powComplex, real part

Percentage Accurate: 40.1% → 80.3%

Time: 20.8s

Alternatives: 13

Speedup: 7.4×

• 34.6% 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: 40.1% 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: 80.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - t\_0}\\ t_2 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_3 := y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ \mathbf{if}\;y.re \leq -6.8:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq 5000000:\\ \;\;\;\;\frac{\cos t\_2 - t\_3 \cdot \sin t\_2}{\frac{e^{t\_0}}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \cos t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im))
        (t_1
         (exp (- (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im))))) t_0)))
        (t_2 (* y.re (atan2 x.im x.re)))
        (t_3 (* y.im (log (hypot x.im x.re)))))
   (if (<= y.re -6.8)
     t_1
     (if (<= y.re 5000000.0)
       (/
        (- (cos t_2) (* t_3 (sin t_2)))
        (/ (exp t_0) (pow (hypot x.re x.im) y.re)))
       (* t_1 (cos t_3))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_0));
	double t_2 = y_46_re * atan2(x_46_im, x_46_re);
	double t_3 = y_46_im * log(hypot(x_46_im, x_46_re));
	double tmp;
	if (y_46_re <= -6.8) {
		tmp = t_1;
	} else if (y_46_re <= 5000000.0) {
		tmp = (cos(t_2) - (t_3 * sin(t_2))) / (exp(t_0) / pow(hypot(x_46_re, x_46_im), y_46_re));
	} else {
		tmp = t_1 * cos(t_3);
	}
	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.atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = Math.exp(((y_46_re * Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_0));
	double t_2 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double t_3 = y_46_im * Math.log(Math.hypot(x_46_im, x_46_re));
	double tmp;
	if (y_46_re <= -6.8) {
		tmp = t_1;
	} else if (y_46_re <= 5000000.0) {
		tmp = (Math.cos(t_2) - (t_3 * Math.sin(t_2))) / (Math.exp(t_0) / Math.pow(Math.hypot(x_46_re, x_46_im), y_46_re));
	} else {
		tmp = t_1 * Math.cos(t_3);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.atan2(x_46_im, x_46_re) * y_46_im
	t_1 = math.exp(((y_46_re * math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_0))
	t_2 = y_46_re * math.atan2(x_46_im, x_46_re)
	t_3 = y_46_im * math.log(math.hypot(x_46_im, x_46_re))
	tmp = 0
	if y_46_re <= -6.8:
		tmp = t_1
	elif y_46_re <= 5000000.0:
		tmp = (math.cos(t_2) - (t_3 * math.sin(t_2))) / (math.exp(t_0) / math.pow(math.hypot(x_46_re, x_46_im), y_46_re))
	else:
		tmp = t_1 * math.cos(t_3)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_1 = exp(Float64(Float64(y_46_re * log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))) - t_0))
	t_2 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_3 = Float64(y_46_im * log(hypot(x_46_im, x_46_re)))
	tmp = 0.0
	if (y_46_re <= -6.8)
		tmp = t_1;
	elseif (y_46_re <= 5000000.0)
		tmp = Float64(Float64(cos(t_2) - Float64(t_3 * sin(t_2))) / Float64(exp(t_0) / (hypot(x_46_re, x_46_im) ^ y_46_re)));
	else
		tmp = Float64(t_1 * cos(t_3));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	t_1 = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_0));
	t_2 = y_46_re * atan2(x_46_im, x_46_re);
	t_3 = y_46_im * log(hypot(x_46_im, x_46_re));
	tmp = 0.0;
	if (y_46_re <= -6.8)
		tmp = t_1;
	elseif (y_46_re <= 5000000.0)
		tmp = (cos(t_2) - (t_3 * sin(t_2))) / (exp(t_0) / (hypot(x_46_re, x_46_im) ^ y_46_re));
	else
		tmp = t_1 * cos(t_3);
	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[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(N[(y$46$re * N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -6.8], t$95$1, If[LessEqual[y$46$re, 5000000.0], N[(N[(N[Cos[t$95$2], $MachinePrecision] - N[(t$95$3 * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Exp[t$95$0], $MachinePrecision] / N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Cos[t$95$3], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -6.79999999999999982

   1. Initial program 37.5%

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

   3. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \color{blue}{\cos \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{cos.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      3. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right)\right)\right) \]

      6. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right)\right)\right) \]

      7. hypot-lowering-hypot.f64 87.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right)\right) \]

   5. Simplified 87.6%

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

   6. Taylor expanded in y.im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \color{blue}{1}\right) \]
   7. Step-by-step derivation

   8. Simplified 87.6%

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

   if -6.79999999999999982 < y.re < 5e6

   1. Initial program 36.9%

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

      1. exp-diff N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/r/ N/A

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

      6. exp-diff N/A

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

   3. Simplified 78.0%

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

   5. Taylor expanded in y.im around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + -1 \cdot \left(y.im \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right)}, \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \left(-1 \cdot \left(y.im \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)}, \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{cos.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left(-1 \cdot \left(y.im \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)}\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left(-1 \cdot \left(y.im \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{atan2.f64}\left(x.im, x.re\right)}, y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left(-1 \cdot \left(y.im \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left(\mathsf{neg}\left(y.im \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      6. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{neg.f64}\left(\left(y.im \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{neg.f64}\left(\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

   7. Simplified 79.3%

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

   if 5e6 < y.re

   1. Initial program 37.3%

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

   3. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \color{blue}{\cos \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{cos.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      3. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right)\right)\right) \]

      6. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right)\right)\right) \]

      7. hypot-lowering-hypot.f64 77.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right)\right) \]

   5. Simplified 77.7%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -6.8:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;y.re \leq 5000000:\\ \;\;\;\;\frac{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) - \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{\frac{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - t\_0}\\ \mathbf{if}\;y.re \leq -9.8:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq 250000000:\\ \;\;\;\;\frac{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{\frac{e^{t\_0}}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im))
        (t_1
         (exp (- (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im))))) t_0))))
   (if (<= y.re -9.8)
     t_1
     (if (<= y.re 250000000.0)
       (/
        (cos (* y.re (atan2 x.im x.re)))
        (/ (exp t_0) (pow (hypot x.re x.im) y.re)))
       (* t_1 (cos (* y.im (log (hypot x.im x.re)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_0));
	double tmp;
	if (y_46_re <= -9.8) {
		tmp = t_1;
	} else if (y_46_re <= 250000000.0) {
		tmp = cos((y_46_re * atan2(x_46_im, x_46_re))) / (exp(t_0) / pow(hypot(x_46_re, x_46_im), y_46_re));
	} else {
		tmp = t_1 * cos((y_46_im * log(hypot(x_46_im, x_46_re))));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = Math.exp(((y_46_re * Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_0));
	double tmp;
	if (y_46_re <= -9.8) {
		tmp = t_1;
	} else if (y_46_re <= 250000000.0) {
		tmp = Math.cos((y_46_re * Math.atan2(x_46_im, x_46_re))) / (Math.exp(t_0) / Math.pow(Math.hypot(x_46_re, x_46_im), y_46_re));
	} else {
		tmp = t_1 * Math.cos((y_46_im * Math.log(Math.hypot(x_46_im, x_46_re))));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.atan2(x_46_im, x_46_re) * y_46_im
	t_1 = math.exp(((y_46_re * math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_0))
	tmp = 0
	if y_46_re <= -9.8:
		tmp = t_1
	elif y_46_re <= 250000000.0:
		tmp = math.cos((y_46_re * math.atan2(x_46_im, x_46_re))) / (math.exp(t_0) / math.pow(math.hypot(x_46_re, x_46_im), y_46_re))
	else:
		tmp = t_1 * math.cos((y_46_im * math.log(math.hypot(x_46_im, x_46_re))))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_1 = exp(Float64(Float64(y_46_re * log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))) - t_0))
	tmp = 0.0
	if (y_46_re <= -9.8)
		tmp = t_1;
	elseif (y_46_re <= 250000000.0)
		tmp = Float64(cos(Float64(y_46_re * atan(x_46_im, x_46_re))) / Float64(exp(t_0) / (hypot(x_46_re, x_46_im) ^ y_46_re)));
	else
		tmp = Float64(t_1 * cos(Float64(y_46_im * log(hypot(x_46_im, x_46_re)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	t_1 = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_0));
	tmp = 0.0;
	if (y_46_re <= -9.8)
		tmp = t_1;
	elseif (y_46_re <= 250000000.0)
		tmp = cos((y_46_re * atan2(x_46_im, x_46_re))) / (exp(t_0) / (hypot(x_46_re, x_46_im) ^ y_46_re));
	else
		tmp = t_1 * cos((y_46_im * log(hypot(x_46_im, x_46_re))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(N[(y$46$re * N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -9.8], t$95$1, If[LessEqual[y$46$re, 250000000.0], N[(N[Cos[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[Exp[t$95$0], $MachinePrecision] / N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Cos[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -9.8000000000000007

   1. Initial program 37.5%

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

   3. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \color{blue}{\cos \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{cos.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      3. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right)\right)\right) \]

      6. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right)\right)\right) \]

      7. hypot-lowering-hypot.f64 87.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right)\right) \]

   5. Simplified 87.6%

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

   6. Taylor expanded in y.im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \color{blue}{1}\right) \]
   7. Step-by-step derivation

   8. Simplified 87.6%

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

   if -9.8000000000000007 < y.re < 2.5e8

   1. Initial program 36.9%

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

      1. exp-diff N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/r/ N/A

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

      6. exp-diff N/A

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

   3. Simplified 78.0%

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

   5. Taylor expanded in y.im around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}, \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]
   6. Step-by-step derivation

      1. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)}, \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)}\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      3. atan2-lowering-atan2.f64 78.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \color{blue}{y.im}\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

   7. Simplified 78.5%

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

   if 2.5e8 < y.re

   1. Initial program 37.3%

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

   3. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \color{blue}{\cos \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{cos.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      3. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right)\right)\right) \]

      6. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right)\right)\right) \]

      7. hypot-lowering-hypot.f64 77.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right)\right) \]

   5. Simplified 77.7%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -9.8:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;y.re \leq 250000000:\\ \;\;\;\;\frac{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{\frac{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - t\_0}\\ \mathbf{if}\;y.re \leq -6.5:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq 160000000:\\ \;\;\;\;\frac{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{\frac{e^{t\_0}}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im))
        (t_1
         (exp (- (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im))))) t_0))))
   (if (<= y.re -6.5)
     t_1
     (if (<= y.re 160000000.0)
       (/
        (cos (* y.re (atan2 x.im x.re)))
        (/ (exp t_0) (pow (hypot x.re x.im) y.re)))
       t_1))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_0));
	double tmp;
	if (y_46_re <= -6.5) {
		tmp = t_1;
	} else if (y_46_re <= 160000000.0) {
		tmp = cos((y_46_re * atan2(x_46_im, x_46_re))) / (exp(t_0) / pow(hypot(x_46_re, x_46_im), y_46_re));
	} 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.atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = Math.exp(((y_46_re * Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_0));
	double tmp;
	if (y_46_re <= -6.5) {
		tmp = t_1;
	} else if (y_46_re <= 160000000.0) {
		tmp = Math.cos((y_46_re * Math.atan2(x_46_im, x_46_re))) / (Math.exp(t_0) / Math.pow(Math.hypot(x_46_re, x_46_im), y_46_re));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.atan2(x_46_im, x_46_re) * y_46_im
	t_1 = math.exp(((y_46_re * math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_0))
	tmp = 0
	if y_46_re <= -6.5:
		tmp = t_1
	elif y_46_re <= 160000000.0:
		tmp = math.cos((y_46_re * math.atan2(x_46_im, x_46_re))) / (math.exp(t_0) / math.pow(math.hypot(x_46_re, x_46_im), y_46_re))
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_1 = exp(Float64(Float64(y_46_re * log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))) - t_0))
	tmp = 0.0
	if (y_46_re <= -6.5)
		tmp = t_1;
	elseif (y_46_re <= 160000000.0)
		tmp = Float64(cos(Float64(y_46_re * atan(x_46_im, x_46_re))) / Float64(exp(t_0) / (hypot(x_46_re, x_46_im) ^ y_46_re)));
	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 = atan2(x_46_im, x_46_re) * y_46_im;
	t_1 = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_0));
	tmp = 0.0;
	if (y_46_re <= -6.5)
		tmp = t_1;
	elseif (y_46_re <= 160000000.0)
		tmp = cos((y_46_re * atan2(x_46_im, x_46_re))) / (exp(t_0) / (hypot(x_46_re, x_46_im) ^ y_46_re));
	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[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(N[(y$46$re * N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -6.5], t$95$1, If[LessEqual[y$46$re, 160000000.0], N[(N[Cos[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[Exp[t$95$0], $MachinePrecision] / N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -6.5 or 1.6e8 < y.re

   1. Initial program 37.4%

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

   3. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \color{blue}{\cos \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{cos.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      3. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right)\right)\right) \]

      6. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right)\right)\right) \]

      7. hypot-lowering-hypot.f64 82.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right)\right) \]

   5. Simplified 82.2%

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

   6. Taylor expanded in y.im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \color{blue}{1}\right) \]
   7. Step-by-step derivation

   8. Simplified 80.6%

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

   if -6.5 < y.re < 1.6e8

   1. Initial program 36.9%

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

      1. exp-diff N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/r/ N/A

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

      6. exp-diff N/A

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

   3. Simplified 78.0%

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

   5. Taylor expanded in y.im around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}, \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]
   6. Step-by-step derivation

      1. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)}, \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)}\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      3. atan2-lowering-atan2.f64 78.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \color{blue}{y.im}\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

   7. Simplified 78.5%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -6.5:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;y.re \leq 160000000:\\ \;\;\;\;\frac{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{\frac{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - t\_0}\\ \mathbf{if}\;y.re \leq -6.6:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{-14}:\\ \;\;\;\;\frac{1}{\frac{e^{t\_0}}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im))
        (t_1
         (exp (- (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im))))) t_0))))
   (if (<= y.re -6.6)
     t_1
     (if (<= y.re 2.4e-14)
       (/ 1.0 (/ (exp t_0) (pow (hypot x.re x.im) y.re)))
       t_1))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_0));
	double tmp;
	if (y_46_re <= -6.6) {
		tmp = t_1;
	} else if (y_46_re <= 2.4e-14) {
		tmp = 1.0 / (exp(t_0) / pow(hypot(x_46_re, x_46_im), y_46_re));
	} 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.atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = Math.exp(((y_46_re * Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_0));
	double tmp;
	if (y_46_re <= -6.6) {
		tmp = t_1;
	} else if (y_46_re <= 2.4e-14) {
		tmp = 1.0 / (Math.exp(t_0) / Math.pow(Math.hypot(x_46_re, x_46_im), y_46_re));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.atan2(x_46_im, x_46_re) * y_46_im
	t_1 = math.exp(((y_46_re * math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_0))
	tmp = 0
	if y_46_re <= -6.6:
		tmp = t_1
	elif y_46_re <= 2.4e-14:
		tmp = 1.0 / (math.exp(t_0) / math.pow(math.hypot(x_46_re, x_46_im), y_46_re))
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_1 = exp(Float64(Float64(y_46_re * log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))) - t_0))
	tmp = 0.0
	if (y_46_re <= -6.6)
		tmp = t_1;
	elseif (y_46_re <= 2.4e-14)
		tmp = Float64(1.0 / Float64(exp(t_0) / (hypot(x_46_re, x_46_im) ^ y_46_re)));
	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 = atan2(x_46_im, x_46_re) * y_46_im;
	t_1 = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_0));
	tmp = 0.0;
	if (y_46_re <= -6.6)
		tmp = t_1;
	elseif (y_46_re <= 2.4e-14)
		tmp = 1.0 / (exp(t_0) / (hypot(x_46_re, x_46_im) ^ y_46_re));
	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[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(N[(y$46$re * N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -6.6], t$95$1, If[LessEqual[y$46$re, 2.4e-14], N[(1.0 / N[(N[Exp[t$95$0], $MachinePrecision] / N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -6.5999999999999996 or 2.4e-14 < y.re

   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.re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \color{blue}{\cos \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{cos.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      3. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right)\right)\right) \]

      6. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right)\right)\right) \]

      7. hypot-lowering-hypot.f64 81.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right)\right) \]

   5. Simplified 81.2%

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

   6. Taylor expanded in y.im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \color{blue}{1}\right) \]
   7. Step-by-step derivation

   8. Simplified 80.4%

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

   if -6.5999999999999996 < y.re < 2.4e-14

   1. Initial program 36.5%

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

      1. exp-diff N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/r/ N/A

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

      6. exp-diff N/A

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

   3. Simplified 78.1%

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

   5. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\cos \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}, \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]
   6. Step-by-step derivation

      1. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)}, \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)}\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      3. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \color{blue}{y.im}\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      6. hypot-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      7. hypot-lowering-hypot.f64 77.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

   7. Simplified 77.1%

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

   8. Taylor expanded in y.im around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 77.0%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -6.6:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{-14}:\\ \;\;\;\;\frac{1}{\frac{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -3:\\ \;\;\;\;{\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{2}\right)}\\ \mathbf{elif}\;y.re \leq 2.9 \cdot 10^{+80}:\\ \;\;\;\;\frac{1}{\frac{e^{t\_0}}{t\_1}}\\ \mathbf{elif}\;y.re \leq 7.2 \cdot 10^{+171}:\\ \;\;\;\;{x.im}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t\_0 + 1}{t\_1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im)) (t_1 (pow (hypot x.re x.im) y.re)))
   (if (<= y.re -3.0)
     (pow (+ (* x.re x.re) (* x.im x.im)) (/ y.re 2.0))
     (if (<= y.re 2.9e+80)
       (/ 1.0 (/ (exp t_0) t_1))
       (if (<= y.re 7.2e+171) (pow x.im y.re) (/ 1.0 (/ (+ t_0 1.0) t_1)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = pow(hypot(x_46_re, x_46_im), y_46_re);
	double tmp;
	if (y_46_re <= -3.0) {
		tmp = pow(((x_46_re * x_46_re) + (x_46_im * x_46_im)), (y_46_re / 2.0));
	} else if (y_46_re <= 2.9e+80) {
		tmp = 1.0 / (exp(t_0) / t_1);
	} else if (y_46_re <= 7.2e+171) {
		tmp = pow(x_46_im, y_46_re);
	} else {
		tmp = 1.0 / ((t_0 + 1.0) / 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.atan2(x_46_im, x_46_re) * 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 <= -3.0) {
		tmp = Math.pow(((x_46_re * x_46_re) + (x_46_im * x_46_im)), (y_46_re / 2.0));
	} else if (y_46_re <= 2.9e+80) {
		tmp = 1.0 / (Math.exp(t_0) / t_1);
	} else if (y_46_re <= 7.2e+171) {
		tmp = Math.pow(x_46_im, y_46_re);
	} else {
		tmp = 1.0 / ((t_0 + 1.0) / t_1);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.atan2(x_46_im, x_46_re) * y_46_im
	t_1 = math.pow(math.hypot(x_46_re, x_46_im), y_46_re)
	tmp = 0
	if y_46_re <= -3.0:
		tmp = math.pow(((x_46_re * x_46_re) + (x_46_im * x_46_im)), (y_46_re / 2.0))
	elif y_46_re <= 2.9e+80:
		tmp = 1.0 / (math.exp(t_0) / t_1)
	elif y_46_re <= 7.2e+171:
		tmp = math.pow(x_46_im, y_46_re)
	else:
		tmp = 1.0 / ((t_0 + 1.0) / t_1)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_1 = hypot(x_46_re, x_46_im) ^ y_46_re
	tmp = 0.0
	if (y_46_re <= -3.0)
		tmp = Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)) ^ Float64(y_46_re / 2.0);
	elseif (y_46_re <= 2.9e+80)
		tmp = Float64(1.0 / Float64(exp(t_0) / t_1));
	elseif (y_46_re <= 7.2e+171)
		tmp = x_46_im ^ y_46_re;
	else
		tmp = Float64(1.0 / Float64(Float64(t_0 + 1.0) / 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 = atan2(x_46_im, x_46_re) * y_46_im;
	t_1 = hypot(x_46_re, x_46_im) ^ y_46_re;
	tmp = 0.0;
	if (y_46_re <= -3.0)
		tmp = ((x_46_re * x_46_re) + (x_46_im * x_46_im)) ^ (y_46_re / 2.0);
	elseif (y_46_re <= 2.9e+80)
		tmp = 1.0 / (exp(t_0) / t_1);
	elseif (y_46_re <= 7.2e+171)
		tmp = x_46_im ^ y_46_re;
	else
		tmp = 1.0 / ((t_0 + 1.0) / 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[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $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, -3.0], N[Power[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision], N[(y$46$re / 2.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[y$46$re, 2.9e+80], N[(1.0 / N[(N[Exp[t$95$0], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 7.2e+171], N[Power[x$46$im, y$46$re], $MachinePrecision], N[(1.0 / N[(N[(t$95$0 + 1.0), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -3

   1. Initial program 37.5%

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

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \cos \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right), \cos \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \cos \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{cos.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      10. atan2-lowering-atan2.f64 77.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   5. Simplified 77.0%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \color{blue}{1}\right) \]
   7. Step-by-step derivation

   8. Simplified 84.1%

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

      1. *-rgt-identity N/A

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

      2. sqrt-pow2 N/A

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

      3. +-commutative N/A

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

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(x.re \cdot x.re + x.im \cdot x.im\right), \color{blue}{\left(\frac{y.re}{2}\right)}\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(x.im \cdot x.im + x.re \cdot x.re\right), \left(\frac{\color{blue}{y.re}}{2}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left(x.re \cdot x.re\right)\right), \left(\frac{\color{blue}{y.re}}{2}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right), \left(\frac{y.re}{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right), \left(\frac{y.re}{2}\right)\right) \]

      9. /-lowering-/.f64 84.1%

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right), \mathsf{/.f64}\left(y.re, \color{blue}{2}\right)\right) \]

   10. Applied egg-rr 84.1%

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

   if -3 < y.re < 2.89999999999999986e80

   1. Initial program 35.6%

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

      1. exp-diff N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/r/ N/A

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

      6. exp-diff N/A

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

   3. Simplified 74.1%

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

   5. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\cos \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}, \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]
   6. Step-by-step derivation

      1. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)}, \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)}\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      3. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \color{blue}{y.im}\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      6. hypot-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      7. hypot-lowering-hypot.f64 74.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

   7. Simplified 74.7%

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

   8. Taylor expanded in y.im around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 75.2%

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

   if 2.89999999999999986e80 < y.re < 7.20000000000000036e171

   1. Initial program 50.0%

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

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \cos \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right), \cos \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \cos \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{cos.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      10. atan2-lowering-atan2.f64 50.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   5. Simplified 50.6%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \color{blue}{1}\right) \]
   7. Step-by-step derivation

   8. Simplified 56.8%

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

   9. Taylor expanded in x.re around 0

      \[\leadsto \color{blue}{{x.im}^{y.re}} \]
   10. Step-by-step derivation

       1. pow-lowering-pow.f64 69.2%

          \[\leadsto \mathsf{pow.f64}\left(x.im, \color{blue}{y.re}\right) \]

   11. Simplified 69.2%

       \[\leadsto \color{blue}{{x.im}^{y.re}} \]

   if 7.20000000000000036e171 < y.re

   1. Initial program 37.5%

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

      1. exp-diff N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/r/ N/A

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

      6. exp-diff N/A

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

   3. Simplified 40.6%

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

   5. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\cos \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}, \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]
   6. Step-by-step derivation

      1. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)}, \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)}\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      3. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \color{blue}{y.im}\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      6. hypot-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      7. hypot-lowering-hypot.f64 56.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

   7. Simplified 56.3%

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

   8. Taylor expanded in y.im around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 50.0%

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

   11. Taylor expanded in y.im around 0

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\color{blue}{\left(1 + y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}, \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re} + 1\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{hypot.f64}\left(x.re, x.im\right)}, y.re\right)\right)\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), 1\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{hypot.f64}\left(x.re, x.im\right)}, y.re\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.im, \tan^{-1}_* \frac{x.im}{x.re}\right), 1\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(\color{blue}{x.re}, x.im\right), y.re\right)\right)\right) \]

       4. atan2-lowering-atan2.f64 78.2%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right), 1\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

   13. Simplified 78.2%

       \[\leadsto \frac{1}{\frac{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re} + 1}}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3:\\ \;\;\;\;{\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{2}\right)}\\ \mathbf{elif}\;y.re \leq 2.9 \cdot 10^{+80}:\\ \;\;\;\;\frac{1}{\frac{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}\\ \mathbf{elif}\;y.re \leq 7.2 \cdot 10^{+171}:\\ \;\;\;\;{x.im}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im + 1}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.3% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.65:\\ \;\;\;\;{\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{2}\right)}\\ \mathbf{elif}\;y.re \leq 550000000:\\ \;\;\;\;\frac{1}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.65)
   (pow (+ (* x.re x.re) (* x.im x.im)) (/ y.re 2.0))
   (if (<= y.re 550000000.0)
     (/ 1.0 (exp (* (atan2 x.im x.re) y.im)))
     (pow (hypot 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 tmp;
	if (y_46_re <= -1.65) {
		tmp = pow(((x_46_re * x_46_re) + (x_46_im * x_46_im)), (y_46_re / 2.0));
	} else if (y_46_re <= 550000000.0) {
		tmp = 1.0 / exp((atan2(x_46_im, x_46_re) * y_46_im));
	} else {
		tmp = pow(hypot(x_46_im, x_46_re), y_46_re);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.65) {
		tmp = Math.pow(((x_46_re * x_46_re) + (x_46_im * x_46_im)), (y_46_re / 2.0));
	} else if (y_46_re <= 550000000.0) {
		tmp = 1.0 / Math.exp((Math.atan2(x_46_im, x_46_re) * y_46_im));
	} else {
		tmp = Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -1.65:
		tmp = math.pow(((x_46_re * x_46_re) + (x_46_im * x_46_im)), (y_46_re / 2.0))
	elif y_46_re <= 550000000.0:
		tmp = 1.0 / math.exp((math.atan2(x_46_im, x_46_re) * y_46_im))
	else:
		tmp = math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -1.65)
		tmp = Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)) ^ Float64(y_46_re / 2.0);
	elseif (y_46_re <= 550000000.0)
		tmp = Float64(1.0 / exp(Float64(atan(x_46_im, x_46_re) * y_46_im)));
	else
		tmp = hypot(x_46_im, x_46_re) ^ y_46_re;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -1.65)
		tmp = ((x_46_re * x_46_re) + (x_46_im * x_46_im)) ^ (y_46_re / 2.0);
	elseif (y_46_re <= 550000000.0)
		tmp = 1.0 / exp((atan2(x_46_im, x_46_re) * y_46_im));
	else
		tmp = hypot(x_46_im, x_46_re) ^ y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -1.65], N[Power[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision], N[(y$46$re / 2.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[y$46$re, 550000000.0], N[(1.0 / N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.6499999999999999

   1. Initial program 37.5%

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

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \cos \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right), \cos \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \cos \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{cos.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      10. atan2-lowering-atan2.f64 77.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   5. Simplified 77.0%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \color{blue}{1}\right) \]
   7. Step-by-step derivation

   8. Simplified 84.1%

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

      1. *-rgt-identity N/A

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

      2. sqrt-pow2 N/A

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

      3. +-commutative N/A

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

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(x.re \cdot x.re + x.im \cdot x.im\right), \color{blue}{\left(\frac{y.re}{2}\right)}\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(x.im \cdot x.im + x.re \cdot x.re\right), \left(\frac{\color{blue}{y.re}}{2}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left(x.re \cdot x.re\right)\right), \left(\frac{\color{blue}{y.re}}{2}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right), \left(\frac{y.re}{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right), \left(\frac{y.re}{2}\right)\right) \]

      9. /-lowering-/.f64 84.1%

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right), \mathsf{/.f64}\left(y.re, \color{blue}{2}\right)\right) \]

   10. Applied egg-rr 84.1%

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

   if -1.6499999999999999 < y.re < 5.5e8

   1. Initial program 37.4%

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

      1. exp-diff N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/r/ N/A

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

      6. exp-diff N/A

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

   3. Simplified 77.4%

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

   5. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\cos \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}, \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]
   6. Step-by-step derivation

      1. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)}, \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)}\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      3. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \color{blue}{y.im}\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      6. hypot-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      7. hypot-lowering-hypot.f64 75.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

   7. Simplified 75.7%

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

   8. Taylor expanded in y.im around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 76.3%

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

   11. Taylor expanded in y.re around 0

       \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)}\right) \]
   12. Step-by-step derivation

       1. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

       3. atan2-lowering-atan2.f64 76.2%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   13. Simplified 76.2%

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

   if 5.5e8 < y.re

   1. Initial program 36.4%

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

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \cos \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right), \cos \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \cos \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{cos.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      10. atan2-lowering-atan2.f64 53.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   5. Simplified 53.3%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \color{blue}{1}\right) \]
   7. Step-by-step derivation

   8. Simplified 65.4%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.65:\\ \;\;\;\;{\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{2}\right)}\\ \mathbf{elif}\;y.re \leq 550000000:\\ \;\;\;\;\frac{1}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.9% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.re \cdot x.re + x.im \cdot x.im\\ t_1 := {\left(t\_0 \cdot t\_0\right)}^{\left(\frac{\frac{y.re}{2}}{2}\right)}\\ \mathbf{if}\;y.im \leq -26500000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq 6.2 \cdot 10^{+103}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (* x.re x.re) (* x.im x.im)))
        (t_1 (pow (* t_0 t_0) (/ (/ y.re 2.0) 2.0))))
   (if (<= y.im -26500000000000.0)
     t_1
     (if (<= y.im 6.2e+103) (pow (hypot x.im x.re) y.re) t_1))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_re * x_46_re) + (x_46_im * x_46_im);
	double t_1 = pow((t_0 * t_0), ((y_46_re / 2.0) / 2.0));
	double tmp;
	if (y_46_im <= -26500000000000.0) {
		tmp = t_1;
	} else if (y_46_im <= 6.2e+103) {
		tmp = pow(hypot(x_46_im, x_46_re), y_46_re);
	} 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 = (x_46_re * x_46_re) + (x_46_im * x_46_im);
	double t_1 = Math.pow((t_0 * t_0), ((y_46_re / 2.0) / 2.0));
	double tmp;
	if (y_46_im <= -26500000000000.0) {
		tmp = t_1;
	} else if (y_46_im <= 6.2e+103) {
		tmp = Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_re * x_46_re) + (x_46_im * x_46_im)
	t_1 = math.pow((t_0 * t_0), ((y_46_re / 2.0) / 2.0))
	tmp = 0
	if y_46_im <= -26500000000000.0:
		tmp = t_1
	elif y_46_im <= 6.2e+103:
		tmp = math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))
	t_1 = Float64(t_0 * t_0) ^ Float64(Float64(y_46_re / 2.0) / 2.0)
	tmp = 0.0
	if (y_46_im <= -26500000000000.0)
		tmp = t_1;
	elseif (y_46_im <= 6.2e+103)
		tmp = hypot(x_46_im, x_46_re) ^ y_46_re;
	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 = (x_46_re * x_46_re) + (x_46_im * x_46_im);
	t_1 = (t_0 * t_0) ^ ((y_46_re / 2.0) / 2.0);
	tmp = 0.0;
	if (y_46_im <= -26500000000000.0)
		tmp = t_1;
	elseif (y_46_im <= 6.2e+103)
		tmp = hypot(x_46_im, x_46_re) ^ y_46_re;
	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[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(t$95$0 * t$95$0), $MachinePrecision], N[(N[(y$46$re / 2.0), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$im, -26500000000000.0], t$95$1, If[LessEqual[y$46$im, 6.2e+103], N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -2.65e13 or 6.2000000000000003e103 < y.im

   1. Initial program 37.2%

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

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \cos \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right), \cos \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \cos \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{cos.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      10. atan2-lowering-atan2.f64 29.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   5. Simplified 29.8%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \color{blue}{1}\right) \]
   7. Step-by-step derivation

   8. Simplified 31.5%

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

      1. *-rgt-identity N/A

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

      2. sqrt-pow2 N/A

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

      3. +-commutative N/A

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

      4. sqr-pow N/A

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

      5. pow-prod-down N/A

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

      6. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\left(x.re \cdot x.re + x.im \cdot x.im\right) \cdot \left(x.re \cdot x.re + x.im \cdot x.im\right)\right), \color{blue}{\left(\frac{\frac{y.re}{2}}{2}\right)}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(x.re \cdot x.re + x.im \cdot x.im\right), \left(x.re \cdot x.re + x.im \cdot x.im\right)\right), \left(\frac{\color{blue}{\frac{y.re}{2}}}{2}\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(x.im \cdot x.im + x.re \cdot x.re\right), \left(x.re \cdot x.re + x.im \cdot x.im\right)\right), \left(\frac{\frac{\color{blue}{y.re}}{2}}{2}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left(x.re \cdot x.re\right)\right), \left(x.re \cdot x.re + x.im \cdot x.im\right)\right), \left(\frac{\frac{\color{blue}{y.re}}{2}}{2}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right), \left(x.re \cdot x.re + x.im \cdot x.im\right)\right), \left(\frac{\frac{y.re}{2}}{2}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right), \left(x.re \cdot x.re + x.im \cdot x.im\right)\right), \left(\frac{\frac{y.re}{2}}{2}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right), \left(x.im \cdot x.im + x.re \cdot x.re\right)\right), \left(\frac{\frac{y.re}{\color{blue}{2}}}{2}\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right), \mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left(x.re \cdot x.re\right)\right)\right), \left(\frac{\frac{y.re}{\color{blue}{2}}}{2}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right)\right), \left(\frac{\frac{y.re}{2}}{2}\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right), \left(\frac{\frac{y.re}{2}}{2}\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right), \mathsf{/.f64}\left(\left(\frac{y.re}{2}\right), \color{blue}{2}\right)\right) \]

      17. /-lowering-/.f64 45.9%

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(y.re, 2\right), 2\right)\right) \]

   10. Applied egg-rr 45.9%

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

   if -2.65e13 < y.im < 6.2000000000000003e103

   1. Initial program 37.1%

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

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \cos \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right), \cos \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \cos \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{cos.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      10. atan2-lowering-atan2.f64 74.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   5. Simplified 74.1%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \color{blue}{1}\right) \]
   7. Step-by-step derivation

   8. Simplified 79.6%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -26500000000000:\\ \;\;\;\;{\left(\left(x.re \cdot x.re + x.im \cdot x.im\right) \cdot \left(x.re \cdot x.re + x.im \cdot x.im\right)\right)}^{\left(\frac{\frac{y.re}{2}}{2}\right)}\\ \mathbf{elif}\;y.im \leq 6.2 \cdot 10^{+103}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(x.re \cdot x.re + x.im \cdot x.im\right) \cdot \left(x.re \cdot x.re + x.im \cdot x.im\right)\right)}^{\left(\frac{\frac{y.re}{2}}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.2% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.re \cdot x.re + x.im \cdot x.im\\ \mathbf{if}\;y.re \leq -0.025:\\ \;\;\;\;{\left(t\_0 \cdot t\_0\right)}^{\left(\frac{\frac{y.re}{2}}{2}\right)}\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{-14}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;{t\_0}^{\left(\frac{y.re}{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (* x.re x.re) (* x.im x.im))))
   (if (<= y.re -0.025)
     (pow (* t_0 t_0) (/ (/ y.re 2.0) 2.0))
     (if (<= y.re 2.4e-14) 1.0 (pow t_0 (/ y.re 2.0))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_re * x_46_re) + (x_46_im * x_46_im);
	double tmp;
	if (y_46_re <= -0.025) {
		tmp = pow((t_0 * t_0), ((y_46_re / 2.0) / 2.0));
	} else if (y_46_re <= 2.4e-14) {
		tmp = 1.0;
	} else {
		tmp = pow(t_0, (y_46_re / 2.0));
	}
	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 * x_46re) + (x_46im * x_46im)
    if (y_46re <= (-0.025d0)) then
        tmp = (t_0 * t_0) ** ((y_46re / 2.0d0) / 2.0d0)
    else if (y_46re <= 2.4d-14) then
        tmp = 1.0d0
    else
        tmp = t_0 ** (y_46re / 2.0d0)
    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 t_0 = (x_46_re * x_46_re) + (x_46_im * x_46_im);
	double tmp;
	if (y_46_re <= -0.025) {
		tmp = Math.pow((t_0 * t_0), ((y_46_re / 2.0) / 2.0));
	} else if (y_46_re <= 2.4e-14) {
		tmp = 1.0;
	} else {
		tmp = Math.pow(t_0, (y_46_re / 2.0));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_re * x_46_re) + (x_46_im * x_46_im)
	tmp = 0
	if y_46_re <= -0.025:
		tmp = math.pow((t_0 * t_0), ((y_46_re / 2.0) / 2.0))
	elif y_46_re <= 2.4e-14:
		tmp = 1.0
	else:
		tmp = math.pow(t_0, (y_46_re / 2.0))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))
	tmp = 0.0
	if (y_46_re <= -0.025)
		tmp = Float64(t_0 * t_0) ^ Float64(Float64(y_46_re / 2.0) / 2.0);
	elseif (y_46_re <= 2.4e-14)
		tmp = 1.0;
	else
		tmp = t_0 ^ Float64(y_46_re / 2.0);
	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 * x_46_re) + (x_46_im * x_46_im);
	tmp = 0.0;
	if (y_46_re <= -0.025)
		tmp = (t_0 * t_0) ^ ((y_46_re / 2.0) / 2.0);
	elseif (y_46_re <= 2.4e-14)
		tmp = 1.0;
	else
		tmp = t_0 ^ (y_46_re / 2.0);
	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[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -0.025], N[Power[N[(t$95$0 * t$95$0), $MachinePrecision], N[(N[(y$46$re / 2.0), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[y$46$re, 2.4e-14], 1.0, N[Power[t$95$0, N[(y$46$re / 2.0), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -0.025000000000000001

   1. Initial program 38.6%

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

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \cos \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right), \cos \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \cos \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{cos.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      10. atan2-lowering-atan2.f64 75.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   5. Simplified 75.7%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \color{blue}{1}\right) \]
   7. Step-by-step derivation

   8. Simplified 82.7%

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

      1. *-rgt-identity N/A

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

      2. sqrt-pow2 N/A

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

      3. +-commutative N/A

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

      4. sqr-pow N/A

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

      5. pow-prod-down N/A

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

      6. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\left(x.re \cdot x.re + x.im \cdot x.im\right) \cdot \left(x.re \cdot x.re + x.im \cdot x.im\right)\right), \color{blue}{\left(\frac{\frac{y.re}{2}}{2}\right)}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(x.re \cdot x.re + x.im \cdot x.im\right), \left(x.re \cdot x.re + x.im \cdot x.im\right)\right), \left(\frac{\color{blue}{\frac{y.re}{2}}}{2}\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(x.im \cdot x.im + x.re \cdot x.re\right), \left(x.re \cdot x.re + x.im \cdot x.im\right)\right), \left(\frac{\frac{\color{blue}{y.re}}{2}}{2}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left(x.re \cdot x.re\right)\right), \left(x.re \cdot x.re + x.im \cdot x.im\right)\right), \left(\frac{\frac{\color{blue}{y.re}}{2}}{2}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right), \left(x.re \cdot x.re + x.im \cdot x.im\right)\right), \left(\frac{\frac{y.re}{2}}{2}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right), \left(x.re \cdot x.re + x.im \cdot x.im\right)\right), \left(\frac{\frac{y.re}{2}}{2}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right), \left(x.im \cdot x.im + x.re \cdot x.re\right)\right), \left(\frac{\frac{y.re}{\color{blue}{2}}}{2}\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right), \mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left(x.re \cdot x.re\right)\right)\right), \left(\frac{\frac{y.re}{\color{blue}{2}}}{2}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right)\right), \left(\frac{\frac{y.re}{2}}{2}\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right), \left(\frac{\frac{y.re}{2}}{2}\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right), \mathsf{/.f64}\left(\left(\frac{y.re}{2}\right), \color{blue}{2}\right)\right) \]

      17. /-lowering-/.f64 84.4%

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(y.re, 2\right), 2\right)\right) \]

   10. Applied egg-rr 84.4%

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

   if -0.025000000000000001 < y.re < 2.4e-14

   1. Initial program 36.0%

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

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \cos \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right), \cos \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \cos \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{cos.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      10. atan2-lowering-atan2.f64 45.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   5. Simplified 45.1%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 43.9%

      \[\leadsto \color{blue}{1} \]

   if 2.4e-14 < y.re

   1. Initial program 38.0%

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

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \cos \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right), \cos \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \cos \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{cos.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      10. atan2-lowering-atan2.f64 53.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   5. Simplified 53.8%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \color{blue}{1}\right) \]
   7. Step-by-step derivation

   8. Simplified 63.7%

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

      1. *-rgt-identity N/A

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

      2. sqrt-pow2 N/A

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

      3. +-commutative N/A

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

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(x.re \cdot x.re + x.im \cdot x.im\right), \color{blue}{\left(\frac{y.re}{2}\right)}\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(x.im \cdot x.im + x.re \cdot x.re\right), \left(\frac{\color{blue}{y.re}}{2}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left(x.re \cdot x.re\right)\right), \left(\frac{\color{blue}{y.re}}{2}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right), \left(\frac{y.re}{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right), \left(\frac{y.re}{2}\right)\right) \]

      9. /-lowering-/.f64 65.1%

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right), \mathsf{/.f64}\left(y.re, \color{blue}{2}\right)\right) \]

   10. Applied egg-rr 65.1%

       \[\leadsto \color{blue}{{\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\left(\frac{y.re}{2}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -0.025:\\ \;\;\;\;{\left(\left(x.re \cdot x.re + x.im \cdot x.im\right) \cdot \left(x.re \cdot x.re + x.im \cdot x.im\right)\right)}^{\left(\frac{\frac{y.re}{2}}{2}\right)}\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{-14}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;{\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.3% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{2}\right)}\\ \mathbf{if}\;y.re \leq -3.1 \cdot 10^{-12}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 4.8 \cdot 10^{-18}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (pow (+ (* x.re x.re) (* x.im x.im)) (/ y.re 2.0))))
   (if (<= y.re -3.1e-12) t_0 (if (<= y.re 4.8e-18) 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 * x_46_re) + (x_46_im * x_46_im)), (y_46_re / 2.0));
	double tmp;
	if (y_46_re <= -3.1e-12) {
		tmp = t_0;
	} else if (y_46_re <= 4.8e-18) {
		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 * x_46re) + (x_46im * x_46im)) ** (y_46re / 2.0d0)
    if (y_46re <= (-3.1d-12)) then
        tmp = t_0
    else if (y_46re <= 4.8d-18) 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 * x_46_re) + (x_46_im * x_46_im)), (y_46_re / 2.0));
	double tmp;
	if (y_46_re <= -3.1e-12) {
		tmp = t_0;
	} else if (y_46_re <= 4.8e-18) {
		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 * x_46_re) + (x_46_im * x_46_im)), (y_46_re / 2.0))
	tmp = 0
	if y_46_re <= -3.1e-12:
		tmp = t_0
	elif y_46_re <= 4.8e-18:
		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(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)) ^ Float64(y_46_re / 2.0)
	tmp = 0.0
	if (y_46_re <= -3.1e-12)
		tmp = t_0;
	elseif (y_46_re <= 4.8e-18)
		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 * x_46_re) + (x_46_im * x_46_im)) ^ (y_46_re / 2.0);
	tmp = 0.0;
	if (y_46_re <= -3.1e-12)
		tmp = t_0;
	elseif (y_46_re <= 4.8e-18)
		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[Power[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision], N[(y$46$re / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -3.1e-12], t$95$0, If[LessEqual[y$46$re, 4.8e-18], 1.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -3.1000000000000001e-12 or 4.79999999999999988e-18 < y.re

   1. Initial program 38.5%

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

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \cos \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right), \cos \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \cos \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{cos.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      10. atan2-lowering-atan2.f64 64.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   5. Simplified 64.0%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \color{blue}{1}\right) \]
   7. Step-by-step derivation

   8. Simplified 71.6%

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

      1. *-rgt-identity N/A

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

      2. sqrt-pow2 N/A

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

      3. +-commutative N/A

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

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(x.re \cdot x.re + x.im \cdot x.im\right), \color{blue}{\left(\frac{y.re}{2}\right)}\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(x.im \cdot x.im + x.re \cdot x.re\right), \left(\frac{\color{blue}{y.re}}{2}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left(x.re \cdot x.re\right)\right), \left(\frac{\color{blue}{y.re}}{2}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right), \left(\frac{y.re}{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right), \left(\frac{y.re}{2}\right)\right) \]

      9. /-lowering-/.f64 72.1%

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right), \mathsf{/.f64}\left(y.re, \color{blue}{2}\right)\right) \]

   10. Applied egg-rr 72.1%

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

   if -3.1000000000000001e-12 < y.re < 4.79999999999999988e-18

   1. Initial program 35.8%

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

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \cos \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right), \cos \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \cos \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{cos.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      10. atan2-lowering-atan2.f64 44.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   5. Simplified 44.4%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 44.4%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.1 \cdot 10^{-12}:\\ \;\;\;\;{\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{2}\right)}\\ \mathbf{elif}\;y.re \leq 4.8 \cdot 10^{-18}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;{\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 52.9% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -0.35:\\ \;\;\;\;{x.im}^{y.re}\\ \mathbf{elif}\;y.re \leq 6200000:\\ \;\;\;\;1\\ \mathbf{elif}\;y.re \leq 3.2 \cdot 10^{+80}:\\ \;\;\;\;{x.re}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;{x.im}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -0.35)
   (pow x.im y.re)
   (if (<= y.re 6200000.0)
     1.0
     (if (<= y.re 3.2e+80) (pow x.re y.re) (pow x.im y.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -0.35) {
		tmp = pow(x_46_im, y_46_re);
	} else if (y_46_re <= 6200000.0) {
		tmp = 1.0;
	} else if (y_46_re <= 3.2e+80) {
		tmp = pow(x_46_re, y_46_re);
	} else {
		tmp = pow(x_46_im, y_46_re);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-0.35d0)) then
        tmp = x_46im ** y_46re
    else if (y_46re <= 6200000.0d0) then
        tmp = 1.0d0
    else if (y_46re <= 3.2d+80) then
        tmp = x_46re ** y_46re
    else
        tmp = x_46im ** y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -0.35) {
		tmp = Math.pow(x_46_im, y_46_re);
	} else if (y_46_re <= 6200000.0) {
		tmp = 1.0;
	} else if (y_46_re <= 3.2e+80) {
		tmp = Math.pow(x_46_re, y_46_re);
	} else {
		tmp = Math.pow(x_46_im, y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -0.35:
		tmp = math.pow(x_46_im, y_46_re)
	elif y_46_re <= 6200000.0:
		tmp = 1.0
	elif y_46_re <= 3.2e+80:
		tmp = math.pow(x_46_re, y_46_re)
	else:
		tmp = math.pow(x_46_im, y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -0.35)
		tmp = x_46_im ^ y_46_re;
	elseif (y_46_re <= 6200000.0)
		tmp = 1.0;
	elseif (y_46_re <= 3.2e+80)
		tmp = x_46_re ^ y_46_re;
	else
		tmp = x_46_im ^ y_46_re;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -0.35)
		tmp = x_46_im ^ y_46_re;
	elseif (y_46_re <= 6200000.0)
		tmp = 1.0;
	elseif (y_46_re <= 3.2e+80)
		tmp = x_46_re ^ y_46_re;
	else
		tmp = x_46_im ^ y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -0.35], N[Power[x$46$im, y$46$re], $MachinePrecision], If[LessEqual[y$46$re, 6200000.0], 1.0, If[LessEqual[y$46$re, 3.2e+80], N[Power[x$46$re, y$46$re], $MachinePrecision], N[Power[x$46$im, y$46$re], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -0.34999999999999998 or 3.1999999999999999e80 < y.re

   1. Initial program 39.4%

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

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \cos \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right), \cos \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \cos \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{cos.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      10. atan2-lowering-atan2.f64 65.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   5. Simplified 65.6%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \color{blue}{1}\right) \]
   7. Step-by-step derivation

   8. Simplified 75.2%

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

   9. Taylor expanded in x.re around 0

      \[\leadsto \color{blue}{{x.im}^{y.re}} \]
   10. Step-by-step derivation

       1. pow-lowering-pow.f64 61.0%

          \[\leadsto \mathsf{pow.f64}\left(x.im, \color{blue}{y.re}\right) \]

   11. Simplified 61.0%

       \[\leadsto \color{blue}{{x.im}^{y.re}} \]

   if -0.34999999999999998 < y.re < 6.2e6

   1. Initial program 36.9%

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

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \cos \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right), \cos \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \cos \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{cos.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      10. atan2-lowering-atan2.f64 45.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   5. Simplified 45.7%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 43.3%

      \[\leadsto \color{blue}{1} \]

   if 6.2e6 < y.re < 3.1999999999999999e80

   1. Initial program 26.3%

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

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \cos \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right), \cos \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \cos \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{cos.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      10. atan2-lowering-atan2.f64 53.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   5. Simplified 53.0%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \color{blue}{1}\right) \]
   7. Step-by-step derivation

   8. Simplified 63.6%

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

   9. Taylor expanded in x.im around 0

      \[\leadsto \color{blue}{{x.re}^{y.re}} \]
   10. Step-by-step derivation

       1. pow-lowering-pow.f64 63.6%

          \[\leadsto \mathsf{pow.f64}\left(x.re, \color{blue}{y.re}\right) \]

   11. Simplified 63.6%

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

Alternative 11: 55.3% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -2.1 \cdot 10^{-15}:\\ \;\;\;\;{\left(0 - x.im\right)}^{y.re}\\ \mathbf{elif}\;x.im \leq 1.85 \cdot 10^{-88}:\\ \;\;\;\;{x.re}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;{x.im}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.im -2.1e-15)
   (pow (- 0.0 x.im) y.re)
   (if (<= x.im 1.85e-88) (pow x.re y.re) (pow x.im y.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_im <= -2.1e-15) {
		tmp = pow((0.0 - x_46_im), y_46_re);
	} else if (x_46_im <= 1.85e-88) {
		tmp = pow(x_46_re, y_46_re);
	} else {
		tmp = pow(x_46_im, y_46_re);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (x_46im <= (-2.1d-15)) then
        tmp = (0.0d0 - x_46im) ** y_46re
    else if (x_46im <= 1.85d-88) then
        tmp = x_46re ** y_46re
    else
        tmp = x_46im ** y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_im <= -2.1e-15) {
		tmp = Math.pow((0.0 - x_46_im), y_46_re);
	} else if (x_46_im <= 1.85e-88) {
		tmp = Math.pow(x_46_re, y_46_re);
	} else {
		tmp = Math.pow(x_46_im, y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if x_46_im <= -2.1e-15:
		tmp = math.pow((0.0 - x_46_im), y_46_re)
	elif x_46_im <= 1.85e-88:
		tmp = math.pow(x_46_re, y_46_re)
	else:
		tmp = math.pow(x_46_im, y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (x_46_im <= -2.1e-15)
		tmp = Float64(0.0 - x_46_im) ^ y_46_re;
	elseif (x_46_im <= 1.85e-88)
		tmp = x_46_re ^ y_46_re;
	else
		tmp = x_46_im ^ y_46_re;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (x_46_im <= -2.1e-15)
		tmp = (0.0 - x_46_im) ^ y_46_re;
	elseif (x_46_im <= 1.85e-88)
		tmp = x_46_re ^ y_46_re;
	else
		tmp = x_46_im ^ y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[x$46$im, -2.1e-15], N[Power[N[(0.0 - x$46$im), $MachinePrecision], y$46$re], $MachinePrecision], If[LessEqual[x$46$im, 1.85e-88], N[Power[x$46$re, y$46$re], $MachinePrecision], N[Power[x$46$im, y$46$re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -2.09999999999999981e-15

   1. Initial program 25.4%

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

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \cos \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right), \cos \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \cos \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{cos.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      10. atan2-lowering-atan2.f64 56.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   5. Simplified 56.5%

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

   6. Taylor expanded in x.im around -inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\color{blue}{\left(-1 \cdot x.im\right)}, y.re\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{neg}\left(x.im\right)\right), y.re\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\color{blue}{y.re}, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      2. neg-lowering-neg.f64 56.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{neg.f64}\left(x.im\right), y.re\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\color{blue}{y.re}, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   8. Simplified 56.5%

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

   9. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{neg.f64}\left(x.im\right), y.re\right), \color{blue}{1}\right) \]
   10. Step-by-step derivation

   11. Simplified 57.8%

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

   if -2.09999999999999981e-15 < x.im < 1.8499999999999999e-88

   1. Initial program 47.5%

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

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \cos \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right), \cos \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \cos \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{cos.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      10. atan2-lowering-atan2.f64 58.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   5. Simplified 58.1%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \color{blue}{1}\right) \]
   7. Step-by-step derivation

   8. Simplified 62.3%

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

   9. Taylor expanded in x.im around 0

      \[\leadsto \color{blue}{{x.re}^{y.re}} \]
   10. Step-by-step derivation

       1. pow-lowering-pow.f64 52.4%

          \[\leadsto \mathsf{pow.f64}\left(x.re, \color{blue}{y.re}\right) \]

   11. Simplified 52.4%

       \[\leadsto \color{blue}{{x.re}^{y.re}} \]

   if 1.8499999999999999e-88 < x.im

   1. Initial program 34.8%

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

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \cos \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right), \cos \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \cos \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{cos.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      10. atan2-lowering-atan2.f64 48.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   5. Simplified 48.8%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \color{blue}{1}\right) \]
   7. Step-by-step derivation

   8. Simplified 55.3%

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

   9. Taylor expanded in x.re around 0

      \[\leadsto \color{blue}{{x.im}^{y.re}} \]
   10. Step-by-step derivation

       1. pow-lowering-pow.f64 52.1%

          \[\leadsto \mathsf{pow.f64}\left(x.im, \color{blue}{y.re}\right) \]

   11. Simplified 52.1%

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

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -2.1 \cdot 10^{-15}:\\ \;\;\;\;{\left(0 - x.im\right)}^{y.re}\\ \mathbf{elif}\;x.im \leq 1.85 \cdot 10^{-88}:\\ \;\;\;\;{x.re}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;{x.im}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 52.6% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -0.49:\\ \;\;\;\;{x.im}^{y.re}\\ \mathbf{elif}\;y.re \leq 2.2 \cdot 10^{-15}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;{x.im}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -0.49)
   (pow x.im y.re)
   (if (<= y.re 2.2e-15) 1.0 (pow x.im y.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -0.49) {
		tmp = pow(x_46_im, y_46_re);
	} else if (y_46_re <= 2.2e-15) {
		tmp = 1.0;
	} else {
		tmp = pow(x_46_im, y_46_re);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-0.49d0)) then
        tmp = x_46im ** y_46re
    else if (y_46re <= 2.2d-15) then
        tmp = 1.0d0
    else
        tmp = x_46im ** y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -0.49) {
		tmp = Math.pow(x_46_im, y_46_re);
	} else if (y_46_re <= 2.2e-15) {
		tmp = 1.0;
	} else {
		tmp = Math.pow(x_46_im, y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -0.49:
		tmp = math.pow(x_46_im, y_46_re)
	elif y_46_re <= 2.2e-15:
		tmp = 1.0
	else:
		tmp = math.pow(x_46_im, y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -0.49)
		tmp = x_46_im ^ y_46_re;
	elseif (y_46_re <= 2.2e-15)
		tmp = 1.0;
	else
		tmp = x_46_im ^ y_46_re;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -0.49)
		tmp = x_46_im ^ y_46_re;
	elseif (y_46_re <= 2.2e-15)
		tmp = 1.0;
	else
		tmp = x_46_im ^ y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -0.49], N[Power[x$46$im, y$46$re], $MachinePrecision], If[LessEqual[y$46$re, 2.2e-15], 1.0, N[Power[x$46$im, y$46$re], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -0.48999999999999999 or 2.19999999999999986e-15 < y.re

   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

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \cos \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right), \cos \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \cos \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{cos.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      10. atan2-lowering-atan2.f64 64.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   5. Simplified 64.0%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \color{blue}{1}\right) \]
   7. Step-by-step derivation

   8. Simplified 72.7%

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

   9. Taylor expanded in x.re around 0

      \[\leadsto \color{blue}{{x.im}^{y.re}} \]
   10. Step-by-step derivation

       1. pow-lowering-pow.f64 57.2%

          \[\leadsto \mathsf{pow.f64}\left(x.im, \color{blue}{y.re}\right) \]

   11. Simplified 57.2%

       \[\leadsto \color{blue}{{x.im}^{y.re}} \]

   if -0.48999999999999999 < y.re < 2.19999999999999986e-15

   1. Initial program 36.5%

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

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \cos \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right), \cos \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \cos \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{cos.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      10. atan2-lowering-atan2.f64 44.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   5. Simplified 44.8%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 43.6%

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

Alternative 13: 26.3% accurate, 829.0× 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 37.1%

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

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

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

   3. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \cos \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   6. hypot-define N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right), \cos \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   7. hypot-lowering-hypot.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \cos \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   8. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{cos.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

   10. atan2-lowering-atan2.f64 54.3%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

5. Simplified 54.3%

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

6. Taylor expanded in y.re around 0

   \[\leadsto \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 23.8%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024170 
(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)))))