math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 14.2s

Alternatives: 20

Speedup: 1.0×

• 23.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ e^{re} \cdot \cos im \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (exp re) (cos im)))

C

double code(double re, double im) {
	return exp(re) * cos(im);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = exp(re) * cos(im)
end function

Java

public static double code(double re, double im) {
	return Math.exp(re) * Math.cos(im);
}

Python

def code(re, im):
	return math.exp(re) * math.cos(im)

Julia

function code(re, im)
	return Float64(exp(re) * cos(im))
end

MATLAB

function tmp = code(re, im)
	tmp = exp(re) * cos(im);
end

Wolfram

code[re_, im_] := N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{re} \cdot \cos im \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (exp re) (cos im)))

C

double code(double re, double im) {
	return exp(re) * cos(im);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = exp(re) * cos(im)
end function

Java

public static double code(double re, double im) {
	return Math.exp(re) * Math.cos(im);
}

Python

def code(re, im):
	return math.exp(re) * math.cos(im)

Julia

function code(re, im)
	return Float64(exp(re) * cos(im))
end

MATLAB

function tmp = code(re, im)
	tmp = exp(re) * cos(im);
end

Wolfram

code[re_, im_] := N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{re} \cdot \cos im \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (exp re) (cos im)))

C

double code(double re, double im) {
	return exp(re) * cos(im);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = exp(re) * cos(im)
end function

Java

public static double code(double re, double im) {
	return Math.exp(re) * Math.cos(im);
}

Python

def code(re, im):
	return math.exp(re) * math.cos(im)

Julia

function code(re, im)
	return Float64(exp(re) * cos(im))
end

MATLAB

function tmp = code(re, im)
	tmp = exp(re) * cos(im);
end

Wolfram

code[re_, im_] := N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \cos im \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 97.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{if}\;re \leq -0.026:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 1.3 \cdot 10^{-8}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;{e}^{re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (*
          (cos im)
          (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666)))))))))
   (if (<= re -0.026)
     (exp re)
     (if (<= re 1.3e-8) t_0 (if (<= re 1.05e+103) (pow E re) t_0)))))

C

double code(double re, double im) {
	double t_0 = cos(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	double tmp;
	if (re <= -0.026) {
		tmp = exp(re);
	} else if (re <= 1.3e-8) {
		tmp = t_0;
	} else if (re <= 1.05e+103) {
		tmp = pow(((double) M_E), re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = Math.cos(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	double tmp;
	if (re <= -0.026) {
		tmp = Math.exp(re);
	} else if (re <= 1.3e-8) {
		tmp = t_0;
	} else if (re <= 1.05e+103) {
		tmp = Math.pow(Math.E, re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.cos(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))))
	tmp = 0
	if re <= -0.026:
		tmp = math.exp(re)
	elif re <= 1.3e-8:
		tmp = t_0
	elif re <= 1.05e+103:
		tmp = math.pow(math.e, re)
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(cos(im) * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666)))))))
	tmp = 0.0
	if (re <= -0.026)
		tmp = exp(re);
	elseif (re <= 1.3e-8)
		tmp = t_0;
	elseif (re <= 1.05e+103)
		tmp = exp(1) ^ re;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = cos(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	tmp = 0.0;
	if (re <= -0.026)
		tmp = exp(re);
	elseif (re <= 1.3e-8)
		tmp = t_0;
	elseif (re <= 1.05e+103)
		tmp = 2.71828182845904523536 ^ re;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Cos[im], $MachinePrecision] * N[(1.0 + N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -0.026], N[Exp[re], $MachinePrecision], If[LessEqual[re, 1.3e-8], t$95$0, If[LessEqual[re, 1.05e+103], N[Power[E, re], $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.0259999999999999988

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. exp-lowering-exp.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(re\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{e^{re}} \]

   if -0.0259999999999999988 < re < 1.3000000000000001e-8 or 1.0500000000000001e103 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      7. *-lowering-*.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

   5. Simplified 99.9%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)} \cdot \cos im \]

   if 1.3000000000000001e-8 < re < 1.0500000000000001e103

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. exp-lowering-exp.f64 75.9%

         \[\leadsto \mathsf{exp.f64}\left(re\right) \]

   5. Simplified 75.9%

      \[\leadsto \color{blue}{e^{re}} \]
   6. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto e^{1 \cdot re} \]

      2. exp-prod N/A

         \[\leadsto {\left(e^{1}\right)}^{\color{blue}{re}} \]

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

         \[\leadsto \mathsf{pow.f64}\left(\left(e^{1}\right), \color{blue}{re}\right) \]

      4. exp-lowering-exp.f64 75.9%

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{exp.f64}\left(1\right), re\right) \]

   7. Applied egg-rr 75.9%

      \[\leadsto \color{blue}{{\left(e^{1}\right)}^{re}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.026:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 1.3 \cdot 10^{-8}:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;re \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;{e}^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{if}\;re \leq -0.0074:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 1.3 \cdot 10^{-8}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (*
          (cos im)
          (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666)))))))))
   (if (<= re -0.0074)
     (exp re)
     (if (<= re 1.3e-8) t_0 (if (<= re 1.05e+103) (exp re) t_0)))))

C

double code(double re, double im) {
	double t_0 = cos(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	double tmp;
	if (re <= -0.0074) {
		tmp = exp(re);
	} else if (re <= 1.3e-8) {
		tmp = t_0;
	} else if (re <= 1.05e+103) {
		tmp = exp(re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(im) * (1.0d0 + (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0))))))
    if (re <= (-0.0074d0)) then
        tmp = exp(re)
    else if (re <= 1.3d-8) then
        tmp = t_0
    else if (re <= 1.05d+103) then
        tmp = exp(re)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.cos(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	double tmp;
	if (re <= -0.0074) {
		tmp = Math.exp(re);
	} else if (re <= 1.3e-8) {
		tmp = t_0;
	} else if (re <= 1.05e+103) {
		tmp = Math.exp(re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.cos(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))))
	tmp = 0
	if re <= -0.0074:
		tmp = math.exp(re)
	elif re <= 1.3e-8:
		tmp = t_0
	elif re <= 1.05e+103:
		tmp = math.exp(re)
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(cos(im) * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666)))))))
	tmp = 0.0
	if (re <= -0.0074)
		tmp = exp(re);
	elseif (re <= 1.3e-8)
		tmp = t_0;
	elseif (re <= 1.05e+103)
		tmp = exp(re);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = cos(im) * (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666))))));
	tmp = 0.0;
	if (re <= -0.0074)
		tmp = exp(re);
	elseif (re <= 1.3e-8)
		tmp = t_0;
	elseif (re <= 1.05e+103)
		tmp = exp(re);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Cos[im], $MachinePrecision] * N[(1.0 + N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -0.0074], N[Exp[re], $MachinePrecision], If[LessEqual[re, 1.3e-8], t$95$0, If[LessEqual[re, 1.05e+103], N[Exp[re], $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if re < -0.0074000000000000003 or 1.3000000000000001e-8 < re < 1.0500000000000001e103

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. exp-lowering-exp.f64 92.8%

         \[\leadsto \mathsf{exp.f64}\left(re\right) \]

   5. Simplified 92.8%

      \[\leadsto \color{blue}{e^{re}} \]

   if -0.0074000000000000003 < re < 1.3000000000000001e-8 or 1.0500000000000001e103 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      7. *-lowering-*.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

   5. Simplified 99.9%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)} \cdot \cos im \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0074:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 1.3 \cdot 10^{-8}:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;re \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.018:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 1.3 \cdot 10^{-8}:\\ \;\;\;\;\cos im \cdot \left(re \cdot \left(re \cdot 0.5\right) + \left(re + 1\right)\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re} \cdot \left(1 + im \cdot \left(im \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.018)
   (exp re)
   (if (<= re 1.3e-8)
     (* (cos im) (+ (* re (* re 0.5)) (+ re 1.0)))
     (if (<= re 1.9e+154)
       (* (exp re) (+ 1.0 (* im (* im -0.5))))
       (* (cos im) (+ 1.0 (* re (+ 1.0 (* re 0.5)))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.018) {
		tmp = exp(re);
	} else if (re <= 1.3e-8) {
		tmp = cos(im) * ((re * (re * 0.5)) + (re + 1.0));
	} else if (re <= 1.9e+154) {
		tmp = exp(re) * (1.0 + (im * (im * -0.5)));
	} else {
		tmp = cos(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-0.018d0)) then
        tmp = exp(re)
    else if (re <= 1.3d-8) then
        tmp = cos(im) * ((re * (re * 0.5d0)) + (re + 1.0d0))
    else if (re <= 1.9d+154) then
        tmp = exp(re) * (1.0d0 + (im * (im * (-0.5d0))))
    else
        tmp = cos(im) * (1.0d0 + (re * (1.0d0 + (re * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.018) {
		tmp = Math.exp(re);
	} else if (re <= 1.3e-8) {
		tmp = Math.cos(im) * ((re * (re * 0.5)) + (re + 1.0));
	} else if (re <= 1.9e+154) {
		tmp = Math.exp(re) * (1.0 + (im * (im * -0.5)));
	} else {
		tmp = Math.cos(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.018:
		tmp = math.exp(re)
	elif re <= 1.3e-8:
		tmp = math.cos(im) * ((re * (re * 0.5)) + (re + 1.0))
	elif re <= 1.9e+154:
		tmp = math.exp(re) * (1.0 + (im * (im * -0.5)))
	else:
		tmp = math.cos(im) * (1.0 + (re * (1.0 + (re * 0.5))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.018)
		tmp = exp(re);
	elseif (re <= 1.3e-8)
		tmp = Float64(cos(im) * Float64(Float64(re * Float64(re * 0.5)) + Float64(re + 1.0)));
	elseif (re <= 1.9e+154)
		tmp = Float64(exp(re) * Float64(1.0 + Float64(im * Float64(im * -0.5))));
	else
		tmp = Float64(cos(im) * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.018)
		tmp = exp(re);
	elseif (re <= 1.3e-8)
		tmp = cos(im) * ((re * (re * 0.5)) + (re + 1.0));
	elseif (re <= 1.9e+154)
		tmp = exp(re) * (1.0 + (im * (im * -0.5)));
	else
		tmp = cos(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.018], N[Exp[re], $MachinePrecision], If[LessEqual[re, 1.3e-8], N[(N[Cos[im], $MachinePrecision] * N[(N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision] + N[(re + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.9e+154], N[(N[Exp[re], $MachinePrecision] * N[(1.0 + N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[im], $MachinePrecision] * N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -0.0179999999999999986

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. exp-lowering-exp.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(re\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{e^{re}} \]

   if -0.0179999999999999986 < re < 1.3000000000000001e-8

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + \frac{1}{2} \cdot re\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot re\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \frac{1}{2}\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      5. *-lowering-*.f64 99.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

   5. Simplified 99.6%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)} \cdot \cos im \]
   6. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 + \left(1 \cdot re + \left(re \cdot \frac{1}{2}\right) \cdot re\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      2. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 + \left(re + \left(re \cdot \frac{1}{2}\right) \cdot re\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      3. associate-+r+ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(1 + re\right) + \left(re \cdot \frac{1}{2}\right) \cdot re\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(re + 1\right) + \left(re \cdot \frac{1}{2}\right) \cdot re\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(re + 1\right), \left(\left(re \cdot \frac{1}{2}\right) \cdot re\right)\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(1 + re\right), \left(\left(re \cdot \frac{1}{2}\right) \cdot re\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, re\right), \left(\left(re \cdot \frac{1}{2}\right) \cdot re\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, re\right), \left(re \cdot \left(re \cdot \frac{1}{2}\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, re\right), \mathsf{*.f64}\left(re, \left(re \cdot \frac{1}{2}\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      10. *-lowering-*.f64 99.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, re\right), \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

   7. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\left(\left(1 + re\right) + re \cdot \left(re \cdot 0.5\right)\right)} \cdot \cos im \]

   if 1.3000000000000001e-8 < re < 1.8999999999999999e154

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re} + \frac{-1}{2} \cdot \left({im}^{2} \cdot e^{re}\right)} \]
   4. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto 1 \cdot e^{re} + \color{blue}{\frac{-1}{2}} \cdot \left({im}^{2} \cdot e^{re}\right) \]

      2. associate-*r* N/A

         \[\leadsto 1 \cdot e^{re} + \left(\frac{-1}{2} \cdot {im}^{2}\right) \cdot \color{blue}{e^{re}} \]

      3. distribute-rgt-in N/A

         \[\leadsto e^{re} \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(e^{re}\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \left(\color{blue}{1} + \frac{-1}{2} \cdot {im}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2}\right)}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \left(\frac{-1}{2} \cdot \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \left(\left(\frac{-1}{2} \cdot im\right) \cdot \color{blue}{im}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(\frac{-1}{2} \cdot im\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{2} \cdot im\right)}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 75.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

   5. Simplified 75.0%

      \[\leadsto \color{blue}{e^{re} \cdot \left(1 + im \cdot \left(im \cdot -0.5\right)\right)} \]

   if 1.8999999999999999e154 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + \frac{1}{2} \cdot re\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot re\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \frac{1}{2}\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      5. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)} \cdot \cos im \]
3. Recombined 4 regimes into one program.

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.018:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 1.3 \cdot 10^{-8}:\\ \;\;\;\;\cos im \cdot \left(re \cdot \left(re \cdot 0.5\right) + \left(re + 1\right)\right)\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;e^{re} \cdot \left(1 + im \cdot \left(im \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{if}\;re \leq -0.0042:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 1.3 \cdot 10^{-8}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 10^{+150}:\\ \;\;\;\;e^{re} \cdot \left(1 + im \cdot \left(im \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (cos im) (+ 1.0 (* re (+ 1.0 (* re 0.5)))))))
   (if (<= re -0.0042)
     (exp re)
     (if (<= re 1.3e-8)
       t_0
       (if (<= re 1e+150) (* (exp re) (+ 1.0 (* im (* im -0.5)))) t_0)))))

C

double code(double re, double im) {
	double t_0 = cos(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	double tmp;
	if (re <= -0.0042) {
		tmp = exp(re);
	} else if (re <= 1.3e-8) {
		tmp = t_0;
	} else if (re <= 1e+150) {
		tmp = exp(re) * (1.0 + (im * (im * -0.5)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(im) * (1.0d0 + (re * (1.0d0 + (re * 0.5d0))))
    if (re <= (-0.0042d0)) then
        tmp = exp(re)
    else if (re <= 1.3d-8) then
        tmp = t_0
    else if (re <= 1d+150) then
        tmp = exp(re) * (1.0d0 + (im * (im * (-0.5d0))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.cos(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	double tmp;
	if (re <= -0.0042) {
		tmp = Math.exp(re);
	} else if (re <= 1.3e-8) {
		tmp = t_0;
	} else if (re <= 1e+150) {
		tmp = Math.exp(re) * (1.0 + (im * (im * -0.5)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.cos(im) * (1.0 + (re * (1.0 + (re * 0.5))))
	tmp = 0
	if re <= -0.0042:
		tmp = math.exp(re)
	elif re <= 1.3e-8:
		tmp = t_0
	elif re <= 1e+150:
		tmp = math.exp(re) * (1.0 + (im * (im * -0.5)))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(cos(im) * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5)))))
	tmp = 0.0
	if (re <= -0.0042)
		tmp = exp(re);
	elseif (re <= 1.3e-8)
		tmp = t_0;
	elseif (re <= 1e+150)
		tmp = Float64(exp(re) * Float64(1.0 + Float64(im * Float64(im * -0.5))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = cos(im) * (1.0 + (re * (1.0 + (re * 0.5))));
	tmp = 0.0;
	if (re <= -0.0042)
		tmp = exp(re);
	elseif (re <= 1.3e-8)
		tmp = t_0;
	elseif (re <= 1e+150)
		tmp = exp(re) * (1.0 + (im * (im * -0.5)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Cos[im], $MachinePrecision] * N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -0.0042], N[Exp[re], $MachinePrecision], If[LessEqual[re, 1.3e-8], t$95$0, If[LessEqual[re, 1e+150], N[(N[Exp[re], $MachinePrecision] * N[(1.0 + N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.00419999999999999974

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. exp-lowering-exp.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(re\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{e^{re}} \]

   if -0.00419999999999999974 < re < 1.3000000000000001e-8 or 9.99999999999999981e149 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + \frac{1}{2} \cdot re\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot re\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \frac{1}{2}\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      5. *-lowering-*.f64 99.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

   5. Simplified 99.7%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)} \cdot \cos im \]

   if 1.3000000000000001e-8 < re < 9.99999999999999981e149

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re} + \frac{-1}{2} \cdot \left({im}^{2} \cdot e^{re}\right)} \]
   4. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto 1 \cdot e^{re} + \color{blue}{\frac{-1}{2}} \cdot \left({im}^{2} \cdot e^{re}\right) \]

      2. associate-*r* N/A

         \[\leadsto 1 \cdot e^{re} + \left(\frac{-1}{2} \cdot {im}^{2}\right) \cdot \color{blue}{e^{re}} \]

      3. distribute-rgt-in N/A

         \[\leadsto e^{re} \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(e^{re}\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \left(\color{blue}{1} + \frac{-1}{2} \cdot {im}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2}\right)}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \left(\frac{-1}{2} \cdot \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \left(\left(\frac{-1}{2} \cdot im\right) \cdot \color{blue}{im}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(\frac{-1}{2} \cdot im\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{2} \cdot im\right)}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 75.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

   5. Simplified 75.0%

      \[\leadsto \color{blue}{e^{re} \cdot \left(1 + im \cdot \left(im \cdot -0.5\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0042:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 1.3 \cdot 10^{-8}:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \mathbf{elif}\;re \leq 10^{+150}:\\ \;\;\;\;e^{re} \cdot \left(1 + im \cdot \left(im \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.0014:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 1.3 \cdot 10^{-8}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot \left(1 + im \cdot \left(im \cdot -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.0014)
   (exp re)
   (if (<= re 1.3e-8)
     (* (cos im) (+ re 1.0))
     (* (exp re) (+ 1.0 (* im (* im -0.5)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.0014) {
		tmp = exp(re);
	} else if (re <= 1.3e-8) {
		tmp = cos(im) * (re + 1.0);
	} else {
		tmp = exp(re) * (1.0 + (im * (im * -0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-0.0014d0)) then
        tmp = exp(re)
    else if (re <= 1.3d-8) then
        tmp = cos(im) * (re + 1.0d0)
    else
        tmp = exp(re) * (1.0d0 + (im * (im * (-0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.0014) {
		tmp = Math.exp(re);
	} else if (re <= 1.3e-8) {
		tmp = Math.cos(im) * (re + 1.0);
	} else {
		tmp = Math.exp(re) * (1.0 + (im * (im * -0.5)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.0014:
		tmp = math.exp(re)
	elif re <= 1.3e-8:
		tmp = math.cos(im) * (re + 1.0)
	else:
		tmp = math.exp(re) * (1.0 + (im * (im * -0.5)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.0014)
		tmp = exp(re);
	elseif (re <= 1.3e-8)
		tmp = Float64(cos(im) * Float64(re + 1.0));
	else
		tmp = Float64(exp(re) * Float64(1.0 + Float64(im * Float64(im * -0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.0014)
		tmp = exp(re);
	elseif (re <= 1.3e-8)
		tmp = cos(im) * (re + 1.0);
	else
		tmp = exp(re) * (1.0 + (im * (im * -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.0014], N[Exp[re], $MachinePrecision], If[LessEqual[re, 1.3e-8], N[(N[Cos[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * N[(1.0 + N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.00139999999999999999

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. exp-lowering-exp.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(re\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{e^{re}} \]

   if -0.00139999999999999999 < re < 1.3000000000000001e-8

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. +-lowering-+.f64 99.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

   5. Simplified 99.3%

      \[\leadsto \color{blue}{\left(re + 1\right)} \cdot \cos im \]

   if 1.3000000000000001e-8 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re} + \frac{-1}{2} \cdot \left({im}^{2} \cdot e^{re}\right)} \]
   4. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto 1 \cdot e^{re} + \color{blue}{\frac{-1}{2}} \cdot \left({im}^{2} \cdot e^{re}\right) \]

      2. associate-*r* N/A

         \[\leadsto 1 \cdot e^{re} + \left(\frac{-1}{2} \cdot {im}^{2}\right) \cdot \color{blue}{e^{re}} \]

      3. distribute-rgt-in N/A

         \[\leadsto e^{re} \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(e^{re}\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \left(\color{blue}{1} + \frac{-1}{2} \cdot {im}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2}\right)}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \left(\frac{-1}{2} \cdot \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \left(\left(\frac{-1}{2} \cdot im\right) \cdot \color{blue}{im}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(\frac{-1}{2} \cdot im\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{2} \cdot im\right)}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 77.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

   5. Simplified 77.9%

      \[\leadsto \color{blue}{e^{re} \cdot \left(1 + im \cdot \left(im \cdot -0.5\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0014:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 1.3 \cdot 10^{-8}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot \left(1 + im \cdot \left(im \cdot -0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.0014:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 1.3 \cdot 10^{-8}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 10^{+106}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right) \cdot \left(1 + im \cdot \left(im \cdot -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.0014)
   (exp re)
   (if (<= re 1.3e-8)
     (* (cos im) (+ re 1.0))
     (if (<= re 1e+106)
       (exp re)
       (*
        (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666))))))
        (+ 1.0 (* im (* im -0.5))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.0014) {
		tmp = exp(re);
	} else if (re <= 1.3e-8) {
		tmp = cos(im) * (re + 1.0);
	} else if (re <= 1e+106) {
		tmp = exp(re);
	} else {
		tmp = (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))))) * (1.0 + (im * (im * -0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-0.0014d0)) then
        tmp = exp(re)
    else if (re <= 1.3d-8) then
        tmp = cos(im) * (re + 1.0d0)
    else if (re <= 1d+106) then
        tmp = exp(re)
    else
        tmp = (1.0d0 + (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0)))))) * (1.0d0 + (im * (im * (-0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.0014) {
		tmp = Math.exp(re);
	} else if (re <= 1.3e-8) {
		tmp = Math.cos(im) * (re + 1.0);
	} else if (re <= 1e+106) {
		tmp = Math.exp(re);
	} else {
		tmp = (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))))) * (1.0 + (im * (im * -0.5)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.0014:
		tmp = math.exp(re)
	elif re <= 1.3e-8:
		tmp = math.cos(im) * (re + 1.0)
	elif re <= 1e+106:
		tmp = math.exp(re)
	else:
		tmp = (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))))) * (1.0 + (im * (im * -0.5)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.0014)
		tmp = exp(re);
	elseif (re <= 1.3e-8)
		tmp = Float64(cos(im) * Float64(re + 1.0));
	elseif (re <= 1e+106)
		tmp = exp(re);
	else
		tmp = Float64(Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666)))))) * Float64(1.0 + Float64(im * Float64(im * -0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.0014)
		tmp = exp(re);
	elseif (re <= 1.3e-8)
		tmp = cos(im) * (re + 1.0);
	elseif (re <= 1e+106)
		tmp = exp(re);
	else
		tmp = (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))))) * (1.0 + (im * (im * -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.0014], N[Exp[re], $MachinePrecision], If[LessEqual[re, 1.3e-8], N[(N[Cos[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1e+106], N[Exp[re], $MachinePrecision], N[(N[(1.0 + N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.00139999999999999999 or 1.3000000000000001e-8 < re < 1.00000000000000009e106

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. exp-lowering-exp.f64 92.8%

         \[\leadsto \mathsf{exp.f64}\left(re\right) \]

   5. Simplified 92.8%

      \[\leadsto \color{blue}{e^{re}} \]

   if -0.00139999999999999999 < re < 1.3000000000000001e-8

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. +-lowering-+.f64 99.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

   5. Simplified 99.3%

      \[\leadsto \color{blue}{\left(re + 1\right)} \cdot \cos im \]

   if 1.00000000000000009e106 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)} \cdot \cos im \]

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)}\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2}\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left({im}^{2} \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \frac{-1}{2}\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \frac{-1}{2}\right)}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(im \cdot \left(\frac{-1}{2} \cdot \color{blue}{im}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{2} \cdot im\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 84.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

   8. Simplified 84.1%

      \[\leadsto \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right) \cdot \color{blue}{\left(1 + im \cdot \left(im \cdot -0.5\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0014:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 1.3 \cdot 10^{-8}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 10^{+106}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right) \cdot \left(1 + im \cdot \left(im \cdot -0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.00145:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 5.4 \cdot 10^{-10}:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;re \leq 4.3 \cdot 10^{+105}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right) \cdot \left(1 + im \cdot \left(im \cdot -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.00145)
   (exp re)
   (if (<= re 5.4e-10)
     (cos im)
     (if (<= re 4.3e+105)
       (exp re)
       (*
        (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666))))))
        (+ 1.0 (* im (* im -0.5))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.00145) {
		tmp = exp(re);
	} else if (re <= 5.4e-10) {
		tmp = cos(im);
	} else if (re <= 4.3e+105) {
		tmp = exp(re);
	} else {
		tmp = (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))))) * (1.0 + (im * (im * -0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-0.00145d0)) then
        tmp = exp(re)
    else if (re <= 5.4d-10) then
        tmp = cos(im)
    else if (re <= 4.3d+105) then
        tmp = exp(re)
    else
        tmp = (1.0d0 + (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0)))))) * (1.0d0 + (im * (im * (-0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.00145) {
		tmp = Math.exp(re);
	} else if (re <= 5.4e-10) {
		tmp = Math.cos(im);
	} else if (re <= 4.3e+105) {
		tmp = Math.exp(re);
	} else {
		tmp = (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))))) * (1.0 + (im * (im * -0.5)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.00145:
		tmp = math.exp(re)
	elif re <= 5.4e-10:
		tmp = math.cos(im)
	elif re <= 4.3e+105:
		tmp = math.exp(re)
	else:
		tmp = (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))))) * (1.0 + (im * (im * -0.5)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.00145)
		tmp = exp(re);
	elseif (re <= 5.4e-10)
		tmp = cos(im);
	elseif (re <= 4.3e+105)
		tmp = exp(re);
	else
		tmp = Float64(Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666)))))) * Float64(1.0 + Float64(im * Float64(im * -0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.00145)
		tmp = exp(re);
	elseif (re <= 5.4e-10)
		tmp = cos(im);
	elseif (re <= 4.3e+105)
		tmp = exp(re);
	else
		tmp = (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))))) * (1.0 + (im * (im * -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.00145], N[Exp[re], $MachinePrecision], If[LessEqual[re, 5.4e-10], N[Cos[im], $MachinePrecision], If[LessEqual[re, 4.3e+105], N[Exp[re], $MachinePrecision], N[(N[(1.0 + N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.00145 or 5.4e-10 < re < 4.3000000000000002e105

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. exp-lowering-exp.f64 92.8%

         \[\leadsto \mathsf{exp.f64}\left(re\right) \]

   5. Simplified 92.8%

      \[\leadsto \color{blue}{e^{re}} \]

   if -0.00145 < re < 5.4e-10

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\cos im} \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 98.6%

         \[\leadsto \mathsf{cos.f64}\left(im\right) \]

   5. Simplified 98.6%

      \[\leadsto \color{blue}{\cos im} \]

   if 4.3000000000000002e105 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)} \cdot \cos im \]

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)}\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2}\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left({im}^{2} \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \frac{-1}{2}\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \frac{-1}{2}\right)}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(im \cdot \left(\frac{-1}{2} \cdot \color{blue}{im}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{2} \cdot im\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 84.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

   8. Simplified 84.1%

      \[\leadsto \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right) \cdot \color{blue}{\left(1 + im \cdot \left(im \cdot -0.5\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 83.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -9.2 \cdot 10^{+102}:\\ \;\;\;\;\frac{1}{1 + re \cdot \left(-1 + re \cdot \left(0.5 + re \cdot -0.16666666666666666\right)\right)}\\ \mathbf{elif}\;re \leq -560:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot 0.041666666666666664\\ \mathbf{elif}\;re \leq 1.3 \cdot 10^{-8}:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right) \cdot \left(1 + im \cdot \left(im \cdot -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -9.2e+102)
   (/ 1.0 (+ 1.0 (* re (+ -1.0 (* re (+ 0.5 (* re -0.16666666666666666)))))))
   (if (<= re -560.0)
     (* (* (* im im) (* im im)) 0.041666666666666664)
     (if (<= re 1.3e-8)
       (cos im)
       (*
        (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666))))))
        (+ 1.0 (* im (* im -0.5))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -9.2e+102) {
		tmp = 1.0 / (1.0 + (re * (-1.0 + (re * (0.5 + (re * -0.16666666666666666))))));
	} else if (re <= -560.0) {
		tmp = ((im * im) * (im * im)) * 0.041666666666666664;
	} else if (re <= 1.3e-8) {
		tmp = cos(im);
	} else {
		tmp = (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))))) * (1.0 + (im * (im * -0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-9.2d+102)) then
        tmp = 1.0d0 / (1.0d0 + (re * ((-1.0d0) + (re * (0.5d0 + (re * (-0.16666666666666666d0)))))))
    else if (re <= (-560.0d0)) then
        tmp = ((im * im) * (im * im)) * 0.041666666666666664d0
    else if (re <= 1.3d-8) then
        tmp = cos(im)
    else
        tmp = (1.0d0 + (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0)))))) * (1.0d0 + (im * (im * (-0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -9.2e+102) {
		tmp = 1.0 / (1.0 + (re * (-1.0 + (re * (0.5 + (re * -0.16666666666666666))))));
	} else if (re <= -560.0) {
		tmp = ((im * im) * (im * im)) * 0.041666666666666664;
	} else if (re <= 1.3e-8) {
		tmp = Math.cos(im);
	} else {
		tmp = (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))))) * (1.0 + (im * (im * -0.5)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -9.2e+102:
		tmp = 1.0 / (1.0 + (re * (-1.0 + (re * (0.5 + (re * -0.16666666666666666))))))
	elif re <= -560.0:
		tmp = ((im * im) * (im * im)) * 0.041666666666666664
	elif re <= 1.3e-8:
		tmp = math.cos(im)
	else:
		tmp = (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))))) * (1.0 + (im * (im * -0.5)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -9.2e+102)
		tmp = Float64(1.0 / Float64(1.0 + Float64(re * Float64(-1.0 + Float64(re * Float64(0.5 + Float64(re * -0.16666666666666666)))))));
	elseif (re <= -560.0)
		tmp = Float64(Float64(Float64(im * im) * Float64(im * im)) * 0.041666666666666664);
	elseif (re <= 1.3e-8)
		tmp = cos(im);
	else
		tmp = Float64(Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666)))))) * Float64(1.0 + Float64(im * Float64(im * -0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -9.2e+102)
		tmp = 1.0 / (1.0 + (re * (-1.0 + (re * (0.5 + (re * -0.16666666666666666))))));
	elseif (re <= -560.0)
		tmp = ((im * im) * (im * im)) * 0.041666666666666664;
	elseif (re <= 1.3e-8)
		tmp = cos(im);
	else
		tmp = (1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))))) * (1.0 + (im * (im * -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -9.2e+102], N[(1.0 / N[(1.0 + N[(re * N[(-1.0 + N[(re * N[(0.5 + N[(re * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -560.0], N[(N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] * 0.041666666666666664), $MachinePrecision], If[LessEqual[re, 1.3e-8], N[Cos[im], $MachinePrecision], N[(N[(1.0 + N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -9.1999999999999995e102

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      7. *-lowering-*.f64 1.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

   5. Simplified 1.6%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)} \cdot \cos im \]
   6. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1 \cdot 1 - \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right) \cdot \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right)}{1 - re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)}\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\frac{1 - re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)}{1 \cdot 1 - \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right) \cdot \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right)}}\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{1 - re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)}{1 \cdot 1 - \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right) \cdot \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right)}\right)\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{1}{\frac{1 \cdot 1 - \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right) \cdot \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right)}{1 - re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)}}\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      5. flip-+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{1}{1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)}\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      12. *-lowering-*.f64 1.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

   7. Applied egg-rr 1.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)}}} \cdot \cos im \]

   8. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\left(1 + re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right) - 1\right)\right)}\right), \mathsf{cos.f64}\left(im\right)\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right) - 1\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right) - 1\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right) + -1\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(-1 + re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(-1, \left(re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{-1}{6} \cdot re\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      10. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

   10. Simplified 100.0%

       \[\leadsto \frac{1}{\color{blue}{1 + re \cdot \left(-1 + re \cdot \left(0.5 + re \cdot -0.16666666666666666\right)\right)}} \cdot \cos im \]

   11. Taylor expanded in im around 0

       \[\leadsto \color{blue}{\frac{1}{1 + re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right) - 1\right)}} \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right) - 1\right)\right)}\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right) - 1\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right) - 1\right)}\right)\right)\right) \]

       4. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right)\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right) + -1\right)\right)\right)\right) \]

       6. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(-1 + \color{blue}{re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(-1, \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right)\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{1}{2} + \frac{-1}{6} \cdot re\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{-1}{6} \cdot re\right)}\right)\right)\right)\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

       11. *-lowering-*.f64 100.0%

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

   13. Simplified 100.0%

       \[\leadsto \color{blue}{\frac{1}{1 + re \cdot \left(-1 + re \cdot \left(0.5 + re \cdot -0.16666666666666666\right)\right)}} \]

   if -9.1999999999999995e102 < re < -560

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. +-lowering-+.f64 2.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

   5. Simplified 2.8%

      \[\leadsto \color{blue}{\left(re + 1\right)} \cdot \cos im \]

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} + \frac{-1}{2}\right)\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {im}^{2}}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(\left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(im \cdot \color{blue}{\left(im \cdot \frac{1}{24}\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \frac{1}{24}\right)}\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 2.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 2.3%

      \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(1 + \left(im \cdot im\right) \cdot \left(-0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \left({im}^{4} \cdot \left(1 + re\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{24} \cdot {im}^{4}\right) \cdot \color{blue}{\left(1 + re\right)} \]

       2. *-commutative N/A

          \[\leadsto \left({im}^{4} \cdot \frac{1}{24}\right) \cdot \left(\color{blue}{1} + re\right) \]

       3. associate-*r* N/A

          \[\leadsto {im}^{4} \cdot \color{blue}{\left(\frac{1}{24} \cdot \left(1 + re\right)\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{4}\right), \color{blue}{\left(\frac{1}{24} \cdot \left(1 + re\right)\right)}\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{\left(2 \cdot 2\right)}\right), \left(\frac{1}{24} \cdot \left(1 + re\right)\right)\right) \]

       6. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{2} \cdot {im}^{2}\right), \left(\color{blue}{\frac{1}{24}} \cdot \left(1 + re\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({im}^{2}\right), \left({im}^{2}\right)\right), \left(\color{blue}{\frac{1}{24}} \cdot \left(1 + re\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(im \cdot im\right), \left({im}^{2}\right)\right), \left(\frac{1}{24} \cdot \left(1 + re\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2}\right)\right), \left(\frac{1}{24} \cdot \left(1 + re\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(im \cdot im\right)\right), \left(\frac{1}{24} \cdot \left(1 + re\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \left(\frac{1}{24} \cdot \left(1 + re\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\left(1 + re\right)}\right)\right) \]

       13. +-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{*.f64}\left(\frac{1}{24}, \left(re + \color{blue}{1}\right)\right)\right) \]

       14. +-lowering-+.f64 53.8%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{+.f64}\left(re, \color{blue}{1}\right)\right)\right) \]

   11. Simplified 53.8%

       \[\leadsto \color{blue}{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(0.041666666666666664 \cdot \left(re + 1\right)\right)} \]

   12. Taylor expanded in re around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \color{blue}{\frac{1}{24}}\right) \]
   13. Step-by-step derivation

   14. Simplified 53.9%

       \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{0.041666666666666664} \]

   if -560 < re < 1.3000000000000001e-8

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\cos im} \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 98.6%

         \[\leadsto \mathsf{cos.f64}\left(im\right) \]

   5. Simplified 98.6%

      \[\leadsto \color{blue}{\cos im} \]

   if 1.3000000000000001e-8 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      7. *-lowering-*.f64 68.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

   5. Simplified 68.8%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)} \cdot \cos im \]

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)}\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2}\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left({im}^{2} \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \frac{-1}{2}\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \frac{-1}{2}\right)}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(im \cdot \left(\frac{-1}{2} \cdot \color{blue}{im}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{2} \cdot im\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 63.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

   8. Simplified 63.4%

      \[\leadsto \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right) \cdot \color{blue}{\left(1 + im \cdot \left(im \cdot -0.5\right)\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 10: 60.8% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \mathbf{if}\;re \leq -1 \cdot 10^{+103}:\\ \;\;\;\;\frac{1}{1 + re \cdot \left(-1 + re \cdot \left(0.5 + re \cdot -0.16666666666666666\right)\right)}\\ \mathbf{elif}\;re \leq -0.00135:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(re \cdot 0.041666666666666664\right)\\ \mathbf{elif}\;re \leq 4 \cdot 10^{+19}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(1 + im \cdot \left(im \cdot -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666))))))))
   (if (<= re -1e+103)
     (/ 1.0 (+ 1.0 (* re (+ -1.0 (* re (+ 0.5 (* re -0.16666666666666666)))))))
     (if (<= re -0.00135)
       (* (* (* im im) (* im im)) (* re 0.041666666666666664))
       (if (<= re 4e+19) t_0 (* t_0 (+ 1.0 (* im (* im -0.5)))))))))

C

double code(double re, double im) {
	double t_0 = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	double tmp;
	if (re <= -1e+103) {
		tmp = 1.0 / (1.0 + (re * (-1.0 + (re * (0.5 + (re * -0.16666666666666666))))));
	} else if (re <= -0.00135) {
		tmp = ((im * im) * (im * im)) * (re * 0.041666666666666664);
	} else if (re <= 4e+19) {
		tmp = t_0;
	} else {
		tmp = t_0 * (1.0 + (im * (im * -0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0)))))
    if (re <= (-1d+103)) then
        tmp = 1.0d0 / (1.0d0 + (re * ((-1.0d0) + (re * (0.5d0 + (re * (-0.16666666666666666d0)))))))
    else if (re <= (-0.00135d0)) then
        tmp = ((im * im) * (im * im)) * (re * 0.041666666666666664d0)
    else if (re <= 4d+19) then
        tmp = t_0
    else
        tmp = t_0 * (1.0d0 + (im * (im * (-0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	double tmp;
	if (re <= -1e+103) {
		tmp = 1.0 / (1.0 + (re * (-1.0 + (re * (0.5 + (re * -0.16666666666666666))))));
	} else if (re <= -0.00135) {
		tmp = ((im * im) * (im * im)) * (re * 0.041666666666666664);
	} else if (re <= 4e+19) {
		tmp = t_0;
	} else {
		tmp = t_0 * (1.0 + (im * (im * -0.5)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))))
	tmp = 0
	if re <= -1e+103:
		tmp = 1.0 / (1.0 + (re * (-1.0 + (re * (0.5 + (re * -0.16666666666666666))))))
	elif re <= -0.00135:
		tmp = ((im * im) * (im * im)) * (re * 0.041666666666666664)
	elif re <= 4e+19:
		tmp = t_0
	else:
		tmp = t_0 * (1.0 + (im * (im * -0.5)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666))))))
	tmp = 0.0
	if (re <= -1e+103)
		tmp = Float64(1.0 / Float64(1.0 + Float64(re * Float64(-1.0 + Float64(re * Float64(0.5 + Float64(re * -0.16666666666666666)))))));
	elseif (re <= -0.00135)
		tmp = Float64(Float64(Float64(im * im) * Float64(im * im)) * Float64(re * 0.041666666666666664));
	elseif (re <= 4e+19)
		tmp = t_0;
	else
		tmp = Float64(t_0 * Float64(1.0 + Float64(im * Float64(im * -0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	tmp = 0.0;
	if (re <= -1e+103)
		tmp = 1.0 / (1.0 + (re * (-1.0 + (re * (0.5 + (re * -0.16666666666666666))))));
	elseif (re <= -0.00135)
		tmp = ((im * im) * (im * im)) * (re * 0.041666666666666664);
	elseif (re <= 4e+19)
		tmp = t_0;
	else
		tmp = t_0 * (1.0 + (im * (im * -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(1.0 + N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -1e+103], N[(1.0 / N[(1.0 + N[(re * N[(-1.0 + N[(re * N[(0.5 + N[(re * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -0.00135], N[(N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(re * 0.041666666666666664), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 4e+19], t$95$0, N[(t$95$0 * N[(1.0 + N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -1e103

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      7. *-lowering-*.f64 1.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

   5. Simplified 1.6%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)} \cdot \cos im \]
   6. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1 \cdot 1 - \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right) \cdot \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right)}{1 - re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)}\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\frac{1 - re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)}{1 \cdot 1 - \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right) \cdot \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right)}}\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{1 - re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)}{1 \cdot 1 - \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right) \cdot \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right)}\right)\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{1}{\frac{1 \cdot 1 - \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right) \cdot \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right)}{1 - re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)}}\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      5. flip-+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{1}{1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)}\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      12. *-lowering-*.f64 1.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

   7. Applied egg-rr 1.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)}}} \cdot \cos im \]

   8. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\left(1 + re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right) - 1\right)\right)}\right), \mathsf{cos.f64}\left(im\right)\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right) - 1\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right) - 1\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right) + -1\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(-1 + re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(-1, \left(re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{-1}{6} \cdot re\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      10. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

   10. Simplified 100.0%

       \[\leadsto \frac{1}{\color{blue}{1 + re \cdot \left(-1 + re \cdot \left(0.5 + re \cdot -0.16666666666666666\right)\right)}} \cdot \cos im \]

   11. Taylor expanded in im around 0

       \[\leadsto \color{blue}{\frac{1}{1 + re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right) - 1\right)}} \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right) - 1\right)\right)}\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right) - 1\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right) - 1\right)}\right)\right)\right) \]

       4. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right)\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right) + -1\right)\right)\right)\right) \]

       6. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(-1 + \color{blue}{re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(-1, \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right)\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{1}{2} + \frac{-1}{6} \cdot re\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{-1}{6} \cdot re\right)}\right)\right)\right)\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

       11. *-lowering-*.f64 100.0%

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

   13. Simplified 100.0%

       \[\leadsto \color{blue}{\frac{1}{1 + re \cdot \left(-1 + re \cdot \left(0.5 + re \cdot -0.16666666666666666\right)\right)}} \]

   if -1e103 < re < -0.0013500000000000001

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. +-lowering-+.f64 5.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

   5. Simplified 5.3%

      \[\leadsto \color{blue}{\left(re + 1\right)} \cdot \cos im \]

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} + \frac{-1}{2}\right)\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {im}^{2}}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(\left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(im \cdot \color{blue}{\left(im \cdot \frac{1}{24}\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \frac{1}{24}\right)}\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 2.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 2.2%

      \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(1 + \left(im \cdot im\right) \cdot \left(-0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \left({im}^{4} \cdot \left(1 + re\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{24} \cdot {im}^{4}\right) \cdot \color{blue}{\left(1 + re\right)} \]

       2. *-commutative N/A

          \[\leadsto \left({im}^{4} \cdot \frac{1}{24}\right) \cdot \left(\color{blue}{1} + re\right) \]

       3. associate-*r* N/A

          \[\leadsto {im}^{4} \cdot \color{blue}{\left(\frac{1}{24} \cdot \left(1 + re\right)\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{4}\right), \color{blue}{\left(\frac{1}{24} \cdot \left(1 + re\right)\right)}\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{\left(2 \cdot 2\right)}\right), \left(\frac{1}{24} \cdot \left(1 + re\right)\right)\right) \]

       6. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{2} \cdot {im}^{2}\right), \left(\color{blue}{\frac{1}{24}} \cdot \left(1 + re\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({im}^{2}\right), \left({im}^{2}\right)\right), \left(\color{blue}{\frac{1}{24}} \cdot \left(1 + re\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(im \cdot im\right), \left({im}^{2}\right)\right), \left(\frac{1}{24} \cdot \left(1 + re\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2}\right)\right), \left(\frac{1}{24} \cdot \left(1 + re\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(im \cdot im\right)\right), \left(\frac{1}{24} \cdot \left(1 + re\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \left(\frac{1}{24} \cdot \left(1 + re\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\left(1 + re\right)}\right)\right) \]

       13. +-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{*.f64}\left(\frac{1}{24}, \left(re + \color{blue}{1}\right)\right)\right) \]

       14. +-lowering-+.f64 50.9%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{+.f64}\left(re, \color{blue}{1}\right)\right)\right) \]

   11. Simplified 50.9%

       \[\leadsto \color{blue}{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(0.041666666666666664 \cdot \left(re + 1\right)\right)} \]

   12. Taylor expanded in re around inf

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \color{blue}{\left(\frac{1}{24} \cdot re\right)}\right) \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \left(re \cdot \color{blue}{\frac{1}{24}}\right)\right) \]

       2. *-lowering-*.f64 51.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{*.f64}\left(re, \color{blue}{\frac{1}{24}}\right)\right) \]

   14. Simplified 51.0%

       \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left(re \cdot 0.041666666666666664\right)} \]

   if -0.0013500000000000001 < re < 4e19

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      7. *-lowering-*.f64 96.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

   5. Simplified 96.1%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)} \cdot \cos im \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot re\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{6} \cdot re\right)}\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 51.6%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 51.6%

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)} \]

   if 4e19 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      7. *-lowering-*.f64 73.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

   5. Simplified 73.9%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)} \cdot \cos im \]

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)}\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2}\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left({im}^{2} \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \frac{-1}{2}\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \frac{-1}{2}\right)}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(im \cdot \left(\frac{-1}{2} \cdot \color{blue}{im}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{2} \cdot im\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 68.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

   8. Simplified 68.1%

      \[\leadsto \left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right) \cdot \color{blue}{\left(1 + im \cdot \left(im \cdot -0.5\right)\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 11: 58.6% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1 \cdot 10^{+103}:\\ \;\;\;\;\frac{1}{1 + re \cdot \left(-1 + re \cdot \left(0.5 + re \cdot -0.16666666666666666\right)\right)}\\ \mathbf{elif}\;re \leq -0.00135:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(re \cdot 0.041666666666666664\right)\\ \mathbf{elif}\;re \leq 3 \cdot 10^{+23}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + im \cdot \left(im \cdot -0.5\right)\right) \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1e+103)
   (/ 1.0 (+ 1.0 (* re (+ -1.0 (* re (+ 0.5 (* re -0.16666666666666666)))))))
   (if (<= re -0.00135)
     (* (* (* im im) (* im im)) (* re 0.041666666666666664))
     (if (<= re 3e+23)
       (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666))))))
       (* (+ 1.0 (* im (* im -0.5))) (+ 1.0 (* re (+ 1.0 (* re 0.5)))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1e+103) {
		tmp = 1.0 / (1.0 + (re * (-1.0 + (re * (0.5 + (re * -0.16666666666666666))))));
	} else if (re <= -0.00135) {
		tmp = ((im * im) * (im * im)) * (re * 0.041666666666666664);
	} else if (re <= 3e+23) {
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	} else {
		tmp = (1.0 + (im * (im * -0.5))) * (1.0 + (re * (1.0 + (re * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1d+103)) then
        tmp = 1.0d0 / (1.0d0 + (re * ((-1.0d0) + (re * (0.5d0 + (re * (-0.16666666666666666d0)))))))
    else if (re <= (-0.00135d0)) then
        tmp = ((im * im) * (im * im)) * (re * 0.041666666666666664d0)
    else if (re <= 3d+23) then
        tmp = 1.0d0 + (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0)))))
    else
        tmp = (1.0d0 + (im * (im * (-0.5d0)))) * (1.0d0 + (re * (1.0d0 + (re * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1e+103) {
		tmp = 1.0 / (1.0 + (re * (-1.0 + (re * (0.5 + (re * -0.16666666666666666))))));
	} else if (re <= -0.00135) {
		tmp = ((im * im) * (im * im)) * (re * 0.041666666666666664);
	} else if (re <= 3e+23) {
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	} else {
		tmp = (1.0 + (im * (im * -0.5))) * (1.0 + (re * (1.0 + (re * 0.5))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1e+103:
		tmp = 1.0 / (1.0 + (re * (-1.0 + (re * (0.5 + (re * -0.16666666666666666))))))
	elif re <= -0.00135:
		tmp = ((im * im) * (im * im)) * (re * 0.041666666666666664)
	elif re <= 3e+23:
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))))
	else:
		tmp = (1.0 + (im * (im * -0.5))) * (1.0 + (re * (1.0 + (re * 0.5))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1e+103)
		tmp = Float64(1.0 / Float64(1.0 + Float64(re * Float64(-1.0 + Float64(re * Float64(0.5 + Float64(re * -0.16666666666666666)))))));
	elseif (re <= -0.00135)
		tmp = Float64(Float64(Float64(im * im) * Float64(im * im)) * Float64(re * 0.041666666666666664));
	elseif (re <= 3e+23)
		tmp = Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666))))));
	else
		tmp = Float64(Float64(1.0 + Float64(im * Float64(im * -0.5))) * Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1e+103)
		tmp = 1.0 / (1.0 + (re * (-1.0 + (re * (0.5 + (re * -0.16666666666666666))))));
	elseif (re <= -0.00135)
		tmp = ((im * im) * (im * im)) * (re * 0.041666666666666664);
	elseif (re <= 3e+23)
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	else
		tmp = (1.0 + (im * (im * -0.5))) * (1.0 + (re * (1.0 + (re * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1e+103], N[(1.0 / N[(1.0 + N[(re * N[(-1.0 + N[(re * N[(0.5 + N[(re * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -0.00135], N[(N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(re * 0.041666666666666664), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 3e+23], N[(1.0 + N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -1e103

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      7. *-lowering-*.f64 1.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

   5. Simplified 1.6%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)} \cdot \cos im \]
   6. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1 \cdot 1 - \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right) \cdot \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right)}{1 - re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)}\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\frac{1 - re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)}{1 \cdot 1 - \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right) \cdot \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right)}}\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{1 - re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)}{1 \cdot 1 - \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right) \cdot \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right)}\right)\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{1}{\frac{1 \cdot 1 - \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right) \cdot \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right)}{1 - re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)}}\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      5. flip-+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{1}{1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)}\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      12. *-lowering-*.f64 1.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

   7. Applied egg-rr 1.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)}}} \cdot \cos im \]

   8. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\left(1 + re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right) - 1\right)\right)}\right), \mathsf{cos.f64}\left(im\right)\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right) - 1\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right) - 1\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right) + -1\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(-1 + re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(-1, \left(re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{-1}{6} \cdot re\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      10. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

   10. Simplified 100.0%

       \[\leadsto \frac{1}{\color{blue}{1 + re \cdot \left(-1 + re \cdot \left(0.5 + re \cdot -0.16666666666666666\right)\right)}} \cdot \cos im \]

   11. Taylor expanded in im around 0

       \[\leadsto \color{blue}{\frac{1}{1 + re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right) - 1\right)}} \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right) - 1\right)\right)}\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right) - 1\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right) - 1\right)}\right)\right)\right) \]

       4. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right)\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right) + -1\right)\right)\right)\right) \]

       6. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(-1 + \color{blue}{re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(-1, \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right)\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{1}{2} + \frac{-1}{6} \cdot re\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{-1}{6} \cdot re\right)}\right)\right)\right)\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

       11. *-lowering-*.f64 100.0%

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

   13. Simplified 100.0%

       \[\leadsto \color{blue}{\frac{1}{1 + re \cdot \left(-1 + re \cdot \left(0.5 + re \cdot -0.16666666666666666\right)\right)}} \]

   if -1e103 < re < -0.0013500000000000001

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. +-lowering-+.f64 5.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

   5. Simplified 5.3%

      \[\leadsto \color{blue}{\left(re + 1\right)} \cdot \cos im \]

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} + \frac{-1}{2}\right)\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {im}^{2}}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(\left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(im \cdot \color{blue}{\left(im \cdot \frac{1}{24}\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \frac{1}{24}\right)}\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 2.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 2.2%

      \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(1 + \left(im \cdot im\right) \cdot \left(-0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \left({im}^{4} \cdot \left(1 + re\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{24} \cdot {im}^{4}\right) \cdot \color{blue}{\left(1 + re\right)} \]

       2. *-commutative N/A

          \[\leadsto \left({im}^{4} \cdot \frac{1}{24}\right) \cdot \left(\color{blue}{1} + re\right) \]

       3. associate-*r* N/A

          \[\leadsto {im}^{4} \cdot \color{blue}{\left(\frac{1}{24} \cdot \left(1 + re\right)\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{4}\right), \color{blue}{\left(\frac{1}{24} \cdot \left(1 + re\right)\right)}\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{\left(2 \cdot 2\right)}\right), \left(\frac{1}{24} \cdot \left(1 + re\right)\right)\right) \]

       6. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{2} \cdot {im}^{2}\right), \left(\color{blue}{\frac{1}{24}} \cdot \left(1 + re\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({im}^{2}\right), \left({im}^{2}\right)\right), \left(\color{blue}{\frac{1}{24}} \cdot \left(1 + re\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(im \cdot im\right), \left({im}^{2}\right)\right), \left(\frac{1}{24} \cdot \left(1 + re\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2}\right)\right), \left(\frac{1}{24} \cdot \left(1 + re\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(im \cdot im\right)\right), \left(\frac{1}{24} \cdot \left(1 + re\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \left(\frac{1}{24} \cdot \left(1 + re\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\left(1 + re\right)}\right)\right) \]

       13. +-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{*.f64}\left(\frac{1}{24}, \left(re + \color{blue}{1}\right)\right)\right) \]

       14. +-lowering-+.f64 50.9%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{+.f64}\left(re, \color{blue}{1}\right)\right)\right) \]

   11. Simplified 50.9%

       \[\leadsto \color{blue}{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(0.041666666666666664 \cdot \left(re + 1\right)\right)} \]

   12. Taylor expanded in re around inf

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \color{blue}{\left(\frac{1}{24} \cdot re\right)}\right) \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \left(re \cdot \color{blue}{\frac{1}{24}}\right)\right) \]

       2. *-lowering-*.f64 51.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{*.f64}\left(re, \color{blue}{\frac{1}{24}}\right)\right) \]

   14. Simplified 51.0%

       \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left(re \cdot 0.041666666666666664\right)} \]

   if -0.0013500000000000001 < re < 3.0000000000000001e23

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      7. *-lowering-*.f64 95.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

   5. Simplified 95.5%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)} \cdot \cos im \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot re\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{6} \cdot re\right)}\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 51.2%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 51.2%

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)} \]

   if 3.0000000000000001e23 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re} + \frac{-1}{2} \cdot \left({im}^{2} \cdot e^{re}\right)} \]
   4. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto 1 \cdot e^{re} + \color{blue}{\frac{-1}{2}} \cdot \left({im}^{2} \cdot e^{re}\right) \]

      2. associate-*r* N/A

         \[\leadsto 1 \cdot e^{re} + \left(\frac{-1}{2} \cdot {im}^{2}\right) \cdot \color{blue}{e^{re}} \]

      3. distribute-rgt-in N/A

         \[\leadsto e^{re} \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(e^{re}\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \left(\color{blue}{1} + \frac{-1}{2} \cdot {im}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2}\right)}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \left(\frac{-1}{2} \cdot \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \left(\left(\frac{-1}{2} \cdot im\right) \cdot \color{blue}{im}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(\frac{-1}{2} \cdot im\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{2} \cdot im\right)}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 80.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

   5. Simplified 80.0%

      \[\leadsto \color{blue}{e^{re} \cdot \left(1 + im \cdot \left(im \cdot -0.5\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + \frac{1}{2} \cdot re\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot re\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \frac{1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 58.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \frac{1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]

   8. Simplified 58.1%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)} \cdot \left(1 + im \cdot \left(im \cdot -0.5\right)\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1 \cdot 10^{+103}:\\ \;\;\;\;\frac{1}{1 + re \cdot \left(-1 + re \cdot \left(0.5 + re \cdot -0.16666666666666666\right)\right)}\\ \mathbf{elif}\;re \leq -0.00135:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(re \cdot 0.041666666666666664\right)\\ \mathbf{elif}\;re \leq 3 \cdot 10^{+23}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + im \cdot \left(im \cdot -0.5\right)\right) \cdot \left(1 + re \cdot \left(1 + re \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 59.7% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1 \cdot 10^{+103}:\\ \;\;\;\;\frac{1}{1 + re \cdot \left(-1 + re \cdot \left(0.5 + re \cdot -0.16666666666666666\right)\right)}\\ \mathbf{elif}\;re \leq -0.00135:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(re \cdot 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1e+103)
   (/ 1.0 (+ 1.0 (* re (+ -1.0 (* re (+ 0.5 (* re -0.16666666666666666)))))))
   (if (<= re -0.00135)
     (* (* (* im im) (* im im)) (* re 0.041666666666666664))
     (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666)))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1e+103) {
		tmp = 1.0 / (1.0 + (re * (-1.0 + (re * (0.5 + (re * -0.16666666666666666))))));
	} else if (re <= -0.00135) {
		tmp = ((im * im) * (im * im)) * (re * 0.041666666666666664);
	} else {
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1d+103)) then
        tmp = 1.0d0 / (1.0d0 + (re * ((-1.0d0) + (re * (0.5d0 + (re * (-0.16666666666666666d0)))))))
    else if (re <= (-0.00135d0)) then
        tmp = ((im * im) * (im * im)) * (re * 0.041666666666666664d0)
    else
        tmp = 1.0d0 + (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0)))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1e+103) {
		tmp = 1.0 / (1.0 + (re * (-1.0 + (re * (0.5 + (re * -0.16666666666666666))))));
	} else if (re <= -0.00135) {
		tmp = ((im * im) * (im * im)) * (re * 0.041666666666666664);
	} else {
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1e+103:
		tmp = 1.0 / (1.0 + (re * (-1.0 + (re * (0.5 + (re * -0.16666666666666666))))))
	elif re <= -0.00135:
		tmp = ((im * im) * (im * im)) * (re * 0.041666666666666664)
	else:
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1e+103)
		tmp = Float64(1.0 / Float64(1.0 + Float64(re * Float64(-1.0 + Float64(re * Float64(0.5 + Float64(re * -0.16666666666666666)))))));
	elseif (re <= -0.00135)
		tmp = Float64(Float64(Float64(im * im) * Float64(im * im)) * Float64(re * 0.041666666666666664));
	else
		tmp = Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1e+103)
		tmp = 1.0 / (1.0 + (re * (-1.0 + (re * (0.5 + (re * -0.16666666666666666))))));
	elseif (re <= -0.00135)
		tmp = ((im * im) * (im * im)) * (re * 0.041666666666666664);
	else
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1e+103], N[(1.0 / N[(1.0 + N[(re * N[(-1.0 + N[(re * N[(0.5 + N[(re * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -0.00135], N[(N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(re * 0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1e103

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      7. *-lowering-*.f64 1.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

   5. Simplified 1.6%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)} \cdot \cos im \]
   6. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1 \cdot 1 - \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right) \cdot \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right)}{1 - re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)}\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\frac{1 - re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)}{1 \cdot 1 - \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right) \cdot \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right)}}\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{1 - re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)}{1 \cdot 1 - \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right) \cdot \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right)}\right)\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{1}{\frac{1 \cdot 1 - \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right) \cdot \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right)}{1 - re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)}}\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      5. flip-+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{1}{1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)}\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      12. *-lowering-*.f64 1.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

   7. Applied egg-rr 1.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)}}} \cdot \cos im \]

   8. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\left(1 + re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right) - 1\right)\right)}\right), \mathsf{cos.f64}\left(im\right)\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right) - 1\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right) - 1\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right) + -1\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(-1 + re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(-1, \left(re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{-1}{6} \cdot re\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      10. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

   10. Simplified 100.0%

       \[\leadsto \frac{1}{\color{blue}{1 + re \cdot \left(-1 + re \cdot \left(0.5 + re \cdot -0.16666666666666666\right)\right)}} \cdot \cos im \]

   11. Taylor expanded in im around 0

       \[\leadsto \color{blue}{\frac{1}{1 + re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right) - 1\right)}} \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right) - 1\right)\right)}\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right) - 1\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right) - 1\right)}\right)\right)\right) \]

       4. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right)\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right) + -1\right)\right)\right)\right) \]

       6. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(-1 + \color{blue}{re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(-1, \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right)\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{1}{2} + \frac{-1}{6} \cdot re\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{-1}{6} \cdot re\right)}\right)\right)\right)\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

       11. *-lowering-*.f64 100.0%

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

   13. Simplified 100.0%

       \[\leadsto \color{blue}{\frac{1}{1 + re \cdot \left(-1 + re \cdot \left(0.5 + re \cdot -0.16666666666666666\right)\right)}} \]

   if -1e103 < re < -0.0013500000000000001

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. +-lowering-+.f64 5.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

   5. Simplified 5.3%

      \[\leadsto \color{blue}{\left(re + 1\right)} \cdot \cos im \]

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} + \frac{-1}{2}\right)\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {im}^{2}}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(\left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(im \cdot \color{blue}{\left(im \cdot \frac{1}{24}\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \frac{1}{24}\right)}\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 2.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 2.2%

      \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(1 + \left(im \cdot im\right) \cdot \left(-0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \left({im}^{4} \cdot \left(1 + re\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{24} \cdot {im}^{4}\right) \cdot \color{blue}{\left(1 + re\right)} \]

       2. *-commutative N/A

          \[\leadsto \left({im}^{4} \cdot \frac{1}{24}\right) \cdot \left(\color{blue}{1} + re\right) \]

       3. associate-*r* N/A

          \[\leadsto {im}^{4} \cdot \color{blue}{\left(\frac{1}{24} \cdot \left(1 + re\right)\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{4}\right), \color{blue}{\left(\frac{1}{24} \cdot \left(1 + re\right)\right)}\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{\left(2 \cdot 2\right)}\right), \left(\frac{1}{24} \cdot \left(1 + re\right)\right)\right) \]

       6. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{2} \cdot {im}^{2}\right), \left(\color{blue}{\frac{1}{24}} \cdot \left(1 + re\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({im}^{2}\right), \left({im}^{2}\right)\right), \left(\color{blue}{\frac{1}{24}} \cdot \left(1 + re\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(im \cdot im\right), \left({im}^{2}\right)\right), \left(\frac{1}{24} \cdot \left(1 + re\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2}\right)\right), \left(\frac{1}{24} \cdot \left(1 + re\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(im \cdot im\right)\right), \left(\frac{1}{24} \cdot \left(1 + re\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \left(\frac{1}{24} \cdot \left(1 + re\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\left(1 + re\right)}\right)\right) \]

       13. +-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{*.f64}\left(\frac{1}{24}, \left(re + \color{blue}{1}\right)\right)\right) \]

       14. +-lowering-+.f64 50.9%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{+.f64}\left(re, \color{blue}{1}\right)\right)\right) \]

   11. Simplified 50.9%

       \[\leadsto \color{blue}{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(0.041666666666666664 \cdot \left(re + 1\right)\right)} \]

   12. Taylor expanded in re around inf

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \color{blue}{\left(\frac{1}{24} \cdot re\right)}\right) \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \left(re \cdot \color{blue}{\frac{1}{24}}\right)\right) \]

       2. *-lowering-*.f64 51.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{*.f64}\left(re, \color{blue}{\frac{1}{24}}\right)\right) \]

   14. Simplified 51.0%

       \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left(re \cdot 0.041666666666666664\right)} \]

   if -0.0013500000000000001 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      7. *-lowering-*.f64 89.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

   5. Simplified 89.3%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)} \cdot \cos im \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot re\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{6} \cdot re\right)}\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 51.2%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 51.2%

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 42.4% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.8:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot 0.041666666666666664\\ \mathbf{elif}\;re \leq 1.76 \cdot 10^{+19}:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + im \cdot \left(im \cdot -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4.8)
   (* (* (* im im) (* im im)) 0.041666666666666664)
   (if (<= re 1.76e+19) (+ re 1.0) (* re (+ 1.0 (* im (* im -0.5)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4.8) {
		tmp = ((im * im) * (im * im)) * 0.041666666666666664;
	} else if (re <= 1.76e+19) {
		tmp = re + 1.0;
	} else {
		tmp = re * (1.0 + (im * (im * -0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-4.8d0)) then
        tmp = ((im * im) * (im * im)) * 0.041666666666666664d0
    else if (re <= 1.76d+19) then
        tmp = re + 1.0d0
    else
        tmp = re * (1.0d0 + (im * (im * (-0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -4.8) {
		tmp = ((im * im) * (im * im)) * 0.041666666666666664;
	} else if (re <= 1.76e+19) {
		tmp = re + 1.0;
	} else {
		tmp = re * (1.0 + (im * (im * -0.5)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -4.8:
		tmp = ((im * im) * (im * im)) * 0.041666666666666664
	elif re <= 1.76e+19:
		tmp = re + 1.0
	else:
		tmp = re * (1.0 + (im * (im * -0.5)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4.8)
		tmp = Float64(Float64(Float64(im * im) * Float64(im * im)) * 0.041666666666666664);
	elseif (re <= 1.76e+19)
		tmp = Float64(re + 1.0);
	else
		tmp = Float64(re * Float64(1.0 + Float64(im * Float64(im * -0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -4.8)
		tmp = ((im * im) * (im * im)) * 0.041666666666666664;
	elseif (re <= 1.76e+19)
		tmp = re + 1.0;
	else
		tmp = re * (1.0 + (im * (im * -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4.8], N[(N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] * 0.041666666666666664), $MachinePrecision], If[LessEqual[re, 1.76e+19], N[(re + 1.0), $MachinePrecision], N[(re * N[(1.0 + N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -4.79999999999999982

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. +-lowering-+.f64 2.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

   5. Simplified 2.3%

      \[\leadsto \color{blue}{\left(re + 1\right)} \cdot \cos im \]

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} + \frac{-1}{2}\right)\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {im}^{2}}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(\left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(im \cdot \color{blue}{\left(im \cdot \frac{1}{24}\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \frac{1}{24}\right)}\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 2.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 2.0%

      \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(1 + \left(im \cdot im\right) \cdot \left(-0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \left({im}^{4} \cdot \left(1 + re\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{24} \cdot {im}^{4}\right) \cdot \color{blue}{\left(1 + re\right)} \]

       2. *-commutative N/A

          \[\leadsto \left({im}^{4} \cdot \frac{1}{24}\right) \cdot \left(\color{blue}{1} + re\right) \]

       3. associate-*r* N/A

          \[\leadsto {im}^{4} \cdot \color{blue}{\left(\frac{1}{24} \cdot \left(1 + re\right)\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{4}\right), \color{blue}{\left(\frac{1}{24} \cdot \left(1 + re\right)\right)}\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{\left(2 \cdot 2\right)}\right), \left(\frac{1}{24} \cdot \left(1 + re\right)\right)\right) \]

       6. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{2} \cdot {im}^{2}\right), \left(\color{blue}{\frac{1}{24}} \cdot \left(1 + re\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({im}^{2}\right), \left({im}^{2}\right)\right), \left(\color{blue}{\frac{1}{24}} \cdot \left(1 + re\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(im \cdot im\right), \left({im}^{2}\right)\right), \left(\frac{1}{24} \cdot \left(1 + re\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2}\right)\right), \left(\frac{1}{24} \cdot \left(1 + re\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(im \cdot im\right)\right), \left(\frac{1}{24} \cdot \left(1 + re\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \left(\frac{1}{24} \cdot \left(1 + re\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\left(1 + re\right)}\right)\right) \]

       13. +-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{*.f64}\left(\frac{1}{24}, \left(re + \color{blue}{1}\right)\right)\right) \]

       14. +-lowering-+.f64 42.2%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{+.f64}\left(re, \color{blue}{1}\right)\right)\right) \]

   11. Simplified 42.2%

       \[\leadsto \color{blue}{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(0.041666666666666664 \cdot \left(re + 1\right)\right)} \]

   12. Taylor expanded in re around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \color{blue}{\frac{1}{24}}\right) \]
   13. Step-by-step derivation

   14. Simplified 42.5%

       \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{0.041666666666666664} \]

   if -4.79999999999999982 < re < 1.76e19

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. exp-lowering-exp.f64 54.5%

         \[\leadsto \mathsf{exp.f64}\left(re\right) \]

   5. Simplified 54.5%

      \[\leadsto \color{blue}{e^{re}} \]

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 51.1%

         \[\leadsto \mathsf{+.f64}\left(re, \color{blue}{1}\right) \]

   8. Simplified 51.1%

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

   if 1.76e19 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. +-lowering-+.f64 5.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

   5. Simplified 5.9%

      \[\leadsto \color{blue}{\left(re + 1\right)} \cdot \cos im \]

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)}\right) \]
   7. Step-by-step derivation

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

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

      2. *-commutative N/A

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

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \frac{-1}{2}\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \frac{-1}{2}\right)}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \left(\frac{-1}{2} \cdot \color{blue}{im}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{2} \cdot im\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 28.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

   8. Simplified 28.2%

      \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(1 + im \cdot \left(im \cdot -0.5\right)\right)} \]

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{re}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]
   10. Step-by-step derivation

   11. Simplified 28.2%

       \[\leadsto \color{blue}{re} \cdot \left(1 + im \cdot \left(im \cdot -0.5\right)\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 36.0% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.00135:\\ \;\;\;\;im \cdot \left(im \cdot \left(-0.5 + re \cdot -0.5\right)\right)\\ \mathbf{elif}\;re \leq 9.5 \cdot 10^{+19}:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + im \cdot \left(im \cdot -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.00135)
   (* im (* im (+ -0.5 (* re -0.5))))
   (if (<= re 9.5e+19) (+ re 1.0) (* re (+ 1.0 (* im (* im -0.5)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.00135) {
		tmp = im * (im * (-0.5 + (re * -0.5)));
	} else if (re <= 9.5e+19) {
		tmp = re + 1.0;
	} else {
		tmp = re * (1.0 + (im * (im * -0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-0.00135d0)) then
        tmp = im * (im * ((-0.5d0) + (re * (-0.5d0))))
    else if (re <= 9.5d+19) then
        tmp = re + 1.0d0
    else
        tmp = re * (1.0d0 + (im * (im * (-0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.00135) {
		tmp = im * (im * (-0.5 + (re * -0.5)));
	} else if (re <= 9.5e+19) {
		tmp = re + 1.0;
	} else {
		tmp = re * (1.0 + (im * (im * -0.5)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.00135:
		tmp = im * (im * (-0.5 + (re * -0.5)))
	elif re <= 9.5e+19:
		tmp = re + 1.0
	else:
		tmp = re * (1.0 + (im * (im * -0.5)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.00135)
		tmp = Float64(im * Float64(im * Float64(-0.5 + Float64(re * -0.5))));
	elseif (re <= 9.5e+19)
		tmp = Float64(re + 1.0);
	else
		tmp = Float64(re * Float64(1.0 + Float64(im * Float64(im * -0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.00135)
		tmp = im * (im * (-0.5 + (re * -0.5)));
	elseif (re <= 9.5e+19)
		tmp = re + 1.0;
	else
		tmp = re * (1.0 + (im * (im * -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.00135], N[(im * N[(im * N[(-0.5 + N[(re * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 9.5e+19], N[(re + 1.0), $MachinePrecision], N[(re * N[(1.0 + N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.0013500000000000001

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. +-lowering-+.f64 3.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

   5. Simplified 3.1%

      \[\leadsto \color{blue}{\left(re + 1\right)} \cdot \cos im \]

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)}\right) \]
   7. Step-by-step derivation

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

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

      2. *-commutative N/A

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

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \frac{-1}{2}\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \frac{-1}{2}\right)}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \left(\frac{-1}{2} \cdot \color{blue}{im}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{2} \cdot im\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 2.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

   8. Simplified 2.1%

      \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(1 + im \cdot \left(im \cdot -0.5\right)\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \left({im}^{2} \cdot \left(1 + re\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{2} \cdot {im}^{2}\right) \cdot \color{blue}{\left(1 + re\right)} \]

       2. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \frac{-1}{2}\right) \cdot \left(\color{blue}{1} + re\right) \]

       3. associate-*r* N/A

          \[\leadsto {im}^{2} \cdot \color{blue}{\left(\frac{-1}{2} \cdot \left(1 + re\right)\right)} \]

       4. unpow2 N/A

          \[\leadsto \left(im \cdot im\right) \cdot \left(\color{blue}{\frac{-1}{2}} \cdot \left(1 + re\right)\right) \]

       5. associate-*l* N/A

          \[\leadsto im \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{2} \cdot \left(1 + re\right)\right)\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{-1}{2} \cdot \left(1 + re\right)\right)\right)}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{2} \cdot \left(1 + re\right)\right)}\right)\right) \]

       8. distribute-rgt-in N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(1 \cdot \frac{-1}{2} + \color{blue}{re \cdot \frac{-1}{2}}\right)\right)\right) \]

       9. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{2} + \color{blue}{re} \cdot \frac{-1}{2}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(re \cdot \frac{-1}{2}\right)}\right)\right)\right) \]

       11. *-lowering-*.f64 12.9%

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

   11. Simplified 12.9%

       \[\leadsto \color{blue}{im \cdot \left(im \cdot \left(-0.5 + re \cdot -0.5\right)\right)} \]

   if -0.0013500000000000001 < re < 9.5e19

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. exp-lowering-exp.f64 55.2%

         \[\leadsto \mathsf{exp.f64}\left(re\right) \]

   5. Simplified 55.2%

      \[\leadsto \color{blue}{e^{re}} \]

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 51.1%

         \[\leadsto \mathsf{+.f64}\left(re, \color{blue}{1}\right) \]

   8. Simplified 51.1%

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

   if 9.5e19 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. +-lowering-+.f64 5.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

   5. Simplified 5.9%

      \[\leadsto \color{blue}{\left(re + 1\right)} \cdot \cos im \]

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)}\right) \]
   7. Step-by-step derivation

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

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

      2. *-commutative N/A

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

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \frac{-1}{2}\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \frac{-1}{2}\right)}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \left(\frac{-1}{2} \cdot \color{blue}{im}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{2} \cdot im\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 28.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

   8. Simplified 28.6%

      \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(1 + im \cdot \left(im \cdot -0.5\right)\right)} \]

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{re}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{2}\right)\right)\right)\right) \]
   10. Step-by-step derivation

   11. Simplified 28.6%

       \[\leadsto \color{blue}{re} \cdot \left(1 + im \cdot \left(im \cdot -0.5\right)\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 35.5% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(im \cdot \left(-0.5 + re \cdot -0.5\right)\right)\\ \mathbf{if}\;re \leq -0.00135:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 9.5 \cdot 10^{+19}:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (* im (+ -0.5 (* re -0.5))))))
   (if (<= re -0.00135) t_0 (if (<= re 9.5e+19) (+ re 1.0) t_0))))

C

double code(double re, double im) {
	double t_0 = im * (im * (-0.5 + (re * -0.5)));
	double tmp;
	if (re <= -0.00135) {
		tmp = t_0;
	} else if (re <= 9.5e+19) {
		tmp = re + 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = im * (im * ((-0.5d0) + (re * (-0.5d0))))
    if (re <= (-0.00135d0)) then
        tmp = t_0
    else if (re <= 9.5d+19) then
        tmp = re + 1.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * (im * (-0.5 + (re * -0.5)));
	double tmp;
	if (re <= -0.00135) {
		tmp = t_0;
	} else if (re <= 9.5e+19) {
		tmp = re + 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (im * (-0.5 + (re * -0.5)))
	tmp = 0
	if re <= -0.00135:
		tmp = t_0
	elif re <= 9.5e+19:
		tmp = re + 1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(im * Float64(-0.5 + Float64(re * -0.5))))
	tmp = 0.0
	if (re <= -0.00135)
		tmp = t_0;
	elseif (re <= 9.5e+19)
		tmp = Float64(re + 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (im * (-0.5 + (re * -0.5)));
	tmp = 0.0;
	if (re <= -0.00135)
		tmp = t_0;
	elseif (re <= 9.5e+19)
		tmp = re + 1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(im * N[(-0.5 + N[(re * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -0.00135], t$95$0, If[LessEqual[re, 9.5e+19], N[(re + 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if re < -0.0013500000000000001 or 9.5e19 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. +-lowering-+.f64 4.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

   5. Simplified 4.5%

      \[\leadsto \color{blue}{\left(re + 1\right)} \cdot \cos im \]

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)}\right) \]
   7. Step-by-step derivation

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

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

      2. *-commutative N/A

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

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \frac{-1}{2}\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \frac{-1}{2}\right)}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \left(im \cdot \left(\frac{-1}{2} \cdot \color{blue}{im}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{2} \cdot im\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 15.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

   8. Simplified 15.7%

      \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(1 + im \cdot \left(im \cdot -0.5\right)\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \left({im}^{2} \cdot \left(1 + re\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{2} \cdot {im}^{2}\right) \cdot \color{blue}{\left(1 + re\right)} \]

       2. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \frac{-1}{2}\right) \cdot \left(\color{blue}{1} + re\right) \]

       3. associate-*r* N/A

          \[\leadsto {im}^{2} \cdot \color{blue}{\left(\frac{-1}{2} \cdot \left(1 + re\right)\right)} \]

       4. unpow2 N/A

          \[\leadsto \left(im \cdot im\right) \cdot \left(\color{blue}{\frac{-1}{2}} \cdot \left(1 + re\right)\right) \]

       5. associate-*l* N/A

          \[\leadsto im \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{2} \cdot \left(1 + re\right)\right)\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{-1}{2} \cdot \left(1 + re\right)\right)\right)}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{2} \cdot \left(1 + re\right)\right)}\right)\right) \]

       8. distribute-rgt-in N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(1 \cdot \frac{-1}{2} + \color{blue}{re \cdot \frac{-1}{2}}\right)\right)\right) \]

       9. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{2} + \color{blue}{re} \cdot \frac{-1}{2}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(re \cdot \frac{-1}{2}\right)}\right)\right)\right) \]

       11. *-lowering-*.f64 19.6%

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

   11. Simplified 19.6%

       \[\leadsto \color{blue}{im \cdot \left(im \cdot \left(-0.5 + re \cdot -0.5\right)\right)} \]

   if -0.0013500000000000001 < re < 9.5e19

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. exp-lowering-exp.f64 55.2%

         \[\leadsto \mathsf{exp.f64}\left(re\right) \]

   5. Simplified 55.2%

      \[\leadsto \color{blue}{e^{re}} \]

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 51.1%

         \[\leadsto \mathsf{+.f64}\left(re, \color{blue}{1}\right) \]

   8. Simplified 51.1%

      \[\leadsto \color{blue}{re + 1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 49.5% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -6.8:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot 0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -6.8)
   (* (* (* im im) (* im im)) 0.041666666666666664)
   (+ 1.0 (* re (+ 1.0 (* re (+ 0.5 (* re 0.16666666666666666))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -6.8) {
		tmp = ((im * im) * (im * im)) * 0.041666666666666664;
	} else {
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-6.8d0)) then
        tmp = ((im * im) * (im * im)) * 0.041666666666666664d0
    else
        tmp = 1.0d0 + (re * (1.0d0 + (re * (0.5d0 + (re * 0.16666666666666666d0)))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -6.8) {
		tmp = ((im * im) * (im * im)) * 0.041666666666666664;
	} else {
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -6.8:
		tmp = ((im * im) * (im * im)) * 0.041666666666666664
	else:
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -6.8)
		tmp = Float64(Float64(Float64(im * im) * Float64(im * im)) * 0.041666666666666664);
	else
		tmp = Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * Float64(0.5 + Float64(re * 0.16666666666666666))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -6.8)
		tmp = ((im * im) * (im * im)) * 0.041666666666666664;
	else
		tmp = 1.0 + (re * (1.0 + (re * (0.5 + (re * 0.16666666666666666)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -6.8], N[(N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] * 0.041666666666666664), $MachinePrecision], N[(1.0 + N[(re * N[(1.0 + N[(re * N[(0.5 + N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -6.79999999999999982

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. +-lowering-+.f64 2.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

   5. Simplified 2.3%

      \[\leadsto \color{blue}{\left(re + 1\right)} \cdot \cos im \]

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} + \frac{-1}{2}\right)\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {im}^{2}}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(\left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(im \cdot \color{blue}{\left(im \cdot \frac{1}{24}\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \frac{1}{24}\right)}\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 2.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 2.0%

      \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(1 + \left(im \cdot im\right) \cdot \left(-0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \left({im}^{4} \cdot \left(1 + re\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{24} \cdot {im}^{4}\right) \cdot \color{blue}{\left(1 + re\right)} \]

       2. *-commutative N/A

          \[\leadsto \left({im}^{4} \cdot \frac{1}{24}\right) \cdot \left(\color{blue}{1} + re\right) \]

       3. associate-*r* N/A

          \[\leadsto {im}^{4} \cdot \color{blue}{\left(\frac{1}{24} \cdot \left(1 + re\right)\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{4}\right), \color{blue}{\left(\frac{1}{24} \cdot \left(1 + re\right)\right)}\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{\left(2 \cdot 2\right)}\right), \left(\frac{1}{24} \cdot \left(1 + re\right)\right)\right) \]

       6. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{2} \cdot {im}^{2}\right), \left(\color{blue}{\frac{1}{24}} \cdot \left(1 + re\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({im}^{2}\right), \left({im}^{2}\right)\right), \left(\color{blue}{\frac{1}{24}} \cdot \left(1 + re\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(im \cdot im\right), \left({im}^{2}\right)\right), \left(\frac{1}{24} \cdot \left(1 + re\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2}\right)\right), \left(\frac{1}{24} \cdot \left(1 + re\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(im \cdot im\right)\right), \left(\frac{1}{24} \cdot \left(1 + re\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \left(\frac{1}{24} \cdot \left(1 + re\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\left(1 + re\right)}\right)\right) \]

       13. +-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{*.f64}\left(\frac{1}{24}, \left(re + \color{blue}{1}\right)\right)\right) \]

       14. +-lowering-+.f64 42.2%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{+.f64}\left(re, \color{blue}{1}\right)\right)\right) \]

   11. Simplified 42.2%

       \[\leadsto \color{blue}{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(0.041666666666666664 \cdot \left(re + 1\right)\right)} \]

   12. Taylor expanded in re around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \color{blue}{\frac{1}{24}}\right) \]
   13. Step-by-step derivation

   14. Simplified 42.5%

       \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{0.041666666666666664} \]

   if -6.79999999999999982 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

      7. *-lowering-*.f64 89.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(im\right)\right) \]

   5. Simplified 89.3%

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)\right)} \cdot \cos im \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot re\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{6} \cdot re\right)}\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 51.0%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 51.0%

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + re \cdot \left(0.5 + re \cdot 0.16666666666666666\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 46.6% accurate, 14.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -410:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot 0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;1 + re \cdot \left(1 + re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -410.0)
   (* (* (* im im) (* im im)) 0.041666666666666664)
   (+ 1.0 (* re (+ 1.0 (* re 0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -410.0) {
		tmp = ((im * im) * (im * im)) * 0.041666666666666664;
	} else {
		tmp = 1.0 + (re * (1.0 + (re * 0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-410.0d0)) then
        tmp = ((im * im) * (im * im)) * 0.041666666666666664d0
    else
        tmp = 1.0d0 + (re * (1.0d0 + (re * 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -410.0) {
		tmp = ((im * im) * (im * im)) * 0.041666666666666664;
	} else {
		tmp = 1.0 + (re * (1.0 + (re * 0.5)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -410.0:
		tmp = ((im * im) * (im * im)) * 0.041666666666666664
	else:
		tmp = 1.0 + (re * (1.0 + (re * 0.5)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -410.0)
		tmp = Float64(Float64(Float64(im * im) * Float64(im * im)) * 0.041666666666666664);
	else
		tmp = Float64(1.0 + Float64(re * Float64(1.0 + Float64(re * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -410.0)
		tmp = ((im * im) * (im * im)) * 0.041666666666666664;
	else
		tmp = 1.0 + (re * (1.0 + (re * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -410.0], N[(N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] * 0.041666666666666664), $MachinePrecision], N[(1.0 + N[(re * N[(1.0 + N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -410

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + re\right)}, \mathsf{cos.f64}\left(im\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(re + 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

      2. +-lowering-+.f64 2.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{cos.f64}\left(\color{blue}{im}\right)\right) \]

   5. Simplified 2.3%

      \[\leadsto \color{blue}{\left(re + 1\right)} \cdot \cos im \]

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}\right)\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} \cdot {im}^{2} + \frac{-1}{2}\right)\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {im}^{2}}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(\left(im \cdot im\right) \cdot \frac{1}{24}\right)\right)\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(im \cdot \color{blue}{\left(im \cdot \frac{1}{24}\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \frac{1}{24}\right)}\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 2.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(re, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 2.0%

      \[\leadsto \left(re + 1\right) \cdot \color{blue}{\left(1 + \left(im \cdot im\right) \cdot \left(-0.5 + im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \left({im}^{4} \cdot \left(1 + re\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{24} \cdot {im}^{4}\right) \cdot \color{blue}{\left(1 + re\right)} \]

       2. *-commutative N/A

          \[\leadsto \left({im}^{4} \cdot \frac{1}{24}\right) \cdot \left(\color{blue}{1} + re\right) \]

       3. associate-*r* N/A

          \[\leadsto {im}^{4} \cdot \color{blue}{\left(\frac{1}{24} \cdot \left(1 + re\right)\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{4}\right), \color{blue}{\left(\frac{1}{24} \cdot \left(1 + re\right)\right)}\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{\left(2 \cdot 2\right)}\right), \left(\frac{1}{24} \cdot \left(1 + re\right)\right)\right) \]

       6. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{2} \cdot {im}^{2}\right), \left(\color{blue}{\frac{1}{24}} \cdot \left(1 + re\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({im}^{2}\right), \left({im}^{2}\right)\right), \left(\color{blue}{\frac{1}{24}} \cdot \left(1 + re\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(im \cdot im\right), \left({im}^{2}\right)\right), \left(\frac{1}{24} \cdot \left(1 + re\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2}\right)\right), \left(\frac{1}{24} \cdot \left(1 + re\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(im \cdot im\right)\right), \left(\frac{1}{24} \cdot \left(1 + re\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \left(\frac{1}{24} \cdot \left(1 + re\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\left(1 + re\right)}\right)\right) \]

       13. +-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{*.f64}\left(\frac{1}{24}, \left(re + \color{blue}{1}\right)\right)\right) \]

       14. +-lowering-+.f64 42.2%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{+.f64}\left(re, \color{blue}{1}\right)\right)\right) \]

   11. Simplified 42.2%

       \[\leadsto \color{blue}{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(0.041666666666666664 \cdot \left(re + 1\right)\right)} \]

   12. Taylor expanded in re around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \color{blue}{\frac{1}{24}}\right) \]
   13. Step-by-step derivation

   14. Simplified 42.5%

       \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{0.041666666666666664} \]

   if -410 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. exp-lowering-exp.f64 58.9%

         \[\leadsto \mathsf{exp.f64}\left(re\right) \]

   5. Simplified 58.9%

      \[\leadsto \color{blue}{e^{re}} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\left(1 + \frac{1}{2} \cdot re\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot re\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(re \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 47.6%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

   8. Simplified 47.6%

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + re \cdot 0.5\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 31.2% accurate, 16.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 9.5 \cdot 10^{+19}:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;1 + im \cdot \left(im \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 9.5e+19) (+ re 1.0) (+ 1.0 (* im (* im -0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 9.5e+19) {
		tmp = re + 1.0;
	} else {
		tmp = 1.0 + (im * (im * -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 9.5d+19) then
        tmp = re + 1.0d0
    else
        tmp = 1.0d0 + (im * (im * (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 9.5e+19) {
		tmp = re + 1.0;
	} else {
		tmp = 1.0 + (im * (im * -0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 9.5e+19:
		tmp = re + 1.0
	else:
		tmp = 1.0 + (im * (im * -0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 9.5e+19)
		tmp = Float64(re + 1.0);
	else
		tmp = Float64(1.0 + Float64(im * Float64(im * -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 9.5e+19)
		tmp = re + 1.0;
	else
		tmp = 1.0 + (im * (im * -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 9.5e+19], N[(re + 1.0), $MachinePrecision], N[(1.0 + N[(im * N[(im * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 9.5e19

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. exp-lowering-exp.f64 67.7%

         \[\leadsto \mathsf{exp.f64}\left(re\right) \]

   5. Simplified 67.7%

      \[\leadsto \color{blue}{e^{re}} \]

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 36.9%

         \[\leadsto \mathsf{+.f64}\left(re, \color{blue}{1}\right) \]

   8. Simplified 36.9%

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

   if 9.5e19 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\cos im} \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 3.1%

         \[\leadsto \mathsf{cos.f64}\left(im\right) \]

   5. Simplified 3.1%

      \[\leadsto \color{blue}{\cos im} \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + \frac{-1}{2} \cdot {im}^{2}} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left({im}^{2} \cdot \color{blue}{\frac{-1}{2}}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \frac{-1}{2}\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \frac{-1}{2}\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(im \cdot \left(\frac{-1}{2} \cdot \color{blue}{im}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{2} \cdot im\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

      8. *-lowering-*.f64 19.1%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

   8. Simplified 19.1%

      \[\leadsto \color{blue}{1 + im \cdot \left(im \cdot -0.5\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 29.0% accurate, 67.7× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 (+ re 1.0))

C

double code(double re, double im) {
	return re + 1.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = re + 1.0d0
end function

Java

public static double code(double re, double im) {
	return re + 1.0;
}

Python

def code(re, im):
	return re + 1.0

Julia

function code(re, im)
	return Float64(re + 1.0)
end

MATLAB

function tmp = code(re, im)
	tmp = re + 1.0;
end

Wolfram

code[re_, im_] := N[(re + 1.0), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \cos im \]
2. Add Preprocessing

3. Taylor expanded in im around 0

   \[\leadsto \color{blue}{e^{re}} \]
4. Step-by-step derivation

   1. exp-lowering-exp.f64 67.9%

      \[\leadsto \mathsf{exp.f64}\left(re\right) \]

5. Simplified 67.9%

   \[\leadsto \color{blue}{e^{re}} \]

6. Taylor expanded in re around 0

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

   1. +-commutative N/A

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

   2. +-lowering-+.f64 29.2%

      \[\leadsto \mathsf{+.f64}\left(re, \color{blue}{1}\right) \]

8. Simplified 29.2%

   \[\leadsto \color{blue}{re + 1} \]
9. Add Preprocessing

Alternative 20: 28.6% accurate, 203.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 1.0)

C

double code(double re, double im) {
	return 1.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 1.0d0
end function

Java

public static double code(double re, double im) {
	return 1.0;
}

Python

def code(re, im):
	return 1.0

Julia

function code(re, im)
	return 1.0
end

MATLAB

function tmp = code(re, im)
	tmp = 1.0;
end

Wolfram

code[re_, im_] := 1.0

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \cos im \]
2. Add Preprocessing

3. Taylor expanded in re around 0

   \[\leadsto \color{blue}{\cos im} \]
4. Step-by-step derivation

   1. cos-lowering-cos.f64 52.7%

      \[\leadsto \mathsf{cos.f64}\left(im\right) \]

5. Simplified 52.7%

   \[\leadsto \color{blue}{\cos im} \]

6. Taylor expanded in im around 0

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

8. Simplified 28.7%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024158 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))