powComplex, imaginary part

Average Error: 33.2 → 6.2

Time: 22.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\\ t_2 := e^{t_1 \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(t_1 \cdot y.im + t_0\right)\\ \mathbf{if}\;t_2 \leq 2.8299831771352195 \cdot 10^{-9}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_3 := {\left(\sqrt{\mathsf{hypot}\left(x.im, x.re\right)}\right)}^{y.re}\\ \left(t_3 \cdot \left(t_3 \cdot {\left(e^{\tan^{-1}_* \frac{x.im}{x.re}}\right)}^{\left(-y.im\right)}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, t_0\right)\right) \end{array}\\ \end{array} \]

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)))
  (sin
   (+
    (* (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 (* y.re (atan2 x.im x.re)))
        (t_1 (log (sqrt (+ (* x.re x.re) (* x.im x.im)))))
        (t_2
         (*
          (exp (- (* t_1 y.re) (* (atan2 x.im x.re) y.im)))
          (sin (+ (* t_1 y.im) t_0)))))
   (if (<= t_2 2.8299831771352195e-9)
     t_2
     (let* ((t_3 (pow (sqrt (hypot x.im x.re)) y.re)))
       (*
        (* t_3 (* t_3 (pow (exp (atan2 x.im x.re)) (- y.im))))
        (sin (fma (log (hypot x.re x.im)) y.im t_0)))))))

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)) * sin((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 = y_46_re * atan2(x_46_im, x_46_re);
	double t_1 = log(sqrt((x_46_re * x_46_re) + (x_46_im * x_46_im)));
	double t_2 = exp((t_1 * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im)) * sin((t_1 * y_46_im) + t_0);
	double tmp;
	if (t_2 <= 2.8299831771352195e-9) {
		tmp = t_2;
	} else {
		double t_3 = pow(sqrt(hypot(x_46_im, x_46_re)), y_46_re);
		tmp = (t_3 * (t_3 * pow(exp(atan2(x_46_im, x_46_re)), -y_46_im))) * sin(fma(log(hypot(x_46_re, x_46_im)), y_46_im, t_0));
	}
	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 2 regimes

2. if (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re)))) < 2.82998317713521947e-9

   1. Initial program 1.5

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

   if 2.82998317713521947e-9 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re))))

   1. Initial program 63.2

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

   2. Simplified 10.1

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

   3. Applied *-un-lft-identity_binary64 10.1

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

   4. Applied add-sqr-sqrt_binary64 10.1

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

   5. Applied unpow-prod-down_binary64 10.1

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

   6. Applied times-frac_binary64 10.1

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

   7. Simplified 10.1

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

   8. Simplified 10.1

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

   9. Applied div-inv_binary64 10.1

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

   10. Simplified 10.6

       \[\leadsto \left({\left(\sqrt{\mathsf{hypot}\left(x.im, x.re\right)}\right)}^{y.re} \cdot \left({\left(\sqrt{\mathsf{hypot}\left(x.im, x.re\right)}\right)}^{y.re} \cdot \color{blue}{{\left(e^{\tan^{-1}_* \frac{x.im}{x.re}}\right)}^{\left(-y.im\right)}}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 6.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \leq 2.8299831771352195 \cdot 10^{-9}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\sqrt{\mathsf{hypot}\left(x.im, x.re\right)}\right)}^{y.re} \cdot \left({\left(\sqrt{\mathsf{hypot}\left(x.im, x.re\right)}\right)}^{y.re} \cdot {\left(e^{\tan^{-1}_* \frac{x.im}{x.re}}\right)}^{\left(-y.im\right)}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021225 
(FPCore (x.re x.im y.re y.im)
  :name "powComplex, imaginary part"
  :precision binary64
  (* (exp (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) (* (atan2 x.im x.re) y.im))) (sin (+ (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.im) (* (atan2 x.im x.re) y.re)))))