powComplex, real part

Percentage Accurate: 39.6% → 79.7%

Time: 21.2s

Alternatives: 12

Speedup: 4.0×

• 35.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\\ e^{t_0 \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(t_0 \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (sqrt (+ (* x.re x.re) (* x.im x.im))))))
   (*
    (exp (- (* t_0 y.re) (* (atan2 x.im x.re) y.im)))
    (cos (+ (* t_0 y.im) (* (atan2 x.im x.re) y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))));
	return exp(((t_0 * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * cos(((t_0 * y_46_im) + (atan2(x_46_im, x_46_re) * y_46_re)));
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    t_0 = log(sqrt(((x_46re * x_46re) + (x_46im * x_46im))))
    code = exp(((t_0 * y_46re) - (atan2(x_46im, x_46re) * y_46im))) * cos(((t_0 * y_46im) + (atan2(x_46im, x_46re) * y_46re)))
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))));
	return Math.exp(((t_0 * y_46_re) - (Math.atan2(x_46_im, x_46_re) * y_46_im))) * Math.cos(((t_0 * y_46_im) + (Math.atan2(x_46_im, x_46_re) * y_46_re)));
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))
	return math.exp(((t_0 * y_46_re) - (math.atan2(x_46_im, x_46_re) * y_46_im))) * math.cos(((t_0 * y_46_im) + (math.atan2(x_46_im, x_46_re) * y_46_re)))

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))
	return Float64(exp(Float64(Float64(t_0 * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im))) * cos(Float64(Float64(t_0 * y_46_im) + Float64(atan(x_46_im, x_46_re) * y_46_re))))
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))));
	tmp = exp(((t_0 * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * cos(((t_0 * y_46_im) + (atan2(x_46_im, x_46_re) * y_46_re)));
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, N[(N[Exp[N[(N[(t$95$0 * y$46$re), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(t$95$0 * y$46$im), $MachinePrecision] + N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Initial Program: 39.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\\ e^{t_0 \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(t_0 \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (sqrt (+ (* x.re x.re) (* x.im x.im))))))
   (*
    (exp (- (* t_0 y.re) (* (atan2 x.im x.re) y.im)))
    (cos (+ (* t_0 y.im) (* (atan2 x.im x.re) y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))));
	return exp(((t_0 * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * cos(((t_0 * y_46_im) + (atan2(x_46_im, x_46_re) * y_46_re)));
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    t_0 = log(sqrt(((x_46re * x_46re) + (x_46im * x_46im))))
    code = exp(((t_0 * y_46re) - (atan2(x_46im, x_46re) * y_46im))) * cos(((t_0 * y_46im) + (atan2(x_46im, x_46re) * y_46re)))
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))));
	return Math.exp(((t_0 * y_46_re) - (Math.atan2(x_46_im, x_46_re) * y_46_im))) * Math.cos(((t_0 * y_46_im) + (Math.atan2(x_46_im, x_46_re) * y_46_re)));
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))
	return math.exp(((t_0 * y_46_re) - (math.atan2(x_46_im, x_46_re) * y_46_im))) * math.cos(((t_0 * y_46_im) + (math.atan2(x_46_im, x_46_re) * y_46_re)))

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))
	return Float64(exp(Float64(Float64(t_0 * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im))) * cos(Float64(Float64(t_0 * y_46_im) + Float64(atan(x_46_im, x_46_re) * y_46_re))))
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))));
	tmp = exp(((t_0 * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * cos(((t_0 * y_46_im) + (atan2(x_46_im, x_46_re) * y_46_re)));
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, N[(N[Exp[N[(N[(t$95$0 * y$46$re), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(t$95$0 * y$46$im), $MachinePrecision] + N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Alternative 1: 79.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;x.re \leq -100000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.re \leq -4 \cdot 10^{-310}:\\ \;\;\;\;t_0 \cdot \left(\cos t_1 + \sin t_1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.re}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \cos \left(t_1 + y.im \cdot \log x.re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (exp (- (* (log (hypot x.re x.im)) y.re) (* (atan2 x.im x.re) y.im))))
        (t_1 (* y.re (atan2 x.im x.re))))
   (if (<= x.re -100000000.0)
     t_0
     (if (<= x.re -4e-310)
       (* t_0 (+ (cos t_1) (* (sin t_1) (* y.im (log (/ -1.0 x.re))))))
       (* t_0 (cos (+ t_1 (* y.im (log x.re)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = exp(((log(hypot(x_46_re, x_46_im)) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im)));
	double t_1 = y_46_re * atan2(x_46_im, x_46_re);
	double tmp;
	if (x_46_re <= -100000000.0) {
		tmp = t_0;
	} else if (x_46_re <= -4e-310) {
		tmp = t_0 * (cos(t_1) + (sin(t_1) * (y_46_im * log((-1.0 / x_46_re)))));
	} else {
		tmp = t_0 * cos((t_1 + (y_46_im * log(x_46_re))));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.exp(((Math.log(Math.hypot(x_46_re, x_46_im)) * y_46_re) - (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 tmp;
	if (x_46_re <= -100000000.0) {
		tmp = t_0;
	} else if (x_46_re <= -4e-310) {
		tmp = t_0 * (Math.cos(t_1) + (Math.sin(t_1) * (y_46_im * Math.log((-1.0 / x_46_re)))));
	} else {
		tmp = t_0 * Math.cos((t_1 + (y_46_im * Math.log(x_46_re))));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.exp(((math.log(math.hypot(x_46_re, x_46_im)) * y_46_re) - (math.atan2(x_46_im, x_46_re) * y_46_im)))
	t_1 = y_46_re * math.atan2(x_46_im, x_46_re)
	tmp = 0
	if x_46_re <= -100000000.0:
		tmp = t_0
	elif x_46_re <= -4e-310:
		tmp = t_0 * (math.cos(t_1) + (math.sin(t_1) * (y_46_im * math.log((-1.0 / x_46_re)))))
	else:
		tmp = t_0 * math.cos((t_1 + (y_46_im * math.log(x_46_re))))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = exp(Float64(Float64(log(hypot(x_46_re, x_46_im)) * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im)))
	t_1 = Float64(y_46_re * atan(x_46_im, x_46_re))
	tmp = 0.0
	if (x_46_re <= -100000000.0)
		tmp = t_0;
	elseif (x_46_re <= -4e-310)
		tmp = Float64(t_0 * Float64(cos(t_1) + Float64(sin(t_1) * Float64(y_46_im * log(Float64(-1.0 / x_46_re))))));
	else
		tmp = Float64(t_0 * cos(Float64(t_1 + Float64(y_46_im * log(x_46_re)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = exp(((log(hypot(x_46_re, x_46_im)) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im)));
	t_1 = y_46_re * atan2(x_46_im, x_46_re);
	tmp = 0.0;
	if (x_46_re <= -100000000.0)
		tmp = t_0;
	elseif (x_46_re <= -4e-310)
		tmp = t_0 * (cos(t_1) + (sin(t_1) * (y_46_im * log((-1.0 / x_46_re)))));
	else
		tmp = t_0 * cos((t_1 + (y_46_im * log(x_46_re))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Exp[N[(N[(N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$re, -100000000.0], t$95$0, If[LessEqual[x$46$re, -4e-310], N[(t$95$0 * N[(N[Cos[t$95$1], $MachinePrecision] + N[(N[Sin[t$95$1], $MachinePrecision] * N[(y$46$im * N[Log[N[(-1.0 / x$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Cos[N[(t$95$1 + N[(y$46$im * N[Log[x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -1e8

   1. Initial program 22.7%

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

   3. Simplified 74.2%

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

   4. Taylor expanded in y.im around 0 84.8%

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

   5. Taylor expanded in y.re around 0 92.4%

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

   if -1e8 < x.re < -3.999999999999988e-310

   1. Initial program 51.4%

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

   3. Simplified 81.9%

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

   4. Taylor expanded in x.re around -inf 77.8%

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

      1. +-commutative 77.8%

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

      2. mul-1-neg 77.8%

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

      3. *-commutative 77.8%

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

      4. unsub-neg 77.8%

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

      5. *-commutative 77.8%

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

   6. Simplified 77.8%

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

   7. Taylor expanded in y.im around 0 87.5%

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

   if -3.999999999999988e-310 < x.re

   1. Initial program 40.6%

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

   3. Simplified 79.9%

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

   4. Taylor expanded in x.im around 0 80.9%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -100000000:\\ \;\;\;\;e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;x.re \leq -4 \cdot 10^{-310}:\\ \;\;\;\;e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(y.im \cdot \log \left(\frac{-1}{x.re}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log x.re\right)\\ \end{array} \]

Alternative 2: 79.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ t_1 := e^{t_0 \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{if}\;x.re \leq -0.00024:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \cos \left(\mathsf{fma}\left(t_0, y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (hypot x.re x.im)))
        (t_1 (exp (- (* t_0 y.re) (* (atan2 x.im x.re) y.im)))))
   (if (<= x.re -0.00024)
     t_1
     (* t_1 (cos (fma t_0 y.im (* y.re (atan2 x.im x.re))))))))

C

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 = exp(((t_0 * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im)));
	double tmp;
	if (x_46_re <= -0.00024) {
		tmp = t_1;
	} else {
		tmp = t_1 * cos(fma(t_0, y_46_im, (y_46_re * atan2(x_46_im, x_46_re))));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log(hypot(x_46_re, x_46_im))
	t_1 = exp(Float64(Float64(t_0 * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im)))
	tmp = 0.0
	if (x_46_re <= -0.00024)
		tmp = t_1;
	else
		tmp = Float64(t_1 * cos(fma(t_0, y_46_im, Float64(y_46_re * atan(x_46_im, x_46_re)))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(N[(t$95$0 * y$46$re), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$re, -0.00024], t$95$1, N[(t$95$1 * N[Cos[N[(t$95$0 * y$46$im + N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x.re < -2.40000000000000006e-4

   1. Initial program 22.1%

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

   3. Simplified 72.1%

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

   4. Taylor expanded in y.im around 0 82.4%

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

   5. Taylor expanded in y.re around 0 91.2%

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

   if -2.40000000000000006e-4 < x.re

   1. Initial program 45.1%

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

   3. Simplified 81.5%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -0.00024:\\ \;\;\;\;e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \end{array} \]

Alternative 3: 80.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{if}\;x.re \leq 2 \cdot 10^{-187}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log x.re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (exp
          (- (* (log (hypot x.re x.im)) y.re) (* (atan2 x.im x.re) y.im)))))
   (if (<= x.re 2e-187)
     t_0
     (* t_0 (cos (+ (* y.re (atan2 x.im x.re)) (* y.im (log x.re))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = exp(((log(hypot(x_46_re, x_46_im)) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im)));
	double tmp;
	if (x_46_re <= 2e-187) {
		tmp = t_0;
	} else {
		tmp = t_0 * cos(((y_46_re * atan2(x_46_im, x_46_re)) + (y_46_im * log(x_46_re))));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.exp(((Math.log(Math.hypot(x_46_re, x_46_im)) * y_46_re) - (Math.atan2(x_46_im, x_46_re) * y_46_im)));
	double tmp;
	if (x_46_re <= 2e-187) {
		tmp = t_0;
	} else {
		tmp = t_0 * Math.cos(((y_46_re * Math.atan2(x_46_im, x_46_re)) + (y_46_im * Math.log(x_46_re))));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.exp(((math.log(math.hypot(x_46_re, x_46_im)) * y_46_re) - (math.atan2(x_46_im, x_46_re) * y_46_im)))
	tmp = 0
	if x_46_re <= 2e-187:
		tmp = t_0
	else:
		tmp = t_0 * math.cos(((y_46_re * math.atan2(x_46_im, x_46_re)) + (y_46_im * math.log(x_46_re))))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = exp(Float64(Float64(log(hypot(x_46_re, x_46_im)) * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im)))
	tmp = 0.0
	if (x_46_re <= 2e-187)
		tmp = t_0;
	else
		tmp = Float64(t_0 * cos(Float64(Float64(y_46_re * atan(x_46_im, x_46_re)) + Float64(y_46_im * log(x_46_re)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = exp(((log(hypot(x_46_re, x_46_im)) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im)));
	tmp = 0.0;
	if (x_46_re <= 2e-187)
		tmp = t_0;
	else
		tmp = t_0 * cos(((y_46_re * atan2(x_46_im, x_46_re)) + (y_46_im * log(x_46_re))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Exp[N[(N[(N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$re, 2e-187], t$95$0, N[(t$95$0 * N[Cos[N[(N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision] + N[(y$46$im * N[Log[x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x.re < 2e-187

   1. Initial program 39.5%

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

   3. Simplified 79.6%

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

   4. Taylor expanded in y.im around 0 82.7%

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

   5. Taylor expanded in y.re around 0 85.8%

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

   if 2e-187 < x.re

   1. Initial program 38.2%

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

   3. Simplified 78.0%

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

   4. Taylor expanded in x.im around 0 79.2%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 2 \cdot 10^{-187}:\\ \;\;\;\;e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log x.re\right)\\ \end{array} \]

Alternative 4: 79.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{if}\;x.re \leq -0.0001:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (exp
          (- (* (log (hypot x.re x.im)) y.re) (* (atan2 x.im x.re) y.im)))))
   (if (<= x.re -0.0001) t_0 (* t_0 (cos (* y.re (atan2 x.im x.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = exp(((log(hypot(x_46_re, x_46_im)) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im)));
	double tmp;
	if (x_46_re <= -0.0001) {
		tmp = t_0;
	} else {
		tmp = t_0 * cos((y_46_re * atan2(x_46_im, x_46_re)));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.exp(((Math.log(Math.hypot(x_46_re, x_46_im)) * y_46_re) - (Math.atan2(x_46_im, x_46_re) * y_46_im)));
	double tmp;
	if (x_46_re <= -0.0001) {
		tmp = t_0;
	} else {
		tmp = t_0 * Math.cos((y_46_re * Math.atan2(x_46_im, x_46_re)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.exp(((math.log(math.hypot(x_46_re, x_46_im)) * y_46_re) - (math.atan2(x_46_im, x_46_re) * y_46_im)))
	tmp = 0
	if x_46_re <= -0.0001:
		tmp = t_0
	else:
		tmp = t_0 * math.cos((y_46_re * math.atan2(x_46_im, x_46_re)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = exp(Float64(Float64(log(hypot(x_46_re, x_46_im)) * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im)))
	tmp = 0.0
	if (x_46_re <= -0.0001)
		tmp = t_0;
	else
		tmp = Float64(t_0 * cos(Float64(y_46_re * atan(x_46_im, x_46_re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = exp(((log(hypot(x_46_re, x_46_im)) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im)));
	tmp = 0.0;
	if (x_46_re <= -0.0001)
		tmp = t_0;
	else
		tmp = t_0 * cos((y_46_re * atan2(x_46_im, x_46_re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Exp[N[(N[(N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$re, -0.0001], t$95$0, N[(t$95$0 * N[Cos[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x.re < -1.00000000000000005e-4

   1. Initial program 22.1%

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

   3. Simplified 72.1%

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

   4. Taylor expanded in y.im around 0 82.4%

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

   5. Taylor expanded in y.re around 0 91.2%

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

   if -1.00000000000000005e-4 < x.re

   1. Initial program 45.1%

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

   3. Simplified 81.5%

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

   4. Taylor expanded in y.im around 0 80.1%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -0.0001:\\ \;\;\;\;e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \end{array} \]

Alternative 5: 80.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (exp (- (* (log (hypot x.re x.im)) y.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(hypot(x_46_re, x_46_im)) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im)));
}

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.hypot(x_46_re, x_46_im)) * y_46_re) - (Math.atan2(x_46_im, x_46_re) * y_46_im)));
}

Python

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

Julia

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

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = exp(((log(hypot(x_46_re, x_46_im)) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im)));
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[Exp[N[(N[(N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 39.0%

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

3. Simplified 79.0%

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

4. Taylor expanded in y.im around 0 80.7%

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

5. Taylor expanded in y.re around 0 81.0%

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

6. Final simplification 81.0%

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

Alternative 6: 62.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{y.re \cdot \left(-\log \left(\frac{-1}{x.im}\right)\right)}\\ \mathbf{if}\;x.im \leq -9.5 \cdot 10^{+181}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.im \leq -1.42 \cdot 10^{+156}:\\ \;\;\;\;e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{elif}\;x.im \leq -1 \cdot 10^{-310}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (exp (* y.re (- (log (/ -1.0 x.im)))))))
   (if (<= x.im -9.5e+181)
     t_0
     (if (<= x.im -1.42e+156)
       (exp (* (atan2 x.im x.re) (- y.im)))
       (if (<= x.im -1e-310)
         t_0
         (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) {
	double t_0 = exp((y_46_re * -log((-1.0 / x_46_im))));
	double tmp;
	if (x_46_im <= -9.5e+181) {
		tmp = t_0;
	} else if (x_46_im <= -1.42e+156) {
		tmp = exp((atan2(x_46_im, x_46_re) * -y_46_im));
	} else if (x_46_im <= -1e-310) {
		tmp = t_0;
	} else {
		tmp = exp(((y_46_re * log(x_46_im)) - (atan2(x_46_im, x_46_re) * y_46_im)));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp((y_46re * -log(((-1.0d0) / x_46im))))
    if (x_46im <= (-9.5d+181)) then
        tmp = t_0
    else if (x_46im <= (-1.42d+156)) then
        tmp = exp((atan2(x_46im, x_46re) * -y_46im))
    else if (x_46im <= (-1d-310)) then
        tmp = t_0
    else
        tmp = exp(((y_46re * log(x_46im)) - (atan2(x_46im, x_46re) * y_46im)))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.exp((y_46_re * -Math.log((-1.0 / x_46_im))));
	double tmp;
	if (x_46_im <= -9.5e+181) {
		tmp = t_0;
	} else if (x_46_im <= -1.42e+156) {
		tmp = Math.exp((Math.atan2(x_46_im, x_46_re) * -y_46_im));
	} else if (x_46_im <= -1e-310) {
		tmp = t_0;
	} else {
		tmp = Math.exp(((y_46_re * Math.log(x_46_im)) - (Math.atan2(x_46_im, x_46_re) * y_46_im)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.exp((y_46_re * -math.log((-1.0 / x_46_im))))
	tmp = 0
	if x_46_im <= -9.5e+181:
		tmp = t_0
	elif x_46_im <= -1.42e+156:
		tmp = math.exp((math.atan2(x_46_im, x_46_re) * -y_46_im))
	elif x_46_im <= -1e-310:
		tmp = t_0
	else:
		tmp = math.exp(((y_46_re * math.log(x_46_im)) - (math.atan2(x_46_im, x_46_re) * y_46_im)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = exp(Float64(y_46_re * Float64(-log(Float64(-1.0 / x_46_im)))))
	tmp = 0.0
	if (x_46_im <= -9.5e+181)
		tmp = t_0;
	elseif (x_46_im <= -1.42e+156)
		tmp = exp(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im)));
	elseif (x_46_im <= -1e-310)
		tmp = t_0;
	else
		tmp = exp(Float64(Float64(y_46_re * log(x_46_im)) - Float64(atan(x_46_im, x_46_re) * y_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = exp((y_46_re * -log((-1.0 / x_46_im))));
	tmp = 0.0;
	if (x_46_im <= -9.5e+181)
		tmp = t_0;
	elseif (x_46_im <= -1.42e+156)
		tmp = exp((atan2(x_46_im, x_46_re) * -y_46_im));
	elseif (x_46_im <= -1e-310)
		tmp = t_0;
	else
		tmp = exp(((y_46_re * log(x_46_im)) - (atan2(x_46_im, x_46_re) * y_46_im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Exp[N[(y$46$re * (-N[Log[N[(-1.0 / x$46$im), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$im, -9.5e+181], t$95$0, If[LessEqual[x$46$im, -1.42e+156], N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision], If[LessEqual[x$46$im, -1e-310], t$95$0, N[Exp[N[(N[(y$46$re * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -9.50000000000000032e181 or -1.41999999999999998e156 < x.im < -9.999999999999969e-311

   1. Initial program 45.7%

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

   3. Simplified 83.6%

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

   4. Taylor expanded in y.im around 0 82.5%

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

   5. Taylor expanded in y.re around 0 83.1%

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

   6. Taylor expanded in y.re around inf 60.2%

      \[\leadsto e^{\color{blue}{y.re \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}} \cdot 1 \]
   7. Step-by-step derivation

      1. unpow2 60.2%

         \[\leadsto e^{y.re \cdot \log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)} \cdot 1 \]

      2. unpow2 60.2%

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

      3. hypot-def 68.6%

         \[\leadsto e^{y.re \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}} \cdot 1 \]

      4. hypot-def 60.2%

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

      5. unpow2 60.2%

         \[\leadsto e^{y.re \cdot \log \left(\sqrt{\color{blue}{{x.im}^{2}} + x.re \cdot x.re}\right)} \cdot 1 \]

      6. unpow2 60.2%

         \[\leadsto e^{y.re \cdot \log \left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)} \cdot 1 \]

      7. +-commutative 60.2%

         \[\leadsto e^{y.re \cdot \log \left(\sqrt{\color{blue}{{x.re}^{2} + {x.im}^{2}}}\right)} \cdot 1 \]

      8. unpow2 60.2%

         \[\leadsto e^{y.re \cdot \log \left(\sqrt{\color{blue}{x.re \cdot x.re} + {x.im}^{2}}\right)} \cdot 1 \]

      9. unpow2 60.2%

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

      10. hypot-def 68.6%

          \[\leadsto e^{y.re \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}} \cdot 1 \]

      11. log-pow 68.6%

          \[\leadsto e^{\color{blue}{\log \left({\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\right)}} \cdot 1 \]

      12. hypot-def 60.2%

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

      13. unpow2 60.2%

          \[\leadsto e^{\log \left({\left(\sqrt{\color{blue}{{x.re}^{2}} + x.im \cdot x.im}\right)}^{y.re}\right)} \cdot 1 \]

      14. unpow2 60.2%

          \[\leadsto e^{\log \left({\left(\sqrt{{x.re}^{2} + \color{blue}{{x.im}^{2}}}\right)}^{y.re}\right)} \cdot 1 \]

      15. +-commutative 60.2%

          \[\leadsto e^{\log \left({\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re}\right)} \cdot 1 \]

      16. unpow2 60.2%

          \[\leadsto e^{\log \left({\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re}\right)} \cdot 1 \]

      17. unpow2 60.2%

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

      18. hypot-def 68.6%

          \[\leadsto e^{\log \left({\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re}\right)} \cdot 1 \]

   8. Simplified 68.6%

      \[\leadsto e^{\color{blue}{\log \left({\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)}} \cdot 1 \]

   9. Taylor expanded in x.im around -inf 58.8%

      \[\leadsto e^{\color{blue}{-1 \cdot \left(\log \left(\frac{-1}{x.im}\right) \cdot y.re\right)}} \cdot 1 \]
   10. Step-by-step derivation

       1. mul-1-neg 58.8%

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

       2. *-commutative 58.8%

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

       3. distribute-rgt-neg-in 58.8%

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

   11. Simplified 58.8%

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

   if -9.50000000000000032e181 < x.im < -1.41999999999999998e156

   1. Initial program 0.0%

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

   3. Simplified 76.9%

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

   4. Taylor expanded in y.im around 0 100.0%

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

   5. Taylor expanded in y.re around 0 100.0%

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

   6. Taylor expanded in y.re around 0 92.5%

      \[\leadsto e^{\color{blue}{-1 \cdot \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot 1 \]
   7. Step-by-step derivation

      1. mul-1-neg 92.5%

         \[\leadsto e^{\color{blue}{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot 1 \]

      2. *-commutative 92.5%

         \[\leadsto e^{-\color{blue}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot 1 \]

      3. distribute-lft-neg-in 92.5%

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

      4. *-commutative 92.5%

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

   8. Simplified 92.5%

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

   if -9.999999999999969e-311 < x.im

   1. Initial program 36.1%

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

   2. Taylor expanded in y.im around 0 62.1%

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

   3. Taylor expanded in y.re around 0 62.1%

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

   4. Taylor expanded in x.re around 0 70.6%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -9.5 \cdot 10^{+181}:\\ \;\;\;\;e^{y.re \cdot \left(-\log \left(\frac{-1}{x.im}\right)\right)}\\ \mathbf{elif}\;x.im \leq -1.42 \cdot 10^{+156}:\\ \;\;\;\;e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{elif}\;x.im \leq -1 \cdot 10^{-310}:\\ \;\;\;\;e^{y.re \cdot \left(-\log \left(\frac{-1}{x.im}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array} \]

Alternative 7: 76.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -30500 \lor \neg \left(y.re \leq 10\right):\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -30500.0) (not (<= y.re 10.0)))
   (exp (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im))))))
   (exp (* (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) {
	double tmp;
	if ((y_46_re <= -30500.0) || !(y_46_re <= 10.0)) {
		tmp = exp((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))));
	} else {
		tmp = exp((atan2(x_46_im, x_46_re) * -y_46_im));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-30500.0d0)) .or. (.not. (y_46re <= 10.0d0))) then
        tmp = exp((y_46re * log(sqrt(((x_46re * x_46re) + (x_46im * x_46im))))))
    else
        tmp = exp((atan2(x_46im, x_46re) * -y_46im))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -30500.0) || !(y_46_re <= 10.0)) {
		tmp = Math.exp((y_46_re * Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))));
	} else {
		tmp = Math.exp((Math.atan2(x_46_im, x_46_re) * -y_46_im));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -30500.0) or not (y_46_re <= 10.0):
		tmp = math.exp((y_46_re * math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))))
	else:
		tmp = math.exp((math.atan2(x_46_im, x_46_re) * -y_46_im))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -30500.0) || !(y_46_re <= 10.0))
		tmp = exp(Float64(y_46_re * log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))));
	else
		tmp = exp(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -30500.0) || ~((y_46_re <= 10.0)))
		tmp = exp((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))));
	else
		tmp = exp((atan2(x_46_im, x_46_re) * -y_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -30500.0], N[Not[LessEqual[y$46$re, 10.0]], $MachinePrecision]], N[Exp[N[(y$46$re * N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -30500 or 10 < y.re

   1. Initial program 36.5%

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

   3. Simplified 75.4%

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

   4. Taylor expanded in y.im around 0 78.6%

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

   5. Taylor expanded in y.re around 0 79.4%

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

   6. Taylor expanded in y.re around inf 74.8%

      \[\leadsto e^{\color{blue}{y.re \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}} \cdot 1 \]
   7. Step-by-step derivation

      1. +-commutative 74.8%

         \[\leadsto e^{y.re \cdot \log \left(\sqrt{\color{blue}{{x.re}^{2} + {x.im}^{2}}}\right)} \cdot 1 \]

      2. unpow2 74.8%

         \[\leadsto e^{y.re \cdot \log \left(\sqrt{\color{blue}{x.re \cdot x.re} + {x.im}^{2}}\right)} \cdot 1 \]

      3. unpow2 74.8%

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

   8. Simplified 74.8%

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

   if -30500 < y.re < 10

   1. Initial program 41.4%

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

   3. Simplified 82.5%

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

   4. Taylor expanded in y.im around 0 82.7%

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

   5. Taylor expanded in y.re around 0 82.5%

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

   6. Taylor expanded in y.re around 0 81.0%

      \[\leadsto e^{\color{blue}{-1 \cdot \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot 1 \]
   7. Step-by-step derivation

      1. mul-1-neg 81.0%

         \[\leadsto e^{\color{blue}{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot 1 \]

      2. *-commutative 81.0%

         \[\leadsto e^{-\color{blue}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot 1 \]

      3. distribute-lft-neg-in 81.0%

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

      4. *-commutative 81.0%

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

   8. Simplified 81.0%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -30500 \lor \neg \left(y.re \leq 10\right):\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \end{array} \]

Alternative 8: 63.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq 1.02 \cdot 10^{-247}:\\ \;\;\;\;e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.re 1.02e-247)
   (exp (* (atan2 x.im x.re) (- y.im)))
   (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) {
	double tmp;
	if (x_46_re <= 1.02e-247) {
		tmp = exp((atan2(x_46_im, x_46_re) * -y_46_im));
	} else {
		tmp = exp(((y_46_re * log(x_46_re)) - (atan2(x_46_im, x_46_re) * y_46_im)));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (x_46re <= 1.02d-247) then
        tmp = exp((atan2(x_46im, x_46re) * -y_46im))
    else
        tmp = exp(((y_46re * log(x_46re)) - (atan2(x_46im, x_46re) * y_46im)))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_re <= 1.02e-247) {
		tmp = Math.exp((Math.atan2(x_46_im, x_46_re) * -y_46_im));
	} else {
		tmp = Math.exp(((y_46_re * Math.log(x_46_re)) - (Math.atan2(x_46_im, x_46_re) * y_46_im)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if x_46_re <= 1.02e-247:
		tmp = math.exp((math.atan2(x_46_im, x_46_re) * -y_46_im))
	else:
		tmp = math.exp(((y_46_re * math.log(x_46_re)) - (math.atan2(x_46_im, x_46_re) * y_46_im)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (x_46_re <= 1.02e-247)
		tmp = exp(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im)));
	else
		tmp = exp(Float64(Float64(y_46_re * log(x_46_re)) - Float64(atan(x_46_im, x_46_re) * y_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (x_46_re <= 1.02e-247)
		tmp = exp((atan2(x_46_im, x_46_re) * -y_46_im));
	else
		tmp = exp(((y_46_re * log(x_46_re)) - (atan2(x_46_im, x_46_re) * y_46_im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[x$46$re, 1.02e-247], N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision], N[Exp[N[(N[(y$46$re * N[Log[x$46$re], $MachinePrecision]), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.01999999999999994e-247

   1. Initial program 39.2%

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

   3. Simplified 79.1%

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

   4. Taylor expanded in y.im around 0 81.8%

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

   5. Taylor expanded in y.re around 0 85.1%

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

   6. Taylor expanded in y.re around 0 61.6%

      \[\leadsto e^{\color{blue}{-1 \cdot \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot 1 \]
   7. Step-by-step derivation

      1. mul-1-neg 61.6%

         \[\leadsto e^{\color{blue}{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot 1 \]

      2. *-commutative 61.6%

         \[\leadsto e^{-\color{blue}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot 1 \]

      3. distribute-lft-neg-in 61.6%

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

      4. *-commutative 61.6%

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

   8. Simplified 61.6%

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

   if 1.01999999999999994e-247 < x.re

   1. Initial program 38.8%

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

   2. Taylor expanded in y.im around 0 63.5%

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

   3. Taylor expanded in y.re around 0 59.8%

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

   4. Taylor expanded in x.re around inf 67.6%

      \[\leadsto e^{\log \color{blue}{x.re} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 1.02 \cdot 10^{-247}:\\ \;\;\;\;e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array} \]

Alternative 9: 57.8% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq 1.2 \cdot 10^{+28}:\\ \;\;\;\;e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.re 1.2e+28)
   (exp (* (atan2 x.im x.re) (- y.im)))
   (exp (* y.re (log x.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_re <= 1.2e+28) {
		tmp = exp((atan2(x_46_im, x_46_re) * -y_46_im));
	} else {
		tmp = exp((y_46_re * log(x_46_re)));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (x_46re <= 1.2d+28) then
        tmp = exp((atan2(x_46im, x_46re) * -y_46im))
    else
        tmp = exp((y_46re * log(x_46re)))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_re <= 1.2e+28) {
		tmp = Math.exp((Math.atan2(x_46_im, x_46_re) * -y_46_im));
	} else {
		tmp = Math.exp((y_46_re * Math.log(x_46_re)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if x_46_re <= 1.2e+28:
		tmp = math.exp((math.atan2(x_46_im, x_46_re) * -y_46_im))
	else:
		tmp = math.exp((y_46_re * math.log(x_46_re)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (x_46_re <= 1.2e+28)
		tmp = exp(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im)));
	else
		tmp = exp(Float64(y_46_re * log(x_46_re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (x_46_re <= 1.2e+28)
		tmp = exp((atan2(x_46_im, x_46_re) * -y_46_im));
	else
		tmp = exp((y_46_re * log(x_46_re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[x$46$re, 1.2e+28], N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision], N[Exp[N[(y$46$re * N[Log[x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.19999999999999991e28

   1. Initial program 41.7%

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

   3. Simplified 78.5%

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

   4. Taylor expanded in y.im around 0 80.9%

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

   5. Taylor expanded in y.re around 0 82.3%

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

   6. Taylor expanded in y.re around 0 57.1%

      \[\leadsto e^{\color{blue}{-1 \cdot \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot 1 \]
   7. Step-by-step derivation

      1. mul-1-neg 57.1%

         \[\leadsto e^{\color{blue}{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot 1 \]

      2. *-commutative 57.1%

         \[\leadsto e^{-\color{blue}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot 1 \]

      3. distribute-lft-neg-in 57.1%

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

      4. *-commutative 57.1%

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

   8. Simplified 57.1%

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

   if 1.19999999999999991e28 < x.re

   1. Initial program 28.5%

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

   3. Simplified 81.2%

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

   4. Taylor expanded in y.im around 0 80.0%

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

   5. Taylor expanded in y.re around 0 75.6%

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

   6. Taylor expanded in y.re around inf 54.7%

      \[\leadsto e^{\color{blue}{y.re \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}} \cdot 1 \]
   7. Step-by-step derivation

      1. unpow2 54.7%

         \[\leadsto e^{y.re \cdot \log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)} \cdot 1 \]

      2. unpow2 54.7%

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

      3. hypot-def 64.4%

         \[\leadsto e^{y.re \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}} \cdot 1 \]

      4. hypot-def 54.7%

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

      5. unpow2 54.7%

         \[\leadsto e^{y.re \cdot \log \left(\sqrt{\color{blue}{{x.im}^{2}} + x.re \cdot x.re}\right)} \cdot 1 \]

      6. unpow2 54.7%

         \[\leadsto e^{y.re \cdot \log \left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)} \cdot 1 \]

      7. +-commutative 54.7%

         \[\leadsto e^{y.re \cdot \log \left(\sqrt{\color{blue}{{x.re}^{2} + {x.im}^{2}}}\right)} \cdot 1 \]

      8. unpow2 54.7%

         \[\leadsto e^{y.re \cdot \log \left(\sqrt{\color{blue}{x.re \cdot x.re} + {x.im}^{2}}\right)} \cdot 1 \]

      9. unpow2 54.7%

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

      10. hypot-def 64.4%

          \[\leadsto e^{y.re \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}} \cdot 1 \]

      11. log-pow 64.4%

          \[\leadsto e^{\color{blue}{\log \left({\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\right)}} \cdot 1 \]

      12. hypot-def 54.7%

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

      13. unpow2 54.7%

          \[\leadsto e^{\log \left({\left(\sqrt{\color{blue}{{x.re}^{2}} + x.im \cdot x.im}\right)}^{y.re}\right)} \cdot 1 \]

      14. unpow2 54.7%

          \[\leadsto e^{\log \left({\left(\sqrt{{x.re}^{2} + \color{blue}{{x.im}^{2}}}\right)}^{y.re}\right)} \cdot 1 \]

      15. +-commutative 54.7%

          \[\leadsto e^{\log \left({\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re}\right)} \cdot 1 \]

      16. unpow2 54.7%

          \[\leadsto e^{\log \left({\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re}\right)} \cdot 1 \]

      17. unpow2 54.7%

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

      18. hypot-def 64.4%

          \[\leadsto e^{\log \left({\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re}\right)} \cdot 1 \]

   8. Simplified 64.4%

      \[\leadsto e^{\color{blue}{\log \left({\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)}} \cdot 1 \]

   9. Taylor expanded in x.im around 0 63.7%

      \[\leadsto e^{\color{blue}{\log \left({x.re}^{y.re}\right)}} \cdot 1 \]
   10. Step-by-step derivation

       1. log-pow 63.7%

          \[\leadsto e^{\color{blue}{y.re \cdot \log x.re}} \cdot 1 \]

   11. Simplified 63.7%

       \[\leadsto e^{\color{blue}{y.re \cdot \log x.re}} \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 1.2 \cdot 10^{+28}:\\ \;\;\;\;e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.re}\\ \end{array} \]

Alternative 10: 41.7% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -1 \cdot 10^{-310}:\\ \;\;\;\;e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.im -1e-310)
   (exp (* (atan2 x.im x.re) y.im))
   (exp (* y.re (log x.im)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_im <= -1e-310) {
		tmp = exp((atan2(x_46_im, x_46_re) * y_46_im));
	} else {
		tmp = exp((y_46_re * log(x_46_im)));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (x_46im <= (-1d-310)) then
        tmp = exp((atan2(x_46im, x_46re) * y_46im))
    else
        tmp = exp((y_46re * log(x_46im)))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_im <= -1e-310) {
		tmp = Math.exp((Math.atan2(x_46_im, x_46_re) * y_46_im));
	} else {
		tmp = Math.exp((y_46_re * Math.log(x_46_im)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if x_46_im <= -1e-310:
		tmp = math.exp((math.atan2(x_46_im, x_46_re) * y_46_im))
	else:
		tmp = math.exp((y_46_re * math.log(x_46_im)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (x_46_im <= -1e-310)
		tmp = exp(Float64(atan(x_46_im, x_46_re) * y_46_im));
	else
		tmp = exp(Float64(y_46_re * log(x_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (x_46_im <= -1e-310)
		tmp = exp((atan2(x_46_im, x_46_re) * y_46_im));
	else
		tmp = exp((y_46_re * log(x_46_im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[x$46$im, -1e-310], N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision], N[Exp[N[(y$46$re * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -9.999999999999969e-311

   1. Initial program 41.4%

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

   3. Simplified 83.0%

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

   4. Taylor expanded in y.im around 0 84.2%

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

   5. Taylor expanded in y.re around 0 84.7%

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

   6. Taylor expanded in y.re around 0 54.1%

      \[\leadsto e^{\color{blue}{-1 \cdot \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot 1 \]
   7. Step-by-step derivation

      1. mul-1-neg 54.1%

         \[\leadsto e^{\color{blue}{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot 1 \]

      2. *-commutative 54.1%

         \[\leadsto e^{-\color{blue}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot 1 \]

      3. distribute-lft-neg-in 54.1%

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

      4. *-commutative 54.1%

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

   8. Simplified 54.1%

      \[\leadsto e^{\color{blue}{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot 1 \]
   9. Step-by-step derivation

      1. *-commutative 54.1%

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

      2. add-sqr-sqrt 54.1%

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

      3. sqrt-unprod 54.1%

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

      4. sqr-neg 54.1%

         \[\leadsto e^{\sqrt{\color{blue}{\tan^{-1}_* \frac{x.im}{x.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot y.im} \cdot 1 \]

      5. sqrt-unprod 0.2%

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

      6. add-sqr-sqrt 36.6%

         \[\leadsto e^{\color{blue}{\tan^{-1}_* \frac{x.im}{x.re}} \cdot y.im} \cdot 1 \]

      7. expm1-log1p-u 30.6%

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

      8. expm1-udef 30.6%

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

   10. Applied egg-rr 30.6%

       \[\leadsto e^{\color{blue}{e^{\mathsf{log1p}\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right)} - 1}} \cdot 1 \]
   11. Step-by-step derivation

       1. expm1-def 30.6%

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

       2. expm1-log1p 36.6%

          \[\leadsto e^{\color{blue}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot 1 \]

       3. *-commutative 36.6%

          \[\leadsto e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot 1 \]

   12. Simplified 36.6%

       \[\leadsto e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot 1 \]

   if -9.999999999999969e-311 < x.im

   1. Initial program 36.1%

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

   3. Simplified 74.2%

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

   4. Taylor expanded in y.im around 0 76.5%

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

   5. Taylor expanded in y.re around 0 76.5%

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

   6. Taylor expanded in y.re around inf 52.9%

      \[\leadsto e^{\color{blue}{y.re \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}} \cdot 1 \]
   7. Step-by-step derivation

      1. unpow2 52.9%

         \[\leadsto e^{y.re \cdot \log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)} \cdot 1 \]

      2. unpow2 52.9%

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

      3. hypot-def 57.4%

         \[\leadsto e^{y.re \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}} \cdot 1 \]

      4. hypot-def 52.9%

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

      5. unpow2 52.9%

         \[\leadsto e^{y.re \cdot \log \left(\sqrt{\color{blue}{{x.im}^{2}} + x.re \cdot x.re}\right)} \cdot 1 \]

      6. unpow2 52.9%

         \[\leadsto e^{y.re \cdot \log \left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)} \cdot 1 \]

      7. +-commutative 52.9%

         \[\leadsto e^{y.re \cdot \log \left(\sqrt{\color{blue}{{x.re}^{2} + {x.im}^{2}}}\right)} \cdot 1 \]

      8. unpow2 52.9%

         \[\leadsto e^{y.re \cdot \log \left(\sqrt{\color{blue}{x.re \cdot x.re} + {x.im}^{2}}\right)} \cdot 1 \]

      9. unpow2 52.9%

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

      10. hypot-def 57.4%

          \[\leadsto e^{y.re \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}} \cdot 1 \]

      11. log-pow 57.4%

          \[\leadsto e^{\color{blue}{\log \left({\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\right)}} \cdot 1 \]

      12. hypot-def 52.9%

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

      13. unpow2 52.9%

          \[\leadsto e^{\log \left({\left(\sqrt{\color{blue}{{x.re}^{2}} + x.im \cdot x.im}\right)}^{y.re}\right)} \cdot 1 \]

      14. unpow2 52.9%

          \[\leadsto e^{\log \left({\left(\sqrt{{x.re}^{2} + \color{blue}{{x.im}^{2}}}\right)}^{y.re}\right)} \cdot 1 \]

      15. +-commutative 52.9%

          \[\leadsto e^{\log \left({\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re}\right)} \cdot 1 \]

      16. unpow2 52.9%

          \[\leadsto e^{\log \left({\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re}\right)} \cdot 1 \]

      17. unpow2 52.9%

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

      18. hypot-def 57.4%

          \[\leadsto e^{\log \left({\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re}\right)} \cdot 1 \]

   8. Simplified 57.4%

      \[\leadsto e^{\color{blue}{\log \left({\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)}} \cdot 1 \]

   9. Taylor expanded in x.re around 0 51.4%

      \[\leadsto e^{\color{blue}{\log \left({x.im}^{y.re}\right)}} \cdot 1 \]
   10. Step-by-step derivation

       1. log-pow 51.4%

          \[\leadsto e^{\color{blue}{y.re \cdot \log x.im}} \cdot 1 \]

   11. Simplified 51.4%

       \[\leadsto e^{\color{blue}{y.re \cdot \log x.im}} \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -1 \cdot 10^{-310}:\\ \;\;\;\;e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im}\\ \end{array} \]

Alternative 11: 40.2% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq 2 \cdot 10^{-233}:\\ \;\;\;\;e^{y.re \cdot \log x.re}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.im 2e-233) (exp (* y.re (log x.re))) (exp (* y.re (log x.im)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_im <= 2e-233) {
		tmp = exp((y_46_re * log(x_46_re)));
	} else {
		tmp = exp((y_46_re * log(x_46_im)));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (x_46im <= 2d-233) then
        tmp = exp((y_46re * log(x_46re)))
    else
        tmp = exp((y_46re * log(x_46im)))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_im <= 2e-233) {
		tmp = Math.exp((y_46_re * Math.log(x_46_re)));
	} else {
		tmp = Math.exp((y_46_re * Math.log(x_46_im)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if x_46_im <= 2e-233:
		tmp = math.exp((y_46_re * math.log(x_46_re)))
	else:
		tmp = math.exp((y_46_re * math.log(x_46_im)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (x_46_im <= 2e-233)
		tmp = exp(Float64(y_46_re * log(x_46_re)));
	else
		tmp = exp(Float64(y_46_re * log(x_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (x_46_im <= 2e-233)
		tmp = exp((y_46_re * log(x_46_re)));
	else
		tmp = exp((y_46_re * log(x_46_im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[x$46$im, 2e-233], N[Exp[N[(y$46$re * N[Log[x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[N[(y$46$re * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 1.99999999999999992e-233

   1. Initial program 39.4%

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

   3. Simplified 80.8%

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

   4. Taylor expanded in y.im around 0 83.3%

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

   5. Taylor expanded in y.re around 0 83.7%

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

   6. Taylor expanded in y.re around inf 57.4%

      \[\leadsto e^{\color{blue}{y.re \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}} \cdot 1 \]
   7. Step-by-step derivation

      1. unpow2 57.4%

         \[\leadsto e^{y.re \cdot \log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)} \cdot 1 \]

      2. unpow2 57.4%

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

      3. hypot-def 66.8%

         \[\leadsto e^{y.re \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}} \cdot 1 \]

      4. hypot-def 57.4%

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

      5. unpow2 57.4%

         \[\leadsto e^{y.re \cdot \log \left(\sqrt{\color{blue}{{x.im}^{2}} + x.re \cdot x.re}\right)} \cdot 1 \]

      6. unpow2 57.4%

         \[\leadsto e^{y.re \cdot \log \left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)} \cdot 1 \]

      7. +-commutative 57.4%

         \[\leadsto e^{y.re \cdot \log \left(\sqrt{\color{blue}{{x.re}^{2} + {x.im}^{2}}}\right)} \cdot 1 \]

      8. unpow2 57.4%

         \[\leadsto e^{y.re \cdot \log \left(\sqrt{\color{blue}{x.re \cdot x.re} + {x.im}^{2}}\right)} \cdot 1 \]

      9. unpow2 57.4%

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

      10. hypot-def 66.8%

          \[\leadsto e^{y.re \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}} \cdot 1 \]

      11. log-pow 66.8%

          \[\leadsto e^{\color{blue}{\log \left({\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\right)}} \cdot 1 \]

      12. hypot-def 57.4%

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

      13. unpow2 57.4%

          \[\leadsto e^{\log \left({\left(\sqrt{\color{blue}{{x.re}^{2}} + x.im \cdot x.im}\right)}^{y.re}\right)} \cdot 1 \]

      14. unpow2 57.4%

          \[\leadsto e^{\log \left({\left(\sqrt{{x.re}^{2} + \color{blue}{{x.im}^{2}}}\right)}^{y.re}\right)} \cdot 1 \]

      15. +-commutative 57.4%

          \[\leadsto e^{\log \left({\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re}\right)} \cdot 1 \]

      16. unpow2 57.4%

          \[\leadsto e^{\log \left({\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re}\right)} \cdot 1 \]

      17. unpow2 57.4%

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

      18. hypot-def 66.8%

          \[\leadsto e^{\log \left({\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re}\right)} \cdot 1 \]

   8. Simplified 66.8%

      \[\leadsto e^{\color{blue}{\log \left({\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)}} \cdot 1 \]

   9. Taylor expanded in x.im around 0 40.6%

      \[\leadsto e^{\color{blue}{\log \left({x.re}^{y.re}\right)}} \cdot 1 \]
   10. Step-by-step derivation

       1. log-pow 25.0%

          \[\leadsto e^{\color{blue}{y.re \cdot \log x.re}} \cdot 1 \]

   11. Simplified 25.0%

       \[\leadsto e^{\color{blue}{y.re \cdot \log x.re}} \cdot 1 \]

   if 1.99999999999999992e-233 < x.im

   1. Initial program 38.5%

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

   3. Simplified 76.3%

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

   4. Taylor expanded in y.im around 0 76.8%

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

   5. Taylor expanded in y.re around 0 76.8%

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

   6. Taylor expanded in y.re around inf 51.6%

      \[\leadsto e^{\color{blue}{y.re \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}} \cdot 1 \]
   7. Step-by-step derivation

      1. unpow2 51.6%

         \[\leadsto e^{y.re \cdot \log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)} \cdot 1 \]

      2. unpow2 51.6%

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

      3. hypot-def 56.7%

         \[\leadsto e^{y.re \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}} \cdot 1 \]

      4. hypot-def 51.6%

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

      5. unpow2 51.6%

         \[\leadsto e^{y.re \cdot \log \left(\sqrt{\color{blue}{{x.im}^{2}} + x.re \cdot x.re}\right)} \cdot 1 \]

      6. unpow2 51.6%

         \[\leadsto e^{y.re \cdot \log \left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)} \cdot 1 \]

      7. +-commutative 51.6%

         \[\leadsto e^{y.re \cdot \log \left(\sqrt{\color{blue}{{x.re}^{2} + {x.im}^{2}}}\right)} \cdot 1 \]

      8. unpow2 51.6%

         \[\leadsto e^{y.re \cdot \log \left(\sqrt{\color{blue}{x.re \cdot x.re} + {x.im}^{2}}\right)} \cdot 1 \]

      9. unpow2 51.6%

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

      10. hypot-def 56.7%

          \[\leadsto e^{y.re \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}} \cdot 1 \]

      11. log-pow 56.7%

          \[\leadsto e^{\color{blue}{\log \left({\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\right)}} \cdot 1 \]

      12. hypot-def 51.6%

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

      13. unpow2 51.6%

          \[\leadsto e^{\log \left({\left(\sqrt{\color{blue}{{x.re}^{2}} + x.im \cdot x.im}\right)}^{y.re}\right)} \cdot 1 \]

      14. unpow2 51.6%

          \[\leadsto e^{\log \left({\left(\sqrt{{x.re}^{2} + \color{blue}{{x.im}^{2}}}\right)}^{y.re}\right)} \cdot 1 \]

      15. +-commutative 51.6%

          \[\leadsto e^{\log \left({\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re}\right)} \cdot 1 \]

      16. unpow2 51.6%

          \[\leadsto e^{\log \left({\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re}\right)} \cdot 1 \]

      17. unpow2 51.6%

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

      18. hypot-def 56.7%

          \[\leadsto e^{\log \left({\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re}\right)} \cdot 1 \]

   8. Simplified 56.7%

      \[\leadsto e^{\color{blue}{\log \left({\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)}} \cdot 1 \]

   9. Taylor expanded in x.re around 0 54.7%

      \[\leadsto e^{\color{blue}{\log \left({x.im}^{y.re}\right)}} \cdot 1 \]
   10. Step-by-step derivation

       1. log-pow 54.7%

          \[\leadsto e^{\color{blue}{y.re \cdot \log x.im}} \cdot 1 \]

   11. Simplified 54.7%

       \[\leadsto e^{\color{blue}{y.re \cdot \log x.im}} \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 2 \cdot 10^{-233}:\\ \;\;\;\;e^{y.re \cdot \log x.re}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im}\\ \end{array} \]

Alternative 12: 27.0% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ e^{y.re \cdot \log x.im} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im) :precision binary64 (exp (* y.re (log x.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return exp((y_46_re * log(x_46_im)));
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    code = exp((y_46re * log(x_46im)))
end function

Java

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

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return math.exp((y_46_re * math.log(x_46_im)))

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return exp(Float64(y_46_re * log(x_46_im)))
end

MATLAB

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

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[Exp[N[(y$46$re * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 39.0%

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

3. Simplified 79.0%

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

4. Taylor expanded in y.im around 0 80.7%

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

5. Taylor expanded in y.re around 0 81.0%

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

6. Taylor expanded in y.re around inf 55.1%

   \[\leadsto e^{\color{blue}{y.re \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}} \cdot 1 \]
7. Step-by-step derivation

   1. unpow2 55.1%

      \[\leadsto e^{y.re \cdot \log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)} \cdot 1 \]

   2. unpow2 55.1%

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

   3. hypot-def 62.8%

      \[\leadsto e^{y.re \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}} \cdot 1 \]

   4. hypot-def 55.1%

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

   5. unpow2 55.1%

      \[\leadsto e^{y.re \cdot \log \left(\sqrt{\color{blue}{{x.im}^{2}} + x.re \cdot x.re}\right)} \cdot 1 \]

   6. unpow2 55.1%

      \[\leadsto e^{y.re \cdot \log \left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)} \cdot 1 \]

   7. +-commutative 55.1%

      \[\leadsto e^{y.re \cdot \log \left(\sqrt{\color{blue}{{x.re}^{2} + {x.im}^{2}}}\right)} \cdot 1 \]

   8. unpow2 55.1%

      \[\leadsto e^{y.re \cdot \log \left(\sqrt{\color{blue}{x.re \cdot x.re} + {x.im}^{2}}\right)} \cdot 1 \]

   9. unpow2 55.1%

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

   10. hypot-def 62.8%

       \[\leadsto e^{y.re \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}} \cdot 1 \]

   11. log-pow 62.8%

       \[\leadsto e^{\color{blue}{\log \left({\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\right)}} \cdot 1 \]

   12. hypot-def 55.1%

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

   13. unpow2 55.1%

       \[\leadsto e^{\log \left({\left(\sqrt{\color{blue}{{x.re}^{2}} + x.im \cdot x.im}\right)}^{y.re}\right)} \cdot 1 \]

   14. unpow2 55.1%

       \[\leadsto e^{\log \left({\left(\sqrt{{x.re}^{2} + \color{blue}{{x.im}^{2}}}\right)}^{y.re}\right)} \cdot 1 \]

   15. +-commutative 55.1%

       \[\leadsto e^{\log \left({\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re}\right)} \cdot 1 \]

   16. unpow2 55.1%

       \[\leadsto e^{\log \left({\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re}\right)} \cdot 1 \]

   17. unpow2 55.1%

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

   18. hypot-def 62.8%

       \[\leadsto e^{\log \left({\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re}\right)} \cdot 1 \]

8. Simplified 62.8%

   \[\leadsto e^{\color{blue}{\log \left({\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)}} \cdot 1 \]

9. Taylor expanded in x.re around 0 39.0%

   \[\leadsto e^{\color{blue}{\log \left({x.im}^{y.re}\right)}} \cdot 1 \]
10. Step-by-step derivation

    1. log-pow 23.3%

       \[\leadsto e^{\color{blue}{y.re \cdot \log x.im}} \cdot 1 \]

11. Simplified 23.3%

    \[\leadsto e^{\color{blue}{y.re \cdot \log x.im}} \cdot 1 \]

12. Final simplification 23.3%

    \[\leadsto e^{y.re \cdot \log x.im} \]

Reproduce

herbie shell --seed 2023189 
(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)))))