powComplex, real part

Average Error: 37.9 → 13.8

Time: 17.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ t_1 := {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ t_2 := \frac{t_1}{\mathsf{fma}\left(y.im, \tan^{-1}_* \frac{x.im}{x.re}, 1\right)} \cdot \cos \left(y.im \cdot t_0\right)\\ \mathbf{if}\;y.re \leq -10650560034225.063:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_3 := \sqrt[3]{y.im} \cdot \sqrt[3]{y.im}\\ t_4 := \frac{t_1}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{if}\;y.re \leq 8.81112223723101 \cdot 10^{-16}:\\ \;\;\;\;t_4 \cdot \cos \left(\left(\sqrt[3]{y.im} \cdot \sqrt[3]{\sqrt[3]{y.im} \cdot t_3}\right) \cdot \left(t_0 \cdot \sqrt[3]{y.im}\right)\right)\\ \mathbf{elif}\;y.re \leq 7.380232567388229 \cdot 10^{+91}:\\ \;\;\;\;\begin{array}{l} t_5 := \sqrt[3]{\mathsf{hypot}\left(x.im, x.re\right)}\\ t_6 := t_3 \cdot \left(\sqrt[3]{y.im} \cdot \log \left(t_5 \cdot t_5\right)\right)\\ t_7 := t_3 \cdot \left(\sqrt[3]{y.im} \cdot \log t_5\right)\\ t_4 \cdot \left(\cos t_6 \cdot \cos t_7 - \sin t_6 \cdot \sin t_7\right) \end{array}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array}\\ \end{array} \]

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
 (let* ((t_0 (log (hypot x.im x.re)))
        (t_1 (pow (hypot x.re x.im) y.re))
        (t_2 (* (/ t_1 (fma y.im (atan2 x.im x.re) 1.0)) (cos (* y.im t_0)))))
   (if (<= y.re -10650560034225.063)
     t_2
     (let* ((t_3 (* (cbrt y.im) (cbrt y.im)))
            (t_4 (/ t_1 (exp (* y.im (atan2 x.im x.re))))))
       (if (<= y.re 8.81112223723101e-16)
         (*
          t_4
          (cos
           (* (* (cbrt y.im) (cbrt (* (cbrt y.im) t_3))) (* t_0 (cbrt y.im)))))
         (if (<= y.re 7.380232567388229e+91)
           (let* ((t_5 (cbrt (hypot x.im x.re)))
                  (t_6 (* t_3 (* (cbrt y.im) (log (* t_5 t_5)))))
                  (t_7 (* t_3 (* (cbrt y.im) (log t_5)))))
             (* t_4 (- (* (cos t_6) (cos t_7)) (* (sin t_6) (sin t_7)))))
           t_2))))))

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 t_0 = log(hypot(x_46_im, x_46_re));
	double t_1 = pow(hypot(x_46_re, x_46_im), y_46_re);
	double t_2 = (t_1 / fma(y_46_im, atan2(x_46_im, x_46_re), 1.0)) * cos(y_46_im * t_0);
	double tmp;
	if (y_46_re <= -10650560034225.063) {
		tmp = t_2;
	} else {
		double t_3 = cbrt(y_46_im) * cbrt(y_46_im);
		double t_4 = t_1 / exp(y_46_im * atan2(x_46_im, x_46_re));
		double tmp_1;
		if (y_46_re <= 8.81112223723101e-16) {
			tmp_1 = t_4 * cos((cbrt(y_46_im) * cbrt(cbrt(y_46_im) * t_3)) * (t_0 * cbrt(y_46_im)));
		} else if (y_46_re <= 7.380232567388229e+91) {
			double t_5 = cbrt(hypot(x_46_im, x_46_re));
			double t_6 = t_3 * (cbrt(y_46_im) * log(t_5 * t_5));
			double t_7 = t_3 * (cbrt(y_46_im) * log(t_5));
			tmp_1 = t_4 * ((cos(t_6) * cos(t_7)) - (sin(t_6) * sin(t_7)));
		} else {
			tmp_1 = t_2;
		}
		tmp = tmp_1;
	}
	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 3 regimes

2. if y.re < -10650560034225.06 or 7.3802325673882287e91 < y.re

   1. Initial program 39.6

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

   2. Simplified 25.4

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

   3. Taylor expanded in y.re around 0 43.2

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \color{blue}{\cos \left(\log \left(\sqrt{{x.re}^{2} + {x.im}^{2}}\right) \cdot y.im\right)} \]

   4. Simplified 21.8

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

   5. Taylor expanded in y.im around 0 15.5

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

   6. Simplified 15.5

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

   if -10650560034225.06 < y.re < 8.81112223723100955e-16

   1. Initial program 36.5

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

   2. Simplified 10.2

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

   3. Taylor expanded in y.re around 0 36.7

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \color{blue}{\cos \left(\log \left(\sqrt{{x.re}^{2} + {x.im}^{2}}\right) \cdot y.im\right)} \]

   4. Simplified 10.4

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

   5. Applied add-cube-cbrt_binary64 10.8

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

   6. Applied associate-*l*_binary64 11.1

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

   7. Applied add-cbrt-cube_binary64 10.8

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

   if 8.81112223723100955e-16 < y.re < 7.3802325673882287e91

   1. Initial program 37.8

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

   2. Simplified 26.8

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

   3. Taylor expanded in y.re around 0 42.7

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \color{blue}{\cos \left(\log \left(\sqrt{{x.re}^{2} + {x.im}^{2}}\right) \cdot y.im\right)} \]

   4. Simplified 23.8

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

   5. Applied add-cube-cbrt_binary64 23.6

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

   6. Applied associate-*l*_binary64 23.2

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

   7. Applied add-cube-cbrt_binary64 23.5

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

   8. Applied log-prod_binary64 23.0

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

   9. Applied distribute-rgt-in_binary64 23.4

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

   10. Applied distribute-rgt-in_binary64 23.6

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

   11. Applied cos-sum_binary64 23.6

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

4. Final simplification 13.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -10650560034225.063:\\ \;\;\;\;\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\mathsf{fma}\left(y.im, \tan^{-1}_* \frac{x.im}{x.re}, 1\right)} \cdot \cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{elif}\;y.re \leq 8.81112223723101 \cdot 10^{-16}:\\ \;\;\;\;\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot \cos \left(\left(\sqrt[3]{y.im} \cdot \sqrt[3]{\sqrt[3]{y.im} \cdot \left(\sqrt[3]{y.im} \cdot \sqrt[3]{y.im}\right)}\right) \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot \sqrt[3]{y.im}\right)\right)\\ \mathbf{elif}\;y.re \leq 7.380232567388229 \cdot 10^{+91}:\\ \;\;\;\;\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot \left(\cos \left(\left(\sqrt[3]{y.im} \cdot \sqrt[3]{y.im}\right) \cdot \left(\sqrt[3]{y.im} \cdot \log \left(\sqrt[3]{\mathsf{hypot}\left(x.im, x.re\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(x.im, x.re\right)}\right)\right)\right) \cdot \cos \left(\left(\sqrt[3]{y.im} \cdot \sqrt[3]{y.im}\right) \cdot \left(\sqrt[3]{y.im} \cdot \log \left(\sqrt[3]{\mathsf{hypot}\left(x.im, x.re\right)}\right)\right)\right) - \sin \left(\left(\sqrt[3]{y.im} \cdot \sqrt[3]{y.im}\right) \cdot \left(\sqrt[3]{y.im} \cdot \log \left(\sqrt[3]{\mathsf{hypot}\left(x.im, x.re\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(x.im, x.re\right)}\right)\right)\right) \cdot \sin \left(\left(\sqrt[3]{y.im} \cdot \sqrt[3]{y.im}\right) \cdot \left(\sqrt[3]{y.im} \cdot \log \left(\sqrt[3]{\mathsf{hypot}\left(x.im, x.re\right)}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\mathsf{fma}\left(y.im, \tan^{-1}_* \frac{x.im}{x.re}, 1\right)} \cdot \cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022068 
(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)))))