powComplex, real part

Percentage Accurate: 40.6% → 79.4%

Time: 14.8s

Alternatives: 12

Speedup: 5.7×

• 35.7% 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: 40.6% 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: 79.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\\ t_1 := t\_0 \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{if}\;y.re \leq -7.2 \cdot 10^{-18}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq 2.5 \cdot 10^{-12}:\\ \;\;\;\;e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (cos (* (log (hypot x.re x.im)) y.im)))
        (t_1
         (*
          t_0
          (exp
           (-
            (* (log (sqrt (+ (* x.im x.im) (* x.re x.re)))) y.re)
            (* y.im (atan2 x.im x.re)))))))
   (if (<= y.re -7.2e-18)
     t_1
     (if (<= y.re 2.5e-12) (* (exp (* (- y.im) (atan2 x.im x.re))) t_0) t_1))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = cos((log(hypot(x_46_re, x_46_im)) * y_46_im));
	double t_1 = t_0 * exp(((log(sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re)))) * y_46_re) - (y_46_im * atan2(x_46_im, x_46_re))));
	double tmp;
	if (y_46_re <= -7.2e-18) {
		tmp = t_1;
	} else if (y_46_re <= 2.5e-12) {
		tmp = exp((-y_46_im * atan2(x_46_im, x_46_re))) * t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.cos((Math.log(Math.hypot(x_46_re, x_46_im)) * y_46_im));
	double t_1 = t_0 * Math.exp(((Math.log(Math.sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re)))) * y_46_re) - (y_46_im * Math.atan2(x_46_im, x_46_re))));
	double tmp;
	if (y_46_re <= -7.2e-18) {
		tmp = t_1;
	} else if (y_46_re <= 2.5e-12) {
		tmp = Math.exp((-y_46_im * Math.atan2(x_46_im, x_46_re))) * t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.cos((math.log(math.hypot(x_46_re, x_46_im)) * y_46_im))
	t_1 = t_0 * math.exp(((math.log(math.sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re)))) * y_46_re) - (y_46_im * math.atan2(x_46_im, x_46_re))))
	tmp = 0
	if y_46_re <= -7.2e-18:
		tmp = t_1
	elif y_46_re <= 2.5e-12:
		tmp = math.exp((-y_46_im * math.atan2(x_46_im, x_46_re))) * t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = cos(Float64(log(hypot(x_46_re, x_46_im)) * y_46_im))
	t_1 = Float64(t_0 * exp(Float64(Float64(log(sqrt(Float64(Float64(x_46_im * x_46_im) + Float64(x_46_re * x_46_re)))) * y_46_re) - Float64(y_46_im * atan(x_46_im, x_46_re)))))
	tmp = 0.0
	if (y_46_re <= -7.2e-18)
		tmp = t_1;
	elseif (y_46_re <= 2.5e-12)
		tmp = Float64(exp(Float64(Float64(-y_46_im) * atan(x_46_im, x_46_re))) * t_0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = cos((log(hypot(x_46_re, x_46_im)) * y_46_im));
	t_1 = t_0 * exp(((log(sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re)))) * y_46_re) - (y_46_im * atan2(x_46_im, x_46_re))));
	tmp = 0.0;
	if (y_46_re <= -7.2e-18)
		tmp = t_1;
	elseif (y_46_re <= 2.5e-12)
		tmp = exp((-y_46_im * atan2(x_46_im, x_46_re))) * t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Cos[N[(N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[Exp[N[(N[(N[Log[N[Sqrt[N[(N[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -7.2e-18], t$95$1, If[LessEqual[y$46$re, 2.5e-12], N[(N[Exp[N[((-y$46$im) * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -7.20000000000000021e-18 or 2.49999999999999985e-12 < y.re

   1. Initial program 46.8%

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

   3. Taylor expanded in y.im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 82.7

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

   5. Applied rewrites 82.7%

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

   if -7.20000000000000021e-18 < y.re < 2.49999999999999985e-12

   1. Initial program 38.7%

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

   3. Taylor expanded in y.im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 44.9

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

   5. Applied rewrites 44.9%

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

   6. Taylor expanded in y.im around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. neg-mul-1 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-atan2.f64 77.1

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

   8. Applied rewrites 77.1%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -7.2 \cdot 10^{-18}:\\ \;\;\;\;\cos \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;y.re \leq 2.5 \cdot 10^{-12}:\\ \;\;\;\;e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \cos \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\\ \mathbf{if}\;y.re \leq -2.3 \cdot 10^{-23}:\\ \;\;\;\;\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;y.re \leq 3.8 \cdot 10^{-17}:\\ \;\;\;\;e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (cos (* (log (hypot x.re x.im)) y.im))))
   (if (<= y.re -2.3e-23)
     (*
      (cos (* (atan2 x.im x.re) y.re))
      (exp
       (-
        (* (log (sqrt (+ (* x.im x.im) (* x.re x.re)))) y.re)
        (* y.im (atan2 x.im x.re)))))
     (if (<= y.re 3.8e-17)
       (* (exp (* (- y.im) (atan2 x.im x.re))) t_0)
       (* (pow (hypot x.re x.im) y.re) t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = cos((log(hypot(x_46_re, x_46_im)) * y_46_im));
	double tmp;
	if (y_46_re <= -2.3e-23) {
		tmp = cos((atan2(x_46_im, x_46_re) * y_46_re)) * exp(((log(sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re)))) * y_46_re) - (y_46_im * atan2(x_46_im, x_46_re))));
	} else if (y_46_re <= 3.8e-17) {
		tmp = exp((-y_46_im * atan2(x_46_im, x_46_re))) * t_0;
	} else {
		tmp = pow(hypot(x_46_re, x_46_im), y_46_re) * t_0;
	}
	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.cos((Math.log(Math.hypot(x_46_re, x_46_im)) * y_46_im));
	double tmp;
	if (y_46_re <= -2.3e-23) {
		tmp = Math.cos((Math.atan2(x_46_im, x_46_re) * y_46_re)) * Math.exp(((Math.log(Math.sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re)))) * y_46_re) - (y_46_im * Math.atan2(x_46_im, x_46_re))));
	} else if (y_46_re <= 3.8e-17) {
		tmp = Math.exp((-y_46_im * Math.atan2(x_46_im, x_46_re))) * t_0;
	} else {
		tmp = Math.pow(Math.hypot(x_46_re, x_46_im), y_46_re) * t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.cos((math.log(math.hypot(x_46_re, x_46_im)) * y_46_im))
	tmp = 0
	if y_46_re <= -2.3e-23:
		tmp = math.cos((math.atan2(x_46_im, x_46_re) * y_46_re)) * math.exp(((math.log(math.sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re)))) * y_46_re) - (y_46_im * math.atan2(x_46_im, x_46_re))))
	elif y_46_re <= 3.8e-17:
		tmp = math.exp((-y_46_im * math.atan2(x_46_im, x_46_re))) * t_0
	else:
		tmp = math.pow(math.hypot(x_46_re, x_46_im), y_46_re) * t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = cos(Float64(log(hypot(x_46_re, x_46_im)) * y_46_im))
	tmp = 0.0
	if (y_46_re <= -2.3e-23)
		tmp = Float64(cos(Float64(atan(x_46_im, x_46_re) * y_46_re)) * exp(Float64(Float64(log(sqrt(Float64(Float64(x_46_im * x_46_im) + Float64(x_46_re * x_46_re)))) * y_46_re) - Float64(y_46_im * atan(x_46_im, x_46_re)))));
	elseif (y_46_re <= 3.8e-17)
		tmp = Float64(exp(Float64(Float64(-y_46_im) * atan(x_46_im, x_46_re))) * t_0);
	else
		tmp = Float64((hypot(x_46_re, x_46_im) ^ y_46_re) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = cos((log(hypot(x_46_re, x_46_im)) * y_46_im));
	tmp = 0.0;
	if (y_46_re <= -2.3e-23)
		tmp = cos((atan2(x_46_im, x_46_re) * y_46_re)) * exp(((log(sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re)))) * y_46_re) - (y_46_im * atan2(x_46_im, x_46_re))));
	elseif (y_46_re <= 3.8e-17)
		tmp = exp((-y_46_im * atan2(x_46_im, x_46_re))) * t_0;
	else
		tmp = (hypot(x_46_re, x_46_im) ^ y_46_re) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Cos[N[(N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -2.3e-23], N[(N[Cos[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(N[Log[N[Sqrt[N[(N[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 3.8e-17], N[(N[Exp[N[((-y$46$im) * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.3000000000000001e-23

   1. Initial program 43.1%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 84.7

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

   5. Applied rewrites 84.7%

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

   if -2.3000000000000001e-23 < y.re < 3.8000000000000001e-17

   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. Add Preprocessing

   3. Taylor expanded in y.im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 45.9

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

   5. Applied rewrites 45.9%

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

   6. Taylor expanded in y.im around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. neg-mul-1 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-atan2.f64 77.5

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

   8. Applied rewrites 77.5%

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

   if 3.8000000000000001e-17 < y.re

   1. Initial program 48.1%

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

   3. Taylor expanded in y.im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 77.2

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

   5. Applied rewrites 77.2%

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

   6. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 77.5

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

   8. Applied rewrites 77.5%

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

   9. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \cdot \cos \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right) \]
   10. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \cdot \cos \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right) \]

       2. +-commutative N/A

          \[\leadsto {\left(\sqrt{\color{blue}{{x.re}^{2} + {x.im}^{2}}}\right)}^{y.re} \cdot \cos \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right) \]

       3. unpow2 N/A

          \[\leadsto {\left(\sqrt{\color{blue}{x.re \cdot x.re} + {x.im}^{2}}\right)}^{y.re} \cdot \cos \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right) \]

       4. unpow2 N/A

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

       5. lower-hypot.f64 77.5

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

   11. Applied rewrites 77.5%

       \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}} \cdot \cos \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.3 \cdot 10^{-23}:\\ \;\;\;\;\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;y.re \leq 3.8 \cdot 10^{-17}:\\ \;\;\;\;e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \cos \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot \cos \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\\ t_1 := {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -0.24:\\ \;\;\;\;1 \cdot t\_1\\ \mathbf{elif}\;y.re \leq 3.8 \cdot 10^{-17}:\\ \;\;\;\;e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (cos (* (log (hypot x.re x.im)) y.im)))
        (t_1 (pow (hypot x.re x.im) y.re)))
   (if (<= y.re -0.24)
     (* 1.0 t_1)
     (if (<= y.re 3.8e-17)
       (* (exp (* (- y.im) (atan2 x.im x.re))) t_0)
       (* t_1 t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = cos((log(hypot(x_46_re, x_46_im)) * y_46_im));
	double t_1 = pow(hypot(x_46_re, x_46_im), y_46_re);
	double tmp;
	if (y_46_re <= -0.24) {
		tmp = 1.0 * t_1;
	} else if (y_46_re <= 3.8e-17) {
		tmp = exp((-y_46_im * atan2(x_46_im, x_46_re))) * t_0;
	} else {
		tmp = t_1 * t_0;
	}
	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.cos((Math.log(Math.hypot(x_46_re, x_46_im)) * y_46_im));
	double t_1 = Math.pow(Math.hypot(x_46_re, x_46_im), y_46_re);
	double tmp;
	if (y_46_re <= -0.24) {
		tmp = 1.0 * t_1;
	} else if (y_46_re <= 3.8e-17) {
		tmp = Math.exp((-y_46_im * Math.atan2(x_46_im, x_46_re))) * t_0;
	} else {
		tmp = t_1 * t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.cos((math.log(math.hypot(x_46_re, x_46_im)) * y_46_im))
	t_1 = math.pow(math.hypot(x_46_re, x_46_im), y_46_re)
	tmp = 0
	if y_46_re <= -0.24:
		tmp = 1.0 * t_1
	elif y_46_re <= 3.8e-17:
		tmp = math.exp((-y_46_im * math.atan2(x_46_im, x_46_re))) * t_0
	else:
		tmp = t_1 * t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = cos(Float64(log(hypot(x_46_re, x_46_im)) * y_46_im))
	t_1 = hypot(x_46_re, x_46_im) ^ y_46_re
	tmp = 0.0
	if (y_46_re <= -0.24)
		tmp = Float64(1.0 * t_1);
	elseif (y_46_re <= 3.8e-17)
		tmp = Float64(exp(Float64(Float64(-y_46_im) * atan(x_46_im, x_46_re))) * t_0);
	else
		tmp = Float64(t_1 * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = cos((log(hypot(x_46_re, x_46_im)) * y_46_im));
	t_1 = hypot(x_46_re, x_46_im) ^ y_46_re;
	tmp = 0.0;
	if (y_46_re <= -0.24)
		tmp = 1.0 * t_1;
	elseif (y_46_re <= 3.8e-17)
		tmp = exp((-y_46_im * atan2(x_46_im, x_46_re))) * t_0;
	else
		tmp = t_1 * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Cos[N[(N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]}, If[LessEqual[y$46$re, -0.24], N[(1.0 * t$95$1), $MachinePrecision], If[LessEqual[y$46$re, 3.8e-17], N[(N[Exp[N[((-y$46$im) * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(t$95$1 * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -0.23999999999999999

   1. Initial program 44.3%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-atan2.f64 82.1

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

   5. Applied rewrites 82.1%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 82.1%

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

   if -0.23999999999999999 < y.re < 3.8000000000000001e-17

   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. Add Preprocessing

   3. Taylor expanded in y.im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 46.1

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

   5. Applied rewrites 46.1%

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

   6. Taylor expanded in y.im around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. neg-mul-1 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-atan2.f64 77.3

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

   8. Applied rewrites 77.3%

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

   if 3.8000000000000001e-17 < y.re

   1. Initial program 48.1%

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

   3. Taylor expanded in y.im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 77.2

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

   5. Applied rewrites 77.2%

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

   6. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 77.5

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

   8. Applied rewrites 77.5%

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

   9. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \cdot \cos \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right) \]
   10. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \cdot \cos \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right) \]

       2. +-commutative N/A

          \[\leadsto {\left(\sqrt{\color{blue}{{x.re}^{2} + {x.im}^{2}}}\right)}^{y.re} \cdot \cos \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right) \]

       3. unpow2 N/A

          \[\leadsto {\left(\sqrt{\color{blue}{x.re \cdot x.re} + {x.im}^{2}}\right)}^{y.re} \cdot \cos \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right) \]

       4. unpow2 N/A

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

       5. lower-hypot.f64 77.5

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

   11. Applied rewrites 77.5%

       \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}} \cdot \cos \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -0.24:\\ \;\;\;\;1 \cdot {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 3.8 \cdot 10^{-17}:\\ \;\;\;\;e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \cos \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot \cos \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.45 \cdot 10^{+188}:\\ \;\;\;\;\cos \left(\log \left(-x.im\right) \cdot y.im\right) \cdot e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;y.im \leq 3.7 \cdot 10^{+21}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot \cos \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\mathsf{fma}\left(0.5, \frac{x.im \cdot x.im}{x.re}, x.re\right)\right)}^{2}\right)}^{\left(0.5 \cdot y.re\right)} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -1.45e+188)
   (* (cos (* (log (- x.im)) y.im)) (exp (* (- y.im) (atan2 x.im x.re))))
   (if (<= y.im 3.7e+21)
     (* (pow (hypot x.re x.im) y.re) (cos (* (log (hypot x.re x.im)) y.im)))
     (*
      (pow (pow (fma 0.5 (/ (* x.im x.im) x.re) x.re) 2.0) (* 0.5 y.re))
      1.0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -1.45e+188) {
		tmp = cos((log(-x_46_im) * y_46_im)) * exp((-y_46_im * atan2(x_46_im, x_46_re)));
	} else if (y_46_im <= 3.7e+21) {
		tmp = pow(hypot(x_46_re, x_46_im), y_46_re) * cos((log(hypot(x_46_re, x_46_im)) * y_46_im));
	} else {
		tmp = pow(pow(fma(0.5, ((x_46_im * x_46_im) / x_46_re), x_46_re), 2.0), (0.5 * y_46_re)) * 1.0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -1.45e+188)
		tmp = Float64(cos(Float64(log(Float64(-x_46_im)) * y_46_im)) * exp(Float64(Float64(-y_46_im) * atan(x_46_im, x_46_re))));
	elseif (y_46_im <= 3.7e+21)
		tmp = Float64((hypot(x_46_re, x_46_im) ^ y_46_re) * cos(Float64(log(hypot(x_46_re, x_46_im)) * y_46_im)));
	else
		tmp = Float64(((fma(0.5, Float64(Float64(x_46_im * x_46_im) / x_46_re), x_46_re) ^ 2.0) ^ Float64(0.5 * y_46_re)) * 1.0);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -1.45e+188], N[(N[Cos[N[(N[Log[(-x$46$im)], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision] * N[Exp[N[((-y$46$im) * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 3.7e+21], N[(N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision] * N[Cos[N[(N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[Power[N[(0.5 * N[(N[(x$46$im * x$46$im), $MachinePrecision] / x$46$re), $MachinePrecision] + x$46$re), $MachinePrecision], 2.0], $MachinePrecision], N[(0.5 * y$46$re), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.45e188

   1. Initial program 27.9%

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

   3. Taylor expanded in y.im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 47.5

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

   5. Applied rewrites 47.5%

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

   6. Taylor expanded in x.im around -inf

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

   8. Applied rewrites 36.3%

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

   9. Taylor expanded in y.im around inf

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. neg-mul-1 N/A

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

       4. lower-neg.f64 N/A

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

       5. lower-atan2.f64 59.0

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

   11. Applied rewrites 59.0%

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

   if -1.45e188 < y.im < 3.7e21

   1. Initial program 46.9%

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

   3. Taylor expanded in y.im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 70.0

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

   5. Applied rewrites 70.0%

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

   6. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 82.3

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

   8. Applied rewrites 82.3%

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

   9. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \cdot \cos \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right) \]
   10. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \cdot \cos \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right) \]

       2. +-commutative N/A

          \[\leadsto {\left(\sqrt{\color{blue}{{x.re}^{2} + {x.im}^{2}}}\right)}^{y.re} \cdot \cos \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right) \]

       3. unpow2 N/A

          \[\leadsto {\left(\sqrt{\color{blue}{x.re \cdot x.re} + {x.im}^{2}}\right)}^{y.re} \cdot \cos \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right) \]

       4. unpow2 N/A

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

       5. lower-hypot.f64 82.3

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

   11. Applied rewrites 82.3%

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

   if 3.7e21 < y.im

   1. Initial program 37.8%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-atan2.f64 36.1

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

   5. Applied rewrites 36.1%

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

   6. Taylor expanded in x.im around 0

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

   8. Applied rewrites 39.3%

      \[\leadsto {\left(\mathsf{fma}\left(\frac{x.im \cdot x.im}{x.re}, 0.5, x.re\right)\right)}^{y.re} \cdot \cos \left(\color{blue}{\tan^{-1}_* \frac{x.im}{x.re}} \cdot y.re\right) \]

   9. Taylor expanded in y.re around 0

      \[\leadsto {\left(\mathsf{fma}\left(\frac{x.im \cdot x.im}{x.re}, \frac{1}{2}, x.re\right)\right)}^{y.re} \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 41.2%

       \[\leadsto {\left(\mathsf{fma}\left(\frac{x.im \cdot x.im}{x.re}, 0.5, x.re\right)\right)}^{y.re} \cdot 1 \]
   12. Step-by-step derivation

   13. Applied rewrites 46.5%

       \[\leadsto {\left({\left(\mathsf{fma}\left(0.5, \frac{x.im \cdot x.im}{x.re}, x.re\right)\right)}^{2}\right)}^{\left(y.re \cdot 0.5\right)} \cdot 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.45 \cdot 10^{+188}:\\ \;\;\;\;\cos \left(\log \left(-x.im\right) \cdot y.im\right) \cdot e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;y.im \leq 3.7 \cdot 10^{+21}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot \cos \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\mathsf{fma}\left(0.5, \frac{x.im \cdot x.im}{x.re}, x.re\right)\right)}^{2}\right)}^{\left(0.5 \cdot y.re\right)} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\log \left(-x.im\right) \cdot y.im\right) \cdot e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{if}\;y.im \leq -5.8 \cdot 10^{+149}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 4.6 \cdot 10^{+195}:\\ \;\;\;\;1 \cdot {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (*
          (cos (* (log (- x.im)) y.im))
          (exp (* (- y.im) (atan2 x.im x.re))))))
   (if (<= y.im -5.8e+149)
     t_0
     (if (<= y.im 4.6e+195) (* 1.0 (pow (hypot x.re x.im) y.re)) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = cos((log(-x_46_im) * y_46_im)) * exp((-y_46_im * atan2(x_46_im, x_46_re)));
	double tmp;
	if (y_46_im <= -5.8e+149) {
		tmp = t_0;
	} else if (y_46_im <= 4.6e+195) {
		tmp = 1.0 * pow(hypot(x_46_re, x_46_im), y_46_re);
	} else {
		tmp = t_0;
	}
	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.cos((Math.log(-x_46_im) * y_46_im)) * Math.exp((-y_46_im * Math.atan2(x_46_im, x_46_re)));
	double tmp;
	if (y_46_im <= -5.8e+149) {
		tmp = t_0;
	} else if (y_46_im <= 4.6e+195) {
		tmp = 1.0 * Math.pow(Math.hypot(x_46_re, x_46_im), y_46_re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.cos((math.log(-x_46_im) * y_46_im)) * math.exp((-y_46_im * math.atan2(x_46_im, x_46_re)))
	tmp = 0
	if y_46_im <= -5.8e+149:
		tmp = t_0
	elif y_46_im <= 4.6e+195:
		tmp = 1.0 * math.pow(math.hypot(x_46_re, x_46_im), y_46_re)
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(cos(Float64(log(Float64(-x_46_im)) * y_46_im)) * exp(Float64(Float64(-y_46_im) * atan(x_46_im, x_46_re))))
	tmp = 0.0
	if (y_46_im <= -5.8e+149)
		tmp = t_0;
	elseif (y_46_im <= 4.6e+195)
		tmp = Float64(1.0 * (hypot(x_46_re, x_46_im) ^ y_46_re));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = cos((log(-x_46_im) * y_46_im)) * exp((-y_46_im * atan2(x_46_im, x_46_re)));
	tmp = 0.0;
	if (y_46_im <= -5.8e+149)
		tmp = t_0;
	elseif (y_46_im <= 4.6e+195)
		tmp = 1.0 * (hypot(x_46_re, x_46_im) ^ y_46_re);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[Cos[N[(N[Log[(-x$46$im)], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision] * N[Exp[N[((-y$46$im) * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -5.8e+149], t$95$0, If[LessEqual[y$46$im, 4.6e+195], N[(1.0 * N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -5.8000000000000004e149 or 4.6000000000000002e195 < y.im

   1. Initial program 30.1%

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

   3. Taylor expanded in y.im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 54.5

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

   5. Applied rewrites 54.5%

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

   6. Taylor expanded in x.im around -inf

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

   8. Applied rewrites 34.3%

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

   9. Taylor expanded in y.im around inf

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. neg-mul-1 N/A

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

       4. lower-neg.f64 N/A

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

       5. lower-atan2.f64 50.7

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

   11. Applied rewrites 50.7%

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

   if -5.8000000000000004e149 < y.im < 4.6000000000000002e195

   1. Initial program 46.5%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-atan2.f64 75.1

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

   5. Applied rewrites 75.1%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 78.0%

      \[\leadsto {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -5.8 \cdot 10^{+149}:\\ \;\;\;\;\cos \left(\log \left(-x.im\right) \cdot y.im\right) \cdot e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;y.im \leq 4.6 \cdot 10^{+195}:\\ \;\;\;\;1 \cdot {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\log \left(-x.im\right) \cdot y.im\right) \cdot e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im \cdot x.im}{x.re}\\ \mathbf{if}\;y.im \leq -4.2 \cdot 10^{+170}:\\ \;\;\;\;\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot {\left(\mathsf{fma}\left(t\_0, 0.5, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.im \leq 3.1 \cdot 10^{+80}:\\ \;\;\;\;1 \cdot {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\mathsf{fma}\left(0.5, t\_0, x.re\right)\right)}^{2}\right)}^{\left(0.5 \cdot y.re\right)} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (* x.im x.im) x.re)))
   (if (<= y.im -4.2e+170)
     (* (cos (* (atan2 x.im x.re) y.re)) (pow (fma t_0 0.5 x.re) y.re))
     (if (<= y.im 3.1e+80)
       (* 1.0 (pow (hypot x.re x.im) y.re))
       (* (pow (pow (fma 0.5 t_0 x.re) 2.0) (* 0.5 y.re)) 1.0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im * x_46_im) / x_46_re;
	double tmp;
	if (y_46_im <= -4.2e+170) {
		tmp = cos((atan2(x_46_im, x_46_re) * y_46_re)) * pow(fma(t_0, 0.5, x_46_re), y_46_re);
	} else if (y_46_im <= 3.1e+80) {
		tmp = 1.0 * pow(hypot(x_46_re, x_46_im), y_46_re);
	} else {
		tmp = pow(pow(fma(0.5, t_0, x_46_re), 2.0), (0.5 * y_46_re)) * 1.0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im * x_46_im) / x_46_re)
	tmp = 0.0
	if (y_46_im <= -4.2e+170)
		tmp = Float64(cos(Float64(atan(x_46_im, x_46_re) * y_46_re)) * (fma(t_0, 0.5, x_46_re) ^ y_46_re));
	elseif (y_46_im <= 3.1e+80)
		tmp = Float64(1.0 * (hypot(x_46_re, x_46_im) ^ y_46_re));
	else
		tmp = Float64(((fma(0.5, t_0, x_46_re) ^ 2.0) ^ Float64(0.5 * y_46_re)) * 1.0);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im * x$46$im), $MachinePrecision] / x$46$re), $MachinePrecision]}, If[LessEqual[y$46$im, -4.2e+170], N[(N[Cos[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision] * N[Power[N[(t$95$0 * 0.5 + x$46$re), $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 3.1e+80], N[(1.0 * N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[Power[N[(0.5 * t$95$0 + x$46$re), $MachinePrecision], 2.0], $MachinePrecision], N[(0.5 * y$46$re), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -4.19999999999999996e170

   1. Initial program 27.5%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-atan2.f64 25.5

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

   5. Applied rewrites 25.5%

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

   6. Taylor expanded in x.im around 0

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

   8. Applied rewrites 38.2%

      \[\leadsto {\left(\mathsf{fma}\left(\frac{x.im \cdot x.im}{x.re}, 0.5, x.re\right)\right)}^{y.re} \cdot \cos \left(\color{blue}{\tan^{-1}_* \frac{x.im}{x.re}} \cdot y.re\right) \]

   if -4.19999999999999996e170 < y.im < 3.09999999999999988e80

   1. Initial program 48.1%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-atan2.f64 76.8

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

   5. Applied rewrites 76.8%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 79.4%

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

   if 3.09999999999999988e80 < y.im

   1. Initial program 29.8%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-atan2.f64 29.7

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

   5. Applied rewrites 29.7%

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

   6. Taylor expanded in x.im around 0

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

   8. Applied rewrites 34.5%

      \[\leadsto {\left(\mathsf{fma}\left(\frac{x.im \cdot x.im}{x.re}, 0.5, x.re\right)\right)}^{y.re} \cdot \cos \left(\color{blue}{\tan^{-1}_* \frac{x.im}{x.re}} \cdot y.re\right) \]

   9. Taylor expanded in y.re around 0

      \[\leadsto {\left(\mathsf{fma}\left(\frac{x.im \cdot x.im}{x.re}, \frac{1}{2}, x.re\right)\right)}^{y.re} \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 37.2%

       \[\leadsto {\left(\mathsf{fma}\left(\frac{x.im \cdot x.im}{x.re}, 0.5, x.re\right)\right)}^{y.re} \cdot 1 \]
   12. Step-by-step derivation

   13. Applied rewrites 44.6%

       \[\leadsto {\left({\left(\mathsf{fma}\left(0.5, \frac{x.im \cdot x.im}{x.re}, x.re\right)\right)}^{2}\right)}^{\left(y.re \cdot 0.5\right)} \cdot 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.2 \cdot 10^{+170}:\\ \;\;\;\;\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot {\left(\mathsf{fma}\left(\frac{x.im \cdot x.im}{x.re}, 0.5, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.im \leq 3.1 \cdot 10^{+80}:\\ \;\;\;\;1 \cdot {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\mathsf{fma}\left(0.5, \frac{x.im \cdot x.im}{x.re}, x.re\right)\right)}^{2}\right)}^{\left(0.5 \cdot y.re\right)} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left({\left(\mathsf{fma}\left(0.5, \frac{x.im \cdot x.im}{x.re}, x.re\right)\right)}^{2}\right)}^{\left(0.5 \cdot y.re\right)} \cdot 1\\ \mathbf{if}\;y.im \leq -3.8 \cdot 10^{+187}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 3.1 \cdot 10^{+80}:\\ \;\;\;\;1 \cdot {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (*
          (pow (pow (fma 0.5 (/ (* x.im x.im) x.re) x.re) 2.0) (* 0.5 y.re))
          1.0)))
   (if (<= y.im -3.8e+187)
     t_0
     (if (<= y.im 3.1e+80) (* 1.0 (pow (hypot x.re x.im) y.re)) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = pow(pow(fma(0.5, ((x_46_im * x_46_im) / x_46_re), x_46_re), 2.0), (0.5 * y_46_re)) * 1.0;
	double tmp;
	if (y_46_im <= -3.8e+187) {
		tmp = t_0;
	} else if (y_46_im <= 3.1e+80) {
		tmp = 1.0 * pow(hypot(x_46_re, x_46_im), y_46_re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(((fma(0.5, Float64(Float64(x_46_im * x_46_im) / x_46_re), x_46_re) ^ 2.0) ^ Float64(0.5 * y_46_re)) * 1.0)
	tmp = 0.0
	if (y_46_im <= -3.8e+187)
		tmp = t_0;
	elseif (y_46_im <= 3.1e+80)
		tmp = Float64(1.0 * (hypot(x_46_re, x_46_im) ^ y_46_re));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[Power[N[Power[N[(0.5 * N[(N[(x$46$im * x$46$im), $MachinePrecision] / x$46$re), $MachinePrecision] + x$46$re), $MachinePrecision], 2.0], $MachinePrecision], N[(0.5 * y$46$re), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]}, If[LessEqual[y$46$im, -3.8e+187], t$95$0, If[LessEqual[y$46$im, 3.1e+80], N[(1.0 * N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -3.8e187 or 3.09999999999999988e80 < y.im

   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. Add Preprocessing

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-atan2.f64 26.0

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

   5. Applied rewrites 26.0%

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

   6. Taylor expanded in x.im around 0

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

   8. Applied rewrites 34.7%

      \[\leadsto {\left(\mathsf{fma}\left(\frac{x.im \cdot x.im}{x.re}, 0.5, x.re\right)\right)}^{y.re} \cdot \cos \left(\color{blue}{\tan^{-1}_* \frac{x.im}{x.re}} \cdot y.re\right) \]

   9. Taylor expanded in y.re around 0

      \[\leadsto {\left(\mathsf{fma}\left(\frac{x.im \cdot x.im}{x.re}, \frac{1}{2}, x.re\right)\right)}^{y.re} \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 34.7%

       \[\leadsto {\left(\mathsf{fma}\left(\frac{x.im \cdot x.im}{x.re}, 0.5, x.re\right)\right)}^{y.re} \cdot 1 \]
   12. Step-by-step derivation

   13. Applied rewrites 40.6%

       \[\leadsto {\left({\left(\mathsf{fma}\left(0.5, \frac{x.im \cdot x.im}{x.re}, x.re\right)\right)}^{2}\right)}^{\left(y.re \cdot 0.5\right)} \cdot 1 \]

   if -3.8e187 < y.im < 3.09999999999999988e80

   1. Initial program 47.9%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-atan2.f64 76.7

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

   5. Applied rewrites 76.7%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 78.7%

      \[\leadsto {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3.8 \cdot 10^{+187}:\\ \;\;\;\;{\left({\left(\mathsf{fma}\left(0.5, \frac{x.im \cdot x.im}{x.re}, x.re\right)\right)}^{2}\right)}^{\left(0.5 \cdot y.re\right)} \cdot 1\\ \mathbf{elif}\;y.im \leq 3.1 \cdot 10^{+80}:\\ \;\;\;\;1 \cdot {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\mathsf{fma}\left(0.5, \frac{x.im \cdot x.im}{x.re}, x.re\right)\right)}^{2}\right)}^{\left(0.5 \cdot y.re\right)} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.2% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ 1 \cdot {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (* 1.0 (pow (hypot x.re x.im) y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return 1.0 * pow(hypot(x_46_re, x_46_im), y_46_re);
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return 1.0 * Math.pow(Math.hypot(x_46_re, x_46_im), y_46_re);
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return 1.0 * math.pow(math.hypot(x_46_re, x_46_im), y_46_re)

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(1.0 * (hypot(x_46_re, x_46_im) ^ y_46_re))
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 1.0 * (hypot(x_46_re, x_46_im) ^ y_46_re);
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(1.0 * N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 43.1%

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

3. Taylor expanded in y.im around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. +-commutative N/A

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

   5. unpow2 N/A

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

   6. unpow2 N/A

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

   7. lower-hypot.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-atan2.f64 64.0

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

5. Applied rewrites 64.0%

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

6. Taylor expanded in y.re around 0

   \[\leadsto {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot 1 \]
7. Step-by-step derivation

8. Applied rewrites 65.5%

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

9. Final simplification 65.5%

   \[\leadsto 1 \cdot {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \]
10. Add Preprocessing

Alternative 9: 55.6% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -0.92:\\ \;\;\;\;{\left(-x.im\right)}^{y.re} \cdot 1\\ \mathbf{elif}\;x.im \leq 0.015:\\ \;\;\;\;{x.re}^{y.re} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;{x.im}^{y.re} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.im -0.92)
   (* (pow (- x.im) y.re) 1.0)
   (if (<= x.im 0.015) (* (pow x.re y.re) 1.0) (* (pow x.im y.re) 1.0))))

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 <= -0.92) {
		tmp = pow(-x_46_im, y_46_re) * 1.0;
	} else if (x_46_im <= 0.015) {
		tmp = pow(x_46_re, y_46_re) * 1.0;
	} else {
		tmp = pow(x_46_im, y_46_re) * 1.0;
	}
	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 <= (-0.92d0)) then
        tmp = (-x_46im ** y_46re) * 1.0d0
    else if (x_46im <= 0.015d0) then
        tmp = (x_46re ** y_46re) * 1.0d0
    else
        tmp = (x_46im ** y_46re) * 1.0d0
    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 <= -0.92) {
		tmp = Math.pow(-x_46_im, y_46_re) * 1.0;
	} else if (x_46_im <= 0.015) {
		tmp = Math.pow(x_46_re, y_46_re) * 1.0;
	} else {
		tmp = Math.pow(x_46_im, y_46_re) * 1.0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if x_46_im <= -0.92:
		tmp = math.pow(-x_46_im, y_46_re) * 1.0
	elif x_46_im <= 0.015:
		tmp = math.pow(x_46_re, y_46_re) * 1.0
	else:
		tmp = math.pow(x_46_im, y_46_re) * 1.0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (x_46_im <= -0.92)
		tmp = Float64((Float64(-x_46_im) ^ y_46_re) * 1.0);
	elseif (x_46_im <= 0.015)
		tmp = Float64((x_46_re ^ y_46_re) * 1.0);
	else
		tmp = Float64((x_46_im ^ y_46_re) * 1.0);
	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 <= -0.92)
		tmp = (-x_46_im ^ y_46_re) * 1.0;
	elseif (x_46_im <= 0.015)
		tmp = (x_46_re ^ y_46_re) * 1.0;
	else
		tmp = (x_46_im ^ y_46_re) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[x$46$im, -0.92], N[(N[Power[(-x$46$im), y$46$re], $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[x$46$im, 0.015], N[(N[Power[x$46$re, y$46$re], $MachinePrecision] * 1.0), $MachinePrecision], N[(N[Power[x$46$im, y$46$re], $MachinePrecision] * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -0.92000000000000004

   1. Initial program 32.2%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-atan2.f64 60.4

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

   5. Applied rewrites 60.4%

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

   6. Taylor expanded in x.im around -inf

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

   8. Applied rewrites 60.4%

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

   9. Taylor expanded in y.re around 0

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

   11. Applied rewrites 63.6%

       \[\leadsto {\left(-x.im\right)}^{y.re} \cdot 1 \]

   if -0.92000000000000004 < x.im < 0.014999999999999999

   1. Initial program 50.2%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-atan2.f64 63.8

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

   5. Applied rewrites 63.8%

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

   6. Taylor expanded in x.im around 0

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

   8. Applied rewrites 49.4%

      \[\leadsto {\left(\mathsf{fma}\left(\frac{x.im \cdot x.im}{x.re}, 0.5, x.re\right)\right)}^{y.re} \cdot \cos \left(\color{blue}{\tan^{-1}_* \frac{x.im}{x.re}} \cdot y.re\right) \]

   9. Taylor expanded in y.re around 0

      \[\leadsto {\left(\mathsf{fma}\left(\frac{x.im \cdot x.im}{x.re}, \frac{1}{2}, x.re\right)\right)}^{y.re} \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 53.9%

       \[\leadsto {\left(\mathsf{fma}\left(\frac{x.im \cdot x.im}{x.re}, 0.5, x.re\right)\right)}^{y.re} \cdot 1 \]

   12. Taylor expanded in x.im around 0

       \[\leadsto {x.re}^{y.re} \cdot 1 \]
   13. Step-by-step derivation

   14. Applied rewrites 60.6%

       \[\leadsto {x.re}^{y.re} \cdot 1 \]

   if 0.014999999999999999 < x.im

   1. Initial program 38.4%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-atan2.f64 67.7

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

   5. Applied rewrites 67.7%

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

   6. Taylor expanded in x.im around 0

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

   8. Applied rewrites 43.9%

      \[\leadsto {\left(\mathsf{fma}\left(\frac{x.im \cdot x.im}{x.re}, 0.5, x.re\right)\right)}^{y.re} \cdot \cos \left(\color{blue}{\tan^{-1}_* \frac{x.im}{x.re}} \cdot y.re\right) \]

   9. Taylor expanded in y.re around 0

      \[\leadsto {\left(\mathsf{fma}\left(\frac{x.im \cdot x.im}{x.re}, \frac{1}{2}, x.re\right)\right)}^{y.re} \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 42.3%

       \[\leadsto {\left(\mathsf{fma}\left(\frac{x.im \cdot x.im}{x.re}, 0.5, x.re\right)\right)}^{y.re} \cdot 1 \]

   12. Taylor expanded in x.re around 0

       \[\leadsto {x.im}^{y.re} \cdot 1 \]
   13. Step-by-step derivation

   14. Applied rewrites 66.2%

       \[\leadsto {x.im}^{y.re} \cdot 1 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 52.1% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x.re}^{y.re} \cdot 1\\ \mathbf{if}\;y.re \leq -7.2 \cdot 10^{-18}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1.1 \cdot 10^{-13}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (pow x.re y.re) 1.0)))
   (if (<= y.re -7.2e-18) t_0 (if (<= y.re 1.1e-13) 1.0 t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = pow(x_46_re, y_46_re) * 1.0;
	double tmp;
	if (y_46_re <= -7.2e-18) {
		tmp = t_0;
	} else if (y_46_re <= 1.1e-13) {
		tmp = 1.0;
	} else {
		tmp = t_0;
	}
	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 = (x_46re ** y_46re) * 1.0d0
    if (y_46re <= (-7.2d-18)) then
        tmp = t_0
    else if (y_46re <= 1.1d-13) then
        tmp = 1.0d0
    else
        tmp = t_0
    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.pow(x_46_re, y_46_re) * 1.0;
	double tmp;
	if (y_46_re <= -7.2e-18) {
		tmp = t_0;
	} else if (y_46_re <= 1.1e-13) {
		tmp = 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.pow(x_46_re, y_46_re) * 1.0
	tmp = 0
	if y_46_re <= -7.2e-18:
		tmp = t_0
	elif y_46_re <= 1.1e-13:
		tmp = 1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64((x_46_re ^ y_46_re) * 1.0)
	tmp = 0.0
	if (y_46_re <= -7.2e-18)
		tmp = t_0;
	elseif (y_46_re <= 1.1e-13)
		tmp = 1.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_re ^ y_46_re) * 1.0;
	tmp = 0.0;
	if (y_46_re <= -7.2e-18)
		tmp = t_0;
	elseif (y_46_re <= 1.1e-13)
		tmp = 1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[Power[x$46$re, y$46$re], $MachinePrecision] * 1.0), $MachinePrecision]}, If[LessEqual[y$46$re, -7.2e-18], t$95$0, If[LessEqual[y$46$re, 1.1e-13], 1.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -7.20000000000000021e-18 or 1.09999999999999998e-13 < y.re

   1. Initial program 46.4%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-atan2.f64 74.6

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

   5. Applied rewrites 74.6%

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

   6. Taylor expanded in x.im around 0

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

   8. Applied rewrites 65.0%

      \[\leadsto {\left(\mathsf{fma}\left(\frac{x.im \cdot x.im}{x.re}, 0.5, x.re\right)\right)}^{y.re} \cdot \cos \left(\color{blue}{\tan^{-1}_* \frac{x.im}{x.re}} \cdot y.re\right) \]

   9. Taylor expanded in y.re around 0

      \[\leadsto {\left(\mathsf{fma}\left(\frac{x.im \cdot x.im}{x.re}, \frac{1}{2}, x.re\right)\right)}^{y.re} \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 70.0%

       \[\leadsto {\left(\mathsf{fma}\left(\frac{x.im \cdot x.im}{x.re}, 0.5, x.re\right)\right)}^{y.re} \cdot 1 \]

   12. Taylor expanded in x.im around 0

       \[\leadsto {x.re}^{y.re} \cdot 1 \]
   13. Step-by-step derivation

   14. Applied rewrites 64.4%

       \[\leadsto {x.re}^{y.re} \cdot 1 \]

   if -7.20000000000000021e-18 < y.re < 1.09999999999999998e-13

   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. Add Preprocessing

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-atan2.f64 51.2

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

   5. Applied rewrites 51.2%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 51.0%

      \[\leadsto 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 51.4% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x.im}^{y.re} \cdot 1\\ \mathbf{if}\;y.re \leq -7.2 \cdot 10^{-18}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 40000000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (pow x.im y.re) 1.0)))
   (if (<= y.re -7.2e-18) t_0 (if (<= y.re 40000000000.0) 1.0 t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = pow(x_46_im, y_46_re) * 1.0;
	double tmp;
	if (y_46_re <= -7.2e-18) {
		tmp = t_0;
	} else if (y_46_re <= 40000000000.0) {
		tmp = 1.0;
	} else {
		tmp = t_0;
	}
	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 = (x_46im ** y_46re) * 1.0d0
    if (y_46re <= (-7.2d-18)) then
        tmp = t_0
    else if (y_46re <= 40000000000.0d0) then
        tmp = 1.0d0
    else
        tmp = t_0
    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.pow(x_46_im, y_46_re) * 1.0;
	double tmp;
	if (y_46_re <= -7.2e-18) {
		tmp = t_0;
	} else if (y_46_re <= 40000000000.0) {
		tmp = 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.pow(x_46_im, y_46_re) * 1.0
	tmp = 0
	if y_46_re <= -7.2e-18:
		tmp = t_0
	elif y_46_re <= 40000000000.0:
		tmp = 1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64((x_46_im ^ y_46_re) * 1.0)
	tmp = 0.0
	if (y_46_re <= -7.2e-18)
		tmp = t_0;
	elseif (y_46_re <= 40000000000.0)
		tmp = 1.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_im ^ y_46_re) * 1.0;
	tmp = 0.0;
	if (y_46_re <= -7.2e-18)
		tmp = t_0;
	elseif (y_46_re <= 40000000000.0)
		tmp = 1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[Power[x$46$im, y$46$re], $MachinePrecision] * 1.0), $MachinePrecision]}, If[LessEqual[y$46$re, -7.2e-18], t$95$0, If[LessEqual[y$46$re, 40000000000.0], 1.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -7.20000000000000021e-18 or 4e10 < y.re

   1. Initial program 45.0%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-atan2.f64 73.8

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

   5. Applied rewrites 73.8%

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

   6. Taylor expanded in x.im around 0

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

   8. Applied rewrites 66.1%

      \[\leadsto {\left(\mathsf{fma}\left(\frac{x.im \cdot x.im}{x.re}, 0.5, x.re\right)\right)}^{y.re} \cdot \cos \left(\color{blue}{\tan^{-1}_* \frac{x.im}{x.re}} \cdot y.re\right) \]

   9. Taylor expanded in y.re around 0

      \[\leadsto {\left(\mathsf{fma}\left(\frac{x.im \cdot x.im}{x.re}, \frac{1}{2}, x.re\right)\right)}^{y.re} \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 72.3%

       \[\leadsto {\left(\mathsf{fma}\left(\frac{x.im \cdot x.im}{x.re}, 0.5, x.re\right)\right)}^{y.re} \cdot 1 \]

   12. Taylor expanded in x.re around 0

       \[\leadsto {x.im}^{y.re} \cdot 1 \]
   13. Step-by-step derivation

   14. Applied rewrites 59.4%

       \[\leadsto {x.im}^{y.re} \cdot 1 \]

   if -7.20000000000000021e-18 < y.re < 4e10

   1. Initial program 41.1%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-atan2.f64 54.1

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

   5. Applied rewrites 54.1%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 48.7%

      \[\leadsto 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 25.4% accurate, 680.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im) :precision binary64 1.0)

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return 1.0;
}

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 = 1.0d0
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return 1.0;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return 1.0

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return 1.0
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 1.0;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := 1.0

Derivation

1. Initial program 43.1%

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

3. Taylor expanded in y.im around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. +-commutative N/A

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

   5. unpow2 N/A

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

   6. unpow2 N/A

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

   7. lower-hypot.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-atan2.f64 64.0

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

5. Applied rewrites 64.0%

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

6. Taylor expanded in y.re around 0

   \[\leadsto 1 \]
7. Step-by-step derivation

8. Applied rewrites 25.9%

   \[\leadsto 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024270 
(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)))))