{blue}{x.im} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\]
3. Recombined 4 regimes into one program.

4. Final simplification 9.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -326275111805596.6:\\ \;\;\;\;e^{\log \left(-x.im\right) \cdot y.re - \left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}} \cdot \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}\right) \cdot \left(y.im \cdot \left(\sqrt[3]{\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \left(\sqrt[3]{\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sqrt[3]{\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}}\right)\right)\right)} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;x.im \leq -1.0023614876036504 \cdot 10^{-65}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;x.im \leq 4.3354624121682 \cdot 10^{-310}:\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{\log \left(-x.im\right) \cdot y.re - \left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}} \cdot \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}\right) \cdot \left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}} \cdot y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log x.im - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(FPCore (x.re x.im y.re y.im)
  :name "powComplex, real part"
  :precision binary64
  (* (exp (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) (* (atan2 x.im x.re) y.im))) (cos (+ (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.im) (* (atan2 x.im x.re) y.re)))))