powComplex, imaginary part

Average Error: 33.7 → 20.4

Time: 11.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x.im \leq -9.423511434206789 \cdot 10^{+154}:\\ \;\;\;\;e^{\log \left(-x.im\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.im \leq -2.2880081199411094 \cdot 10^{-302}:\\ \;\;\;\;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(\frac{-1}{x.im}\right)\right)\\ \mathbf{elif}\;x.im \leq 5.999233792908306 \cdot 10^{-275}:\\ \;\;\;\;\frac{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}{e^{\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.im \leq 8.893872608083424 \cdot 10^{-234}:\\ \;\;\;\;e^{y.re \cdot \log x.im - \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)}^{3}}\\ \mathbf{elif}\;x.im \leq 9.796633468730483 \cdot 10^{-206}:\\ \;\;\;\;\frac{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}{e^{\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.im \leq 2.8096680029547107 \cdot 10^{-08}:\\ \;\;\;\;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.im\right)\\ \mathbf{elif}\;x.im \leq 1.940140996050804 \cdot 10^{+35}:\\ \;\;\;\;\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) \cdot e^{y.re \cdot \log x.im - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\sin \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)))
  (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.im -9.423511434206789e+154)
   (*
    (exp (- (* (log (- x.im)) y.re) (* (atan2 x.im x.re) y.im)))
    (sin (* y.re (atan2 x.im x.re))))
   (if (<= x.im -2.2880081199411094e-302)
     (*
      (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 (/ -1.0 x.im))))))
     (if (<= x.im 5.999233792908306e-275)
       (*
        (/
         (pow (sqrt (+ (* x.re x.re) (* x.im x.im))) y.re)
         (exp (* (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.im 8.893872608083424e-234)
         (*
          (exp (- (* y.re (log x.im)) (* (atan2 x.im x.re) y.im)))
          (cbrt (pow (sin (* y.re (atan2 x.im x.re))) 3.0)))
         (if (<= x.im 9.796633468730483e-206)
           (*
            (/
             (pow (sqrt (+ (* x.re x.re) (* x.im x.im))) y.re)
             (exp (* (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.im 2.8096680029547107e-08)
             (*
              (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.im)))))
             (if (<= x.im 1.940140996050804e+35)
               (*
                (sin
                 (+
                  (* y.re (atan2 x.im x.re))
                  (* y.im (log (sqrt (+ (* x.re x.re) (* x.im x.im)))))))
                (exp (- (* y.re (log x.im)) (* (atan2 x.im x.re) y.im))))
               (*
                (sin (* 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)) * 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_im <= -9.423511434206789e+154) {
		tmp = exp((log(-x_46_im) * 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_im <= -2.2880081199411094e-302) {
		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(-1.0 / x_46_im)));
	} else if (x_46_im <= 5.999233792908306e-275) {
		tmp = (pow(sqrt((x_46_re * x_46_re) + (x_46_im * x_46_im)), y_46_re) / exp(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_im <= 8.893872608083424e-234) {
		tmp = exp((y_46_re * log(x_46_im)) - (atan2(x_46_im, x_46_re) * y_46_im)) * cbrt(pow(sin(y_46_re * atan2(x_46_im, x_46_re)), 3.0));
	} else if (x_46_im <= 9.796633468730483e-206) {
		tmp = (pow(sqrt((x_46_re * x_46_re) + (x_46_im * x_46_im)), y_46_re) / exp(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_im <= 2.8096680029547107e-08) {
		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_im)));
	} else if (x_46_im <= 1.940140996050804e+35) {
		tmp = 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))))) * exp((y_46_re * log(x_46_im)) - (atan2(x_46_im, x_46_re) * y_46_im));
	} else {
		tmp = sin(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 7 regimes

2. if x.im < -9.4235114342067888e154

   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. Taylor expanded around 0 37.6

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

   3. Taylor expanded around -inf 14.5

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

   if -9.4235114342067888e154 < x.im < -2.2880081199411094e-302

   1. Initial program 24.5

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

   2. Taylor expanded around -inf 20.6

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

   if -2.2880081199411094e-302 < x.im < 5.999233792908306e-275 or 8.89387260808342433e-234 < x.im < 9.7966334687304829e-206

   1. Initial program 35.1

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

   2. Simplified 36.7

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

   if 5.999233792908306e-275 < x.im < 8.89387260808342433e-234

   1. Initial program 35.8

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

   2. Taylor expanded around 0 36.3

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

   4. Applied add-cbrt-cube_binary64 38.1

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

   5. Simplified 38.1

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

   6. Taylor expanded around 0 41.3

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

   if 9.7966334687304829e-206 < x.im < 2.80966800295471072e-8

   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 0 19.9

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

   if 2.80966800295471072e-8 < x.im < 1.9401409960508038e35

   1. Initial program 18.2

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

   2. Taylor expanded around 0 19.1

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

   if 1.9401409960508038e35 < x.im

   1. Initial program 43.6

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

   2. Taylor expanded around 0 29.5

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

   3. Taylor expanded around 0 15.7

      \[\leadsto e^{\log \color{blue}{x.im} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\]
3. Recombined 7 regimes into one program.

4. Final simplification 20.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -9.423511434206789 \cdot 10^{+154}:\\ \;\;\;\;e^{\log \left(-x.im\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.im \leq -2.2880081199411094 \cdot 10^{-302}:\\ \;\;\;\;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(\frac{-1}{x.im}\right)\right)\\ \mathbf{elif}\;x.im \leq 5.999233792908306 \cdot 10^{-275}:\\ \;\;\;\;\frac{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}{e^{\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.im \leq 8.893872608083424 \cdot 10^{-234}:\\ \;\;\;\;e^{y.re \cdot \log x.im - \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)}^{3}}\\ \mathbf{elif}\;x.im \leq 9.796633468730483 \cdot 10^{-206}:\\ \;\;\;\;\frac{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}{e^{\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.im \leq 2.8096680029547107 \cdot 10^{-08}:\\ \;\;\;\;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.im\right)\\ \mathbf{elif}\;x.im \leq 1.940140996050804 \cdot 10^{+35}:\\ \;\;\;\;\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) \cdot e^{y.re \cdot \log x.im - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\sin \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 2021058 
(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)))))