powComplex, imaginary part

Average Error: 38.3 → 22.4

Time: 17.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ t_1 := \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\ \mathbf{if}\;x.re \leq -2.675854181798536 \cdot 10^{+193}:\\ \;\;\;\;\begin{array}{l} t_2 := \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}\\ t_1 \cdot \sin \left(\mathsf{fma}\left(t_0, y.im, t_2 \cdot \left(y.re \cdot \left(t_2 \cdot t_2\right)\right)\right)\right) \end{array}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_3 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_4 := {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \sin \left(\mathsf{fma}\left(t_0, y.im, t_3\right)\right)\\ \mathbf{if}\;x.re \leq -13778.819173053213:\\ \;\;\;\;t_4\\ \mathbf{elif}\;x.re \leq -2.9082658274636224 \cdot 10^{-166}:\\ \;\;\;\;t_1 \cdot \sin \left(\mathsf{fma}\left(t_0, y.im, \mathsf{expm1}\left(\mathsf{log1p}\left(t_3\right)\right)\right)\right)\\ \mathbf{elif}\;x.re \leq 1.0159450429663836 \cdot 10^{-77}:\\ \;\;\;\;\begin{array}{l} t_5 := \sqrt[3]{t_3}\\ t_1 \cdot \sin \left(\mathsf{fma}\left(t_0, y.im, t_5 \cdot \left(t_5 \cdot t_5\right)\right)\right) \end{array}\\ \mathbf{elif}\;x.re \leq 1.3769665612619498 \cdot 10^{+73}:\\ \;\;\;\;\begin{array}{l} t_6 := \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ t_7 := \sin t_3\\ t_1 \cdot \left(\mathsf{fma}\left(\cos t_3, y.im \cdot t_6, t_7\right) - 0.5 \cdot \left({t_6}^{2} \cdot \left(t_7 \cdot \left(y.im \cdot y.im\right)\right)\right)\right) \end{array}\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array}\\ \end{array} \]

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 (log (hypot x.re x.im)))
        (t_1
         (/ (pow (hypot x.re x.im) y.re) (exp (* (atan2 x.im x.re) y.im)))))
   (if (<= x.re -2.675854181798536e+193)
     (let* ((t_2 (cbrt (atan2 x.im x.re))))
       (* t_1 (sin (fma t_0 y.im (* t_2 (* y.re (* t_2 t_2)))))))
     (let* ((t_3 (* y.re (atan2 x.im x.re)))
            (t_4 (* (pow (hypot x.im x.re) y.re) (sin (fma t_0 y.im t_3)))))
       (if (<= x.re -13778.819173053213)
         t_4
         (if (<= x.re -2.9082658274636224e-166)
           (* t_1 (sin (fma t_0 y.im (expm1 (log1p t_3)))))
           (if (<= x.re 1.0159450429663836e-77)
             (let* ((t_5 (cbrt t_3)))
               (* t_1 (sin (fma t_0 y.im (* t_5 (* t_5 t_5))))))
             (if (<= x.re 1.3769665612619498e+73)
               (let* ((t_6 (log (hypot x.im x.re))) (t_7 (sin t_3)))
                 (*
                  t_1
                  (-
                   (fma (cos t_3) (* y.im t_6) t_7)
                   (* 0.5 (* (pow t_6 2.0) (* t_7 (* y.im y.im)))))))
               t_4))))))))

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 = log(hypot(x_46_re, x_46_im));
	double t_1 = pow(hypot(x_46_re, x_46_im), y_46_re) / exp(atan2(x_46_im, x_46_re) * y_46_im);
	double tmp;
	if (x_46_re <= -2.675854181798536e+193) {
		double t_2_1 = cbrt(atan2(x_46_im, x_46_re));
		tmp = t_1 * sin(fma(t_0, y_46_im, (t_2_1 * (y_46_re * (t_2_1 * t_2_1)))));
	} else {
		double t_3 = y_46_re * atan2(x_46_im, x_46_re);
		double t_4 = pow(hypot(x_46_im, x_46_re), y_46_re) * sin(fma(t_0, y_46_im, t_3));
		double tmp_2;
		if (x_46_re <= -13778.819173053213) {
			tmp_2 = t_4;
		} else if (x_46_re <= -2.9082658274636224e-166) {
			tmp_2 = t_1 * sin(fma(t_0, y_46_im, expm1(log1p(t_3))));
		} else if (x_46_re <= 1.0159450429663836e-77) {
			double t_5 = cbrt(t_3);
			tmp_2 = t_1 * sin(fma(t_0, y_46_im, (t_5 * (t_5 * t_5))));
		} else if (x_46_re <= 1.3769665612619498e+73) {
			double t_6 = log(hypot(x_46_im, x_46_re));
			double t_7 = sin(t_3);
			tmp_2 = t_1 * (fma(cos(t_3), (y_46_im * t_6), t_7) - (0.5 * (pow(t_6, 2.0) * (t_7 * (y_46_im * y_46_im)))));
		} else {
			tmp_2 = t_4;
		}
		tmp = tmp_2;
	}
	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 5 regimes

2. if x.re < -2.6758541817985359e193

   1. Initial program 64.0

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

   2. Simplified 18.5

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

   3. Applied add-cube-cbrt_binary64 17.9

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

   4. Applied associate-*r*_binary64 18.5

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

   if -2.6758541817985359e193 < x.re < -13778.819173053213 or 1.3769665612619498e73 < x.re

   1. Initial program 42.5

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

   2. Simplified 17.8

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

   3. Taylor expanded in y.im around 0 31.1

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

   4. Simplified 24.1

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

   if -13778.819173053213 < x.re < -2.9082658274636224e-166

   1. Initial program 26.1

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

   2. Simplified 17.1

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

   3. Applied expm1-log1p-u_binary64 26.2

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)}\right)\right) \]

   if -2.9082658274636224e-166 < x.re < 1.0159450429663836e-77

   1. Initial program 35.9

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

   2. Simplified 19.1

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

   3. Applied add-cube-cbrt_binary64 19.1

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

   if 1.0159450429663836e-77 < x.re < 1.3769665612619498e73

   1. Initial program 27.7

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

   2. Simplified 16.8

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

   3. Taylor expanded in y.im around 0 34.8

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \color{blue}{\left(\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.re}^{2} + {x.im}^{2}}\right) \cdot y.im\right)\right) - 0.5 \cdot \left({\log \left(\sqrt{{x.re}^{2} + {x.im}^{2}}\right)}^{2} \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {y.im}^{2}\right)\right)\right)} \]

   4. Simplified 24.4

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \color{blue}{\left(\mathsf{fma}\left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) - 0.5 \cdot \left({\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{2} \cdot \left(\left(y.im \cdot y.im\right) \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 22.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -2.675854181798536 \cdot 10^{+193}:\\ \;\;\;\;\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, \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}} \cdot \left(y.re \cdot \left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}} \cdot \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right)\right)\right)\\ \mathbf{elif}\;x.re \leq -13778.819173053213:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \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)\\ \mathbf{elif}\;x.re \leq -2.9082658274636224 \cdot 10^{-166}:\\ \;\;\;\;\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, \mathsf{expm1}\left(\mathsf{log1p}\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right)\\ \mathbf{elif}\;x.re \leq 1.0159450429663836 \cdot 10^{-77}:\\ \;\;\;\;\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, \sqrt[3]{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left(\sqrt[3]{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sqrt[3]{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right)\right)\\ \mathbf{elif}\;x.re \leq 1.3769665612619498 \cdot 10^{+73}:\\ \;\;\;\;\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 \left(\mathsf{fma}\left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) - 0.5 \cdot \left({\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{2} \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(y.im \cdot y.im\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \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 2022067 
(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)))))