powComplex, real part

Average Error: 33.9 → 5.2

Time: 12.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\\ t_3 := e^{t_2 \cdot y.re - t_0} \cdot \cos \left(t_2 \cdot y.im + t_1\right)\\ \mathbf{if}\;t_3 \leq 1.0000000000003166:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\sqrt{{t_0}^{2}}}} \cdot \left(\cos t_1 - y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot \sin t_1\right)\right)\\ \end{array} \]

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
 (let* ((t_0 (* (atan2 x.im x.re) y.im))
        (t_1 (* y.re (atan2 x.im x.re)))
        (t_2 (log (sqrt (+ (* x.re x.re) (* x.im x.im)))))
        (t_3 (* (exp (- (* t_2 y.re) t_0)) (cos (+ (* t_2 y.im) t_1)))))
   (if (<= t_3 1.0000000000003166)
     t_3
     (*
      (/ (pow (hypot x.re x.im) y.re) (exp (sqrt (pow t_0 2.0))))
      (- (cos t_1) (* y.im (* (log (hypot x.im x.re)) (sin t_1))))))))

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 t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = y_46_re * atan2(x_46_im, x_46_re);
	double t_2 = log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))));
	double t_3 = exp(((t_2 * y_46_re) - t_0)) * cos(((t_2 * y_46_im) + t_1));
	double tmp;
	if (t_3 <= 1.0000000000003166) {
		tmp = t_3;
	} else {
		tmp = (pow(hypot(x_46_re, x_46_im), y_46_re) / exp(sqrt(pow(t_0, 2.0)))) * (cos(t_1) - (y_46_im * (log(hypot(x_46_im, x_46_re)) * sin(t_1))));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return Math.exp(((Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (Math.atan2(x_46_im, x_46_re) * y_46_im))) * Math.cos(((Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_im) + (Math.atan2(x_46_im, x_46_re) * y_46_re)));
}

↓

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double t_2 = Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))));
	double t_3 = Math.exp(((t_2 * y_46_re) - t_0)) * Math.cos(((t_2 * y_46_im) + t_1));
	double tmp;
	if (t_3 <= 1.0000000000003166) {
		tmp = t_3;
	} else {
		tmp = (Math.pow(Math.hypot(x_46_re, x_46_im), y_46_re) / Math.exp(Math.sqrt(Math.pow(t_0, 2.0)))) * (Math.cos(t_1) - (y_46_im * (Math.log(Math.hypot(x_46_im, x_46_re)) * Math.sin(t_1))));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return math.exp(((math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (math.atan2(x_46_im, x_46_re) * y_46_im))) * math.cos(((math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_im) + (math.atan2(x_46_im, x_46_re) * y_46_re)))

↓

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.atan2(x_46_im, x_46_re) * y_46_im
	t_1 = y_46_re * math.atan2(x_46_im, x_46_re)
	t_2 = math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))
	t_3 = math.exp(((t_2 * y_46_re) - t_0)) * math.cos(((t_2 * y_46_im) + t_1))
	tmp = 0
	if t_3 <= 1.0000000000003166:
		tmp = t_3
	else:
		tmp = (math.pow(math.hypot(x_46_re, x_46_im), y_46_re) / math.exp(math.sqrt(math.pow(t_0, 2.0)))) * (math.cos(t_1) - (y_46_im * (math.log(math.hypot(x_46_im, x_46_re)) * math.sin(t_1))))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(exp(Float64(Float64(log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)))) * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im))) * cos(Float64(Float64(log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)))) * y_46_im) + Float64(atan(x_46_im, x_46_re) * y_46_re))))
end

↓

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_1 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_2 = log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))
	t_3 = Float64(exp(Float64(Float64(t_2 * y_46_re) - t_0)) * cos(Float64(Float64(t_2 * y_46_im) + t_1)))
	tmp = 0.0
	if (t_3 <= 1.0000000000003166)
		tmp = t_3;
	else
		tmp = Float64(Float64((hypot(x_46_re, x_46_im) ^ y_46_re) / exp(sqrt((t_0 ^ 2.0)))) * Float64(cos(t_1) - Float64(y_46_im * Float64(log(hypot(x_46_im, x_46_re)) * sin(t_1)))));
	end
	return tmp
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 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)));
end

↓

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	t_1 = y_46_re * atan2(x_46_im, x_46_re);
	t_2 = log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))));
	t_3 = exp(((t_2 * y_46_re) - t_0)) * cos(((t_2 * y_46_im) + t_1));
	tmp = 0.0;
	if (t_3 <= 1.0000000000003166)
		tmp = t_3;
	else
		tmp = ((hypot(x_46_re, x_46_im) ^ y_46_re) / exp(sqrt((t_0 ^ 2.0)))) * (cos(t_1) - (y_46_im * (log(hypot(x_46_im, x_46_re)) * sin(t_1))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(N[Exp[N[(N[(N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision] + N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Exp[N[(N[(t$95$2 * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(t$95$2 * y$46$im), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 1.0000000000003166], t$95$3, N[(N[(N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision] / N[Exp[N[Sqrt[N[Power[t$95$0, 2.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[t$95$1], $MachinePrecision] - N[(y$46$im * N[(N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

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))) (cos.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)))) < 1.0000000000003166

   1. Initial program 2.9

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

   if 1.0000000000003166 < (*.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))) (cos.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 63.1

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

   2. Simplified 10.1

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

   3. Taylor expanded in x.im around 0 9.6

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

   4. Taylor expanded in y.im around 0 63.0

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

   5. Simplified 9.9

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

   6. Applied egg-rr 7.3

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

4. Final simplification 5.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \leq 1.0000000000003166:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\sqrt{{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right)}^{2}}}} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) - y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022150 
(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)))))