powComplex, imaginary part

Average Error: 33.7 → 21.1

Time: 14.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x.re \leq -2.1308440972502256 \cdot 10^{+67}:\\ \;\;\;\;e^{\log \left(-x.re\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;x.re \leq -9.372599816172701 \cdot 10^{-17}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\right)\\ \mathbf{elif}\;x.re \leq -4.9159159806039255 \cdot 10^{-295}:\\ \;\;\;\;e^{\log \left(-x.re\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;x.re \leq 8.427778181788499 \cdot 10^{-157}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sqrt[3]{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)}\\ \mathbf{elif}\;x.re \leq 0.0003698274412800461:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\right)\\ \mathbf{elif}\;x.re \leq 8.974528308302968 \cdot 10^{+147}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log x.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array}\]

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)))
  (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
 (if (<= x.re -2.1308440972502256e+67)
   (*
    (exp (- (* (log (- x.re)) y.re) (* (atan2 x.im x.re) y.im)))
    (sin (* y.re (atan2 x.im x.re))))
   (if (<= x.re -9.372599816172701e-17)
     (*
      (exp
       (-
        (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im)))))
        (* (atan2 x.im x.re) y.im)))
      (sin
       (+
        (* y.re (atan2 x.im x.re))
        (* y.im (log (sqrt (+ (* x.re x.re) (* x.im x.im))))))))
     (if (<= x.re -4.9159159806039255e-295)
       (*
        (exp (- (* (log (- x.re)) y.re) (* (atan2 x.im x.re) y.im)))
        (sin (* y.re (atan2 x.im x.re))))
       (if (<= x.re 8.427778181788499e-157)
         (*
          (exp
           (-
            (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im)))))
            (* (atan2 x.im x.re) y.im)))
          (cbrt
           (*
            (sin (* y.re (atan2 x.im x.re)))
            (*
             (sin (* y.re (atan2 x.im x.re)))
             (sin (* y.re (atan2 x.im x.re)))))))
         (if (<= x.re 0.0003698274412800461)
           (*
            (exp
             (-
              (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im)))))
              (* (atan2 x.im x.re) y.im)))
            (sin
             (+
              (* y.re (atan2 x.im x.re))
              (* y.im (log (sqrt (+ (* x.re x.re) (* x.im x.im))))))))
           (if (<= x.re 8.974528308302968e+147)
             (*
              (exp
               (-
                (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im)))))
                (* (atan2 x.im x.re) y.im)))
              (sin (+ (* y.re (atan2 x.im x.re)) (* y.im (log x.re)))))
             (*
              (sin (* y.re (atan2 x.im x.re)))
              (exp (- (* y.re (log x.re)) (* (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)) * 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 tmp;
	if (x_46_re <= -2.1308440972502256e+67) {
		tmp = exp((log(-x_46_re) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im)) * sin(y_46_re * atan2(x_46_im, x_46_re));
	} else if (x_46_re <= -9.372599816172701e-17) {
		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)) * sin((y_46_re * atan2(x_46_im, x_46_re)) + (y_46_im * log(sqrt((x_46_re * x_46_re) + (x_46_im * x_46_im)))));
	} else if (x_46_re <= -4.9159159806039255e-295) {
		tmp = exp((log(-x_46_re) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im)) * sin(y_46_re * atan2(x_46_im, x_46_re));
	} else if (x_46_re <= 8.427778181788499e-157) {
		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)) * cbrt(sin(y_46_re * atan2(x_46_im, x_46_re)) * (sin(y_46_re * atan2(x_46_im, x_46_re)) * sin(y_46_re * atan2(x_46_im, x_46_re))));
	} else if (x_46_re <= 0.0003698274412800461) {
		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)) * sin((y_46_re * atan2(x_46_im, x_46_re)) + (y_46_im * log(sqrt((x_46_re * x_46_re) + (x_46_im * x_46_im)))));
	} else if (x_46_re <= 8.974528308302968e+147) {
		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)) * sin((y_46_re * atan2(x_46_im, x_46_re)) + (y_46_im * log(x_46_re)));
	} else {
		tmp = sin(y_46_re * atan2(x_46_im, x_46_re)) * exp((y_46_re * log(x_46_re)) - (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 5 regimes

2. if x.re < -2.1308440972502256e67 or -9.37259981617270117e-17 < x.re < -4.91591598060392548e-295

   1. Initial program 36.4

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

   2. Taylor expanded around 0 25.8

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

   3. Taylor expanded around -inf 18.8

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

   if -2.1308440972502256e67 < x.re < -9.37259981617270117e-17 or 8.4277781817884992e-157 < x.re < 3.6982744128004612e-4

   1. Initial program 16.9

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

   if -4.91591598060392548e-295 < x.re < 8.4277781817884992e-157

   1. Initial program 29.4

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

   2. Taylor expanded around 0 21.7

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

   4. Applied add-cbrt-cube_binary64 24.8

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

   if 3.6982744128004612e-4 < x.re < 8.9745283083029681e147

   1. Initial program 22.5

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

   2. Taylor expanded around inf 18.2

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

   3. Simplified 18.2

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

   if 8.9745283083029681e147 < x.re

   1. Initial program 62.4

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

   2. Taylor expanded around 0 38.8

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

   3. Taylor expanded around inf 32.4

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

4. Final simplification 21.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -2.1308440972502256 \cdot 10^{+67}:\\ \;\;\;\;e^{\log \left(-x.re\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;x.re \leq -9.372599816172701 \cdot 10^{-17}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\right)\\ \mathbf{elif}\;x.re \leq -4.9159159806039255 \cdot 10^{-295}:\\ \;\;\;\;e^{\log \left(-x.re\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;x.re \leq 8.427778181788499 \cdot 10^{-157}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sqrt[3]{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)}\\ \mathbf{elif}\;x.re \leq 0.0003698274412800461:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\right)\\ \mathbf{elif}\;x.re \leq 8.974528308302968 \cdot 10^{+147}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log x.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array}\]

Reproduce

herbie shell --seed 2021028 
(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)))))