powComplex, real part

Average Error: 33.4 → 9.5

Time: 12.2s

Precision: binary64

Cost: 33859

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x.im \leq -339342992904603.25:\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log \left(-x.im\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;x.im \leq -1.0162343325851357 \cdot 10^{-69}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;x.im \leq 4.3354624121682 \cdot 10^{-310}:\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log \left(-x.im\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log x.im - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array}\]

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)))
  (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
 (if (<= x.im -339342992904603.25)
   (*
    (cos (* y.re (atan2 x.im x.re)))
    (exp (- (* y.re (log (- x.im))) (* (atan2 x.im x.re) y.im))))
   (if (<= x.im -1.0162343325851357e-69)
     (exp
      (-
       (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im)))))
       (* (atan2 x.im x.re) y.im)))
     (if (<= x.im 4.3354624121682e-310)
       (*
        (cos (* y.re (atan2 x.im x.re)))
        (exp (- (* y.re (log (- x.im))) (* (atan2 x.im x.re) y.im))))
       (*
        (cos (* 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)) * 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 tmp;
	if (x_46_im <= -339342992904603.25) {
		tmp = cos(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));
	} else if (x_46_im <= -1.0162343325851357e-69) {
		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));
	} else if (x_46_im <= 4.3354624121682e-310) {
		tmp = cos(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));
	} else {
		tmp = cos(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

Alternatives

Alternative 1
Error	9.6
Cost	110785

\[\begin{array}{l} \mathbf{if}\;x.im \leq -326275111805596.6:\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log \left(-x.im\right) - \left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}} \cdot \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}\right) \cdot \left(y.im \cdot \left(\sqrt[3]{\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \left(\sqrt[3]{\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sqrt[3]{\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}}\right)\right)\right)}\\ \mathbf{elif}\;x.im \leq -1.0023614876036504 \cdot 10^{-65}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;x.im \leq 4.3354624121682 \cdot 10^{-310}:\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log \left(-x.im\right) - \left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}} \cdot \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}\right) \cdot \left(y.im \cdot \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log x.im - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array}\]

Alternative 2
Error	9.5
Cost	66243

\[\begin{array}{l} \mathbf{if}\;x.im \leq -925944660526301.8:\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log \left(-x.im\right) - \left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}} \cdot \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}\right) \cdot \left(y.im \cdot \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}\right)}\\ \mathbf{elif}\;x.im \leq -9.834281750438918 \cdot 10^{-70}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;x.im \leq 4.3354624121682 \cdot 10^{-310}:\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log \left(-x.im\right) - \left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}} \cdot \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}\right) \cdot \left(y.im \cdot \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log x.im - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array}\]

Alternative 3
Error	9.6
Cost	33795

\[\begin{array}{l} \mathbf{if}\;x.im \leq -339342992904603.25:\\ \;\;\;\;e^{y.re \cdot \log \left(-x.im\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;x.im \leq -1.8575264693177687 \cdot 10^{-69}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;x.im \leq 4.3354624121682 \cdot 10^{-310}:\\ \;\;\;\;e^{y.re \cdot \log \left(-x.im\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log x.im - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array}\]

Alternative 4
Error	14.5
Cost	27459

\[\begin{array}{l} \mathbf{if}\;y.re \leq -1.7214144504141627 \cdot 10^{-15}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;y.re \leq 3.5192623145897594 \cdot 10^{-65}:\\ \;\;\;\;1\\ \mathbf{elif}\;y.re \leq 4.2003005748055775 \cdot 10^{-29}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;y.re \leq 0.0011869954206574697:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Alternative 5
Error	14.6
Cost	1348

\[\begin{array}{l} \mathbf{if}\;y.re \leq -0.010833561531639842:\\ \;\;\;\;0\\ \mathbf{elif}\;y.re \leq 5.531442779259403 \cdot 10^{-53}:\\ \;\;\;\;1\\ \mathbf{elif}\;y.re \leq 1.2949051645648911 \cdot 10^{-40}:\\ \;\;\;\;0\\ \mathbf{elif}\;y.re \leq 8.825529323851523 \cdot 10^{-05}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Alternative 6
Error	38.4
Cost	64

\[1\]

Derivation

1. Split input into 3 regimes

2. if x.im < -339342992904603.25 or -1.01623433258513573e-69 < x.im < 4.33546241216819e-310

   1. Initial program 35.5

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

   2. Taylor expanded around 0 20.6

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

   3. Simplified 20.6

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

   4. Taylor expanded around -inf 9.0

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

   5. Simplified 9.0

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

   6. Simplified 9.0

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

   if -339342992904603.25 < x.im < -1.01623433258513573e-69

   1. Initial program 20.6

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

   2. Taylor expanded around 0 12.0

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

   3. Simplified 12.0

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

   4. Taylor expanded around 0 11.9

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

   5. Simplified 11.9

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

   if 4.33546241216819e-310 < x.im

   1. Initial program 33.2

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

   2. Taylor expanded around 0 20.0

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

   3. Simplified 20.0

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

   4. Taylor expanded around 0 9.6

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

   5. Simplified 9.6

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

4. Final simplification 9.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -339342992904603.25:\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log \left(-x.im\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;x.im \leq -1.0162343325851357 \cdot 10^{-69}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;x.im \leq 4.3354624121682 \cdot 10^{-310}:\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log \left(-x.im\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log x.im - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(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)))))