math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 8.9s

Alternatives: 16

Speedup: 1.0×

• 25.8% 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. Final simplification 100.0%

   \[\leadsto e^{re} \cdot \cos im \]

Alternative 2: 92.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 2:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (exp re) 0.0)
   (exp re)
   (if (<= (exp re) 2.0) (* (cos im) (+ re 1.0)) (exp re))))

C

double code(double re, double im) {
	double tmp;
	if (exp(re) <= 0.0) {
		tmp = exp(re);
	} else if (exp(re) <= 2.0) {
		tmp = cos(im) * (re + 1.0);
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (exp(re) <= 0.0d0) then
        tmp = exp(re)
    else if (exp(re) <= 2.0d0) then
        tmp = cos(im) * (re + 1.0d0)
    else
        tmp = exp(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.exp(re) <= 0.0) {
		tmp = Math.exp(re);
	} else if (Math.exp(re) <= 2.0) {
		tmp = Math.cos(im) * (re + 1.0);
	} else {
		tmp = Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.exp(re) <= 0.0:
		tmp = math.exp(re)
	elif math.exp(re) <= 2.0:
		tmp = math.cos(im) * (re + 1.0)
	else:
		tmp = math.exp(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (exp(re) <= 0.0)
		tmp = exp(re);
	elseif (exp(re) <= 2.0)
		tmp = Float64(cos(im) * Float64(re + 1.0));
	else
		tmp = exp(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (exp(re) <= 0.0)
		tmp = exp(re);
	elseif (exp(re) <= 2.0)
		tmp = cos(im) * (re + 1.0);
	else
		tmp = exp(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Exp[re], $MachinePrecision], 0.0], N[Exp[re], $MachinePrecision], If[LessEqual[N[Exp[re], $MachinePrecision], 2.0], N[(N[Cos[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], N[Exp[re], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.0 or 2 < (exp.f64 re)

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in im around 0 85.5%

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

   if 0.0 < (exp.f64 re) < 2

   1. Initial program 99.9%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 99.5%

      \[\leadsto \color{blue}{\cos im \cdot re + \cos im} \]
   3. Step-by-step derivation

      1. *-rgt-identity 99.5%

         \[\leadsto \cos im \cdot re + \color{blue}{\cos im \cdot 1} \]

      2. distribute-lft-out 99.5%

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

   4. Simplified 99.5%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 2:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]

Alternative 3: 92.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.999999999998:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 2:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (exp re) 0.999999999998)
   (exp re)
   (if (<= (exp re) 2.0) (cos im) (exp re))))

C

double code(double re, double im) {
	double tmp;
	if (exp(re) <= 0.999999999998) {
		tmp = exp(re);
	} else if (exp(re) <= 2.0) {
		tmp = cos(im);
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (exp(re) <= 0.999999999998d0) then
        tmp = exp(re)
    else if (exp(re) <= 2.0d0) then
        tmp = cos(im)
    else
        tmp = exp(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.exp(re) <= 0.999999999998) {
		tmp = Math.exp(re);
	} else if (Math.exp(re) <= 2.0) {
		tmp = Math.cos(im);
	} else {
		tmp = Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.exp(re) <= 0.999999999998:
		tmp = math.exp(re)
	elif math.exp(re) <= 2.0:
		tmp = math.cos(im)
	else:
		tmp = math.exp(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (exp(re) <= 0.999999999998)
		tmp = exp(re);
	elseif (exp(re) <= 2.0)
		tmp = cos(im);
	else
		tmp = exp(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (exp(re) <= 0.999999999998)
		tmp = exp(re);
	elseif (exp(re) <= 2.0)
		tmp = cos(im);
	else
		tmp = exp(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Exp[re], $MachinePrecision], 0.999999999998], N[Exp[re], $MachinePrecision], If[LessEqual[N[Exp[re], $MachinePrecision], 2.0], N[Cos[im], $MachinePrecision], N[Exp[re], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.99999999999800004 or 2 < (exp.f64 re)

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in im around 0 85.6%

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

   if 0.99999999999800004 < (exp.f64 re) < 2

   1. Initial program 99.9%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 99.3%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.999999999998:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 2:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]

Alternative 4: 96.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos im \cdot \left(0.5 \cdot \left(re \cdot re\right) + \left(re + 1\right)\right)\\ \mathbf{if}\;re \leq -0.0042:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.038:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+65}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 8.2 \cdot 10^{+150}:\\ \;\;\;\;e^{re} \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (cos im) (+ (* 0.5 (* re re)) (+ re 1.0)))))
   (if (<= re -0.0042)
     (exp re)
     (if (<= re 0.038)
       t_0
       (if (<= re 5e+65)
         (exp re)
         (if (<= re 8.2e+150) (* (exp re) (+ 1.0 (* -0.5 (* im im)))) t_0))))))

C

double code(double re, double im) {
	double t_0 = cos(im) * ((0.5 * (re * re)) + (re + 1.0));
	double tmp;
	if (re <= -0.0042) {
		tmp = exp(re);
	} else if (re <= 0.038) {
		tmp = t_0;
	} else if (re <= 5e+65) {
		tmp = exp(re);
	} else if (re <= 8.2e+150) {
		tmp = exp(re) * (1.0 + (-0.5 * (im * im)));
	} 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) * ((0.5d0 * (re * re)) + (re + 1.0d0))
    if (re <= (-0.0042d0)) then
        tmp = exp(re)
    else if (re <= 0.038d0) then
        tmp = t_0
    else if (re <= 5d+65) then
        tmp = exp(re)
    else if (re <= 8.2d+150) then
        tmp = exp(re) * (1.0d0 + ((-0.5d0) * (im * im)))
    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) * ((0.5 * (re * re)) + (re + 1.0));
	double tmp;
	if (re <= -0.0042) {
		tmp = Math.exp(re);
	} else if (re <= 0.038) {
		tmp = t_0;
	} else if (re <= 5e+65) {
		tmp = Math.exp(re);
	} else if (re <= 8.2e+150) {
		tmp = Math.exp(re) * (1.0 + (-0.5 * (im * im)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.cos(im) * ((0.5 * (re * re)) + (re + 1.0))
	tmp = 0
	if re <= -0.0042:
		tmp = math.exp(re)
	elif re <= 0.038:
		tmp = t_0
	elif re <= 5e+65:
		tmp = math.exp(re)
	elif re <= 8.2e+150:
		tmp = math.exp(re) * (1.0 + (-0.5 * (im * im)))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(cos(im) * Float64(Float64(0.5 * Float64(re * re)) + Float64(re + 1.0)))
	tmp = 0.0
	if (re <= -0.0042)
		tmp = exp(re);
	elseif (re <= 0.038)
		tmp = t_0;
	elseif (re <= 5e+65)
		tmp = exp(re);
	elseif (re <= 8.2e+150)
		tmp = Float64(exp(re) * Float64(1.0 + Float64(-0.5 * Float64(im * im))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = cos(im) * ((0.5 * (re * re)) + (re + 1.0));
	tmp = 0.0;
	if (re <= -0.0042)
		tmp = exp(re);
	elseif (re <= 0.038)
		tmp = t_0;
	elseif (re <= 5e+65)
		tmp = exp(re);
	elseif (re <= 8.2e+150)
		tmp = exp(re) * (1.0 + (-0.5 * (im * im)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Cos[im], $MachinePrecision] * N[(N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision] + N[(re + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -0.0042], N[Exp[re], $MachinePrecision], If[LessEqual[re, 0.038], t$95$0, If[LessEqual[re, 5e+65], N[Exp[re], $MachinePrecision], If[LessEqual[re, 8.2e+150], N[(N[Exp[re], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.00419999999999999974 or 0.0379999999999999991 < re < 4.99999999999999973e65

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in im around 0 97.4%

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

   if -0.00419999999999999974 < re < 0.0379999999999999991 or 8.19999999999999988e150 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 99.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos im \cdot {re}^{2}\right) + \left(\cos im \cdot re + \cos im\right)} \]
   3. Step-by-step derivation

      1. +-commutative 99.1%

         \[\leadsto \color{blue}{\left(\cos im \cdot re + \cos im\right) + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right)} \]

      2. +-commutative 99.1%

         \[\leadsto \color{blue}{\left(\cos im + \cos im \cdot re\right)} + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right) \]

      3. *-rgt-identity 99.1%

         \[\leadsto \left(\color{blue}{\cos im \cdot 1} + \cos im \cdot re\right) + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right) \]

      4. distribute-lft-out 99.1%

         \[\leadsto \color{blue}{\cos im \cdot \left(1 + re\right)} + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right) \]

      5. *-commutative 99.1%

         \[\leadsto \cos im \cdot \left(1 + re\right) + \color{blue}{\left(\cos im \cdot {re}^{2}\right) \cdot 0.5} \]

      6. associate-*l* 99.1%

         \[\leadsto \cos im \cdot \left(1 + re\right) + \color{blue}{\cos im \cdot \left({re}^{2} \cdot 0.5\right)} \]

      7. distribute-lft-out 99.1%

         \[\leadsto \color{blue}{\cos im \cdot \left(\left(1 + re\right) + {re}^{2} \cdot 0.5\right)} \]

      8. +-commutative 99.1%

         \[\leadsto \cos im \cdot \left(\color{blue}{\left(re + 1\right)} + {re}^{2} \cdot 0.5\right) \]

      9. *-commutative 99.1%

         \[\leadsto \cos im \cdot \left(\left(re + 1\right) + \color{blue}{0.5 \cdot {re}^{2}}\right) \]

      10. unpow2 99.1%

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

   4. Simplified 99.1%

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

   if 4.99999999999999973e65 < re < 8.19999999999999988e150

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in im around 0 85.0%

      \[\leadsto e^{re} \cdot \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \]
   3. Step-by-step derivation

      1. unpow2 47.7%

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

   4. Simplified 85.0%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0042:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.038:\\ \;\;\;\;\cos im \cdot \left(0.5 \cdot \left(re \cdot re\right) + \left(re + 1\right)\right)\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+65}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 8.2 \cdot 10^{+150}:\\ \;\;\;\;e^{re} \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(0.5 \cdot \left(re \cdot re\right) + \left(re + 1\right)\right)\\ \end{array} \]

Alternative 5: 92.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -6.8 \cdot 10^{-5}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.0295:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+62}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -6.8e-5)
   (exp re)
   (if (<= re 0.0295)
     (* (cos im) (+ re 1.0))
     (if (<= re 5e+62) (exp re) (* (exp re) (+ 1.0 (* -0.5 (* im im))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -6.8e-5) {
		tmp = exp(re);
	} else if (re <= 0.0295) {
		tmp = cos(im) * (re + 1.0);
	} else if (re <= 5e+62) {
		tmp = exp(re);
	} else {
		tmp = exp(re) * (1.0 + (-0.5 * (im * im)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-6.8d-5)) then
        tmp = exp(re)
    else if (re <= 0.0295d0) then
        tmp = cos(im) * (re + 1.0d0)
    else if (re <= 5d+62) then
        tmp = exp(re)
    else
        tmp = exp(re) * (1.0d0 + ((-0.5d0) * (im * im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -6.8e-5) {
		tmp = Math.exp(re);
	} else if (re <= 0.0295) {
		tmp = Math.cos(im) * (re + 1.0);
	} else if (re <= 5e+62) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.exp(re) * (1.0 + (-0.5 * (im * im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -6.8e-5:
		tmp = math.exp(re)
	elif re <= 0.0295:
		tmp = math.cos(im) * (re + 1.0)
	elif re <= 5e+62:
		tmp = math.exp(re)
	else:
		tmp = math.exp(re) * (1.0 + (-0.5 * (im * im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -6.8e-5)
		tmp = exp(re);
	elseif (re <= 0.0295)
		tmp = Float64(cos(im) * Float64(re + 1.0));
	elseif (re <= 5e+62)
		tmp = exp(re);
	else
		tmp = Float64(exp(re) * Float64(1.0 + Float64(-0.5 * Float64(im * im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -6.8e-5)
		tmp = exp(re);
	elseif (re <= 0.0295)
		tmp = cos(im) * (re + 1.0);
	elseif (re <= 5e+62)
		tmp = exp(re);
	else
		tmp = exp(re) * (1.0 + (-0.5 * (im * im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -6.8e-5], N[Exp[re], $MachinePrecision], If[LessEqual[re, 0.0295], N[(N[Cos[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 5e+62], N[Exp[re], $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -6.7999999999999999e-5 or 0.029499999999999998 < re < 5.00000000000000029e62

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in im around 0 97.4%

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

   if -6.7999999999999999e-5 < re < 0.029499999999999998

   1. Initial program 99.9%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 99.5%

      \[\leadsto \color{blue}{\cos im \cdot re + \cos im} \]
   3. Step-by-step derivation

      1. *-rgt-identity 99.5%

         \[\leadsto \cos im \cdot re + \color{blue}{\cos im \cdot 1} \]

      2. distribute-lft-out 99.5%

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

   4. Simplified 99.5%

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

   if 5.00000000000000029e62 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in im around 0 78.7%

      \[\leadsto e^{re} \cdot \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \]
   3. Step-by-step derivation

      1. unpow2 62.9%

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

   4. Simplified 78.7%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -6.8 \cdot 10^{-5}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.0295:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+62}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 6: 67.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left(re \cdot 0.5\right)\\ t_1 := -1 - t_0\\ \mathbf{if}\;re \leq -3400:\\ \;\;\;\;-0.5 \cdot \left(\left(im \cdot im\right) \cdot \left(re + 1\right)\right)\\ \mathbf{elif}\;re \leq 4.3 \cdot 10^{+19}:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;re \leq 1.1 \cdot 10^{+134}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(0.5 \cdot \left(re \cdot re\right) + \left(re + 1\right)\right)\\ \mathbf{elif}\;re \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{re \cdot re + \left(1 + t_0\right) \cdot t_1}{re + t_1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (* re 0.5))) (t_1 (- -1.0 t_0)))
   (if (<= re -3400.0)
     (* -0.5 (* (* im im) (+ re 1.0)))
     (if (<= re 4.3e+19)
       (cos im)
       (if (<= re 1.1e+134)
         (* (+ 1.0 (* -0.5 (* im im))) (+ (* 0.5 (* re re)) (+ re 1.0)))
         (if (<= re 1.4e+154)
           (/ (+ (* re re) (* (+ 1.0 t_0) t_1)) (+ re t_1))
           t_0))))))

C

double code(double re, double im) {
	double t_0 = re * (re * 0.5);
	double t_1 = -1.0 - t_0;
	double tmp;
	if (re <= -3400.0) {
		tmp = -0.5 * ((im * im) * (re + 1.0));
	} else if (re <= 4.3e+19) {
		tmp = cos(im);
	} else if (re <= 1.1e+134) {
		tmp = (1.0 + (-0.5 * (im * im))) * ((0.5 * (re * re)) + (re + 1.0));
	} else if (re <= 1.4e+154) {
		tmp = ((re * re) + ((1.0 + t_0) * t_1)) / (re + t_1);
	} 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) :: t_1
    real(8) :: tmp
    t_0 = re * (re * 0.5d0)
    t_1 = (-1.0d0) - t_0
    if (re <= (-3400.0d0)) then
        tmp = (-0.5d0) * ((im * im) * (re + 1.0d0))
    else if (re <= 4.3d+19) then
        tmp = cos(im)
    else if (re <= 1.1d+134) then
        tmp = (1.0d0 + ((-0.5d0) * (im * im))) * ((0.5d0 * (re * re)) + (re + 1.0d0))
    else if (re <= 1.4d+154) then
        tmp = ((re * re) + ((1.0d0 + t_0) * t_1)) / (re + t_1)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re * (re * 0.5);
	double t_1 = -1.0 - t_0;
	double tmp;
	if (re <= -3400.0) {
		tmp = -0.5 * ((im * im) * (re + 1.0));
	} else if (re <= 4.3e+19) {
		tmp = Math.cos(im);
	} else if (re <= 1.1e+134) {
		tmp = (1.0 + (-0.5 * (im * im))) * ((0.5 * (re * re)) + (re + 1.0));
	} else if (re <= 1.4e+154) {
		tmp = ((re * re) + ((1.0 + t_0) * t_1)) / (re + t_1);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * (re * 0.5)
	t_1 = -1.0 - t_0
	tmp = 0
	if re <= -3400.0:
		tmp = -0.5 * ((im * im) * (re + 1.0))
	elif re <= 4.3e+19:
		tmp = math.cos(im)
	elif re <= 1.1e+134:
		tmp = (1.0 + (-0.5 * (im * im))) * ((0.5 * (re * re)) + (re + 1.0))
	elif re <= 1.4e+154:
		tmp = ((re * re) + ((1.0 + t_0) * t_1)) / (re + t_1)
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(re * 0.5))
	t_1 = Float64(-1.0 - t_0)
	tmp = 0.0
	if (re <= -3400.0)
		tmp = Float64(-0.5 * Float64(Float64(im * im) * Float64(re + 1.0)));
	elseif (re <= 4.3e+19)
		tmp = cos(im);
	elseif (re <= 1.1e+134)
		tmp = Float64(Float64(1.0 + Float64(-0.5 * Float64(im * im))) * Float64(Float64(0.5 * Float64(re * re)) + Float64(re + 1.0)));
	elseif (re <= 1.4e+154)
		tmp = Float64(Float64(Float64(re * re) + Float64(Float64(1.0 + t_0) * t_1)) / Float64(re + t_1));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * (re * 0.5);
	t_1 = -1.0 - t_0;
	tmp = 0.0;
	if (re <= -3400.0)
		tmp = -0.5 * ((im * im) * (re + 1.0));
	elseif (re <= 4.3e+19)
		tmp = cos(im);
	elseif (re <= 1.1e+134)
		tmp = (1.0 + (-0.5 * (im * im))) * ((0.5 * (re * re)) + (re + 1.0));
	elseif (re <= 1.4e+154)
		tmp = ((re * re) + ((1.0 + t_0) * t_1)) / (re + t_1);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-1.0 - t$95$0), $MachinePrecision]}, If[LessEqual[re, -3400.0], N[(-0.5 * N[(N[(im * im), $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 4.3e+19], N[Cos[im], $MachinePrecision], If[LessEqual[re, 1.1e+134], N[(N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision] + N[(re + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.4e+154], N[(N[(N[(re * re), $MachinePrecision] + N[(N[(1.0 + t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(re + t$95$1), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 5 regimes

2. if re < -3400

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 2.2%

      \[\leadsto \color{blue}{\cos im \cdot re + \cos im} \]
   3. Step-by-step derivation

      1. *-rgt-identity 2.2%

         \[\leadsto \cos im \cdot re + \color{blue}{\cos im \cdot 1} \]

      2. distribute-lft-out 2.2%

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

   4. Simplified 2.2%

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

   5. Taylor expanded in im around 0 2.0%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \cdot \left(re + 1\right) \]
   6. Step-by-step derivation

      1. unpow2 1.7%

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

   7. Simplified 2.0%

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

   8. Taylor expanded in im around inf 33.6%

      \[\leadsto \color{blue}{-0.5 \cdot \left(\left(1 + re\right) \cdot {im}^{2}\right)} \]
   9. Step-by-step derivation

      1. unpow2 33.6%

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

      2. +-commutative 33.6%

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

      3. *-commutative 33.6%

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

      4. +-commutative 33.6%

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

   10. Simplified 33.6%

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

   if -3400 < re < 4.3e19

   1. Initial program 99.9%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 94.3%

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

   if 4.3e19 < re < 1.1e134

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 4.8%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos im \cdot {re}^{2}\right) + \left(\cos im \cdot re + \cos im\right)} \]
   3. Step-by-step derivation

      1. +-commutative 4.8%

         \[\leadsto \color{blue}{\left(\cos im \cdot re + \cos im\right) + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right)} \]

      2. +-commutative 4.8%

         \[\leadsto \color{blue}{\left(\cos im + \cos im \cdot re\right)} + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right) \]

      3. *-rgt-identity 4.8%

         \[\leadsto \left(\color{blue}{\cos im \cdot 1} + \cos im \cdot re\right) + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right) \]

      4. distribute-lft-out 4.8%

         \[\leadsto \color{blue}{\cos im \cdot \left(1 + re\right)} + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right) \]

      5. *-commutative 4.8%

         \[\leadsto \cos im \cdot \left(1 + re\right) + \color{blue}{\left(\cos im \cdot {re}^{2}\right) \cdot 0.5} \]

      6. associate-*l* 4.8%

         \[\leadsto \cos im \cdot \left(1 + re\right) + \color{blue}{\cos im \cdot \left({re}^{2} \cdot 0.5\right)} \]

      7. distribute-lft-out 4.8%

         \[\leadsto \color{blue}{\cos im \cdot \left(\left(1 + re\right) + {re}^{2} \cdot 0.5\right)} \]

      8. +-commutative 4.8%

         \[\leadsto \cos im \cdot \left(\color{blue}{\left(re + 1\right)} + {re}^{2} \cdot 0.5\right) \]

      9. *-commutative 4.8%

         \[\leadsto \cos im \cdot \left(\left(re + 1\right) + \color{blue}{0.5 \cdot {re}^{2}}\right) \]

      10. unpow2 4.8%

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

   4. Simplified 4.8%

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

   5. Taylor expanded in im around 0 35.3%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 35.3%

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

   7. Simplified 35.3%

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

   if 1.1e134 < re < 1.4e154

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 9.6%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos im \cdot {re}^{2}\right) + \left(\cos im \cdot re + \cos im\right)} \]
   3. Step-by-step derivation

      1. +-commutative 9.6%

         \[\leadsto \color{blue}{\left(\cos im \cdot re + \cos im\right) + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right)} \]

      2. +-commutative 9.6%

         \[\leadsto \color{blue}{\left(\cos im + \cos im \cdot re\right)} + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right) \]

      3. *-rgt-identity 9.6%

         \[\leadsto \left(\color{blue}{\cos im \cdot 1} + \cos im \cdot re\right) + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right) \]

      4. distribute-lft-out 9.6%

         \[\leadsto \color{blue}{\cos im \cdot \left(1 + re\right)} + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right) \]

      5. *-commutative 9.6%

         \[\leadsto \cos im \cdot \left(1 + re\right) + \color{blue}{\left(\cos im \cdot {re}^{2}\right) \cdot 0.5} \]

      6. associate-*l* 9.6%

         \[\leadsto \cos im \cdot \left(1 + re\right) + \color{blue}{\cos im \cdot \left({re}^{2} \cdot 0.5\right)} \]

      7. distribute-lft-out 9.6%

         \[\leadsto \color{blue}{\cos im \cdot \left(\left(1 + re\right) + {re}^{2} \cdot 0.5\right)} \]

      8. +-commutative 9.6%

         \[\leadsto \cos im \cdot \left(\color{blue}{\left(re + 1\right)} + {re}^{2} \cdot 0.5\right) \]

      9. *-commutative 9.6%

         \[\leadsto \cos im \cdot \left(\left(re + 1\right) + \color{blue}{0.5 \cdot {re}^{2}}\right) \]

      10. unpow2 9.6%

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

   4. Simplified 9.6%

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

   5. Taylor expanded in im around 0 9.6%

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

      1. associate-+l+ 9.6%

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

      2. flip-+ 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-*l* 100.0%

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

      7. *-commutative 100.0%

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

      8. associate-*l* 100.0%

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

   7. Applied egg-rr 100.0%

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

   if 1.4e154 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos im \cdot {re}^{2}\right) + \left(\cos im \cdot re + \cos im\right)} \]
   3. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{\left(\cos im \cdot re + \cos im\right) + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right)} \]

      2. +-commutative 100.0%

         \[\leadsto \color{blue}{\left(\cos im + \cos im \cdot re\right)} + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right) \]

      3. *-rgt-identity 100.0%

         \[\leadsto \left(\color{blue}{\cos im \cdot 1} + \cos im \cdot re\right) + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right) \]

      4. distribute-lft-out 100.0%

         \[\leadsto \color{blue}{\cos im \cdot \left(1 + re\right)} + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right) \]

      5. *-commutative 100.0%

         \[\leadsto \cos im \cdot \left(1 + re\right) + \color{blue}{\left(\cos im \cdot {re}^{2}\right) \cdot 0.5} \]

      6. associate-*l* 100.0%

         \[\leadsto \cos im \cdot \left(1 + re\right) + \color{blue}{\cos im \cdot \left({re}^{2} \cdot 0.5\right)} \]

      7. distribute-lft-out 100.0%

         \[\leadsto \color{blue}{\cos im \cdot \left(\left(1 + re\right) + {re}^{2} \cdot 0.5\right)} \]

      8. +-commutative 100.0%

         \[\leadsto \cos im \cdot \left(\color{blue}{\left(re + 1\right)} + {re}^{2} \cdot 0.5\right) \]

      9. *-commutative 100.0%

         \[\leadsto \cos im \cdot \left(\left(re + 1\right) + \color{blue}{0.5 \cdot {re}^{2}}\right) \]

      10. unpow2 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around 0 76.9%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 76.9%

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

   7. Simplified 76.9%

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

   8. Taylor expanded in re around inf 76.9%

      \[\leadsto \left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left(0.5 \cdot {re}^{2}\right)} \]
   9. Step-by-step derivation

      1. unpow2 76.9%

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

      2. *-commutative 76.9%

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

      3. associate-*r* 76.9%

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

   10. Simplified 76.9%

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

   11. Taylor expanded in im around 0 76.9%

       \[\leadsto \color{blue}{0.5 \cdot {re}^{2}} \]
   12. Step-by-step derivation

       1. unpow2 76.9%

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

       2. *-commutative 76.9%

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

       3. associate-*r* 76.9%

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

   13. Simplified 76.9%

       \[\leadsto \color{blue}{re \cdot \left(re \cdot 0.5\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3400:\\ \;\;\;\;-0.5 \cdot \left(\left(im \cdot im\right) \cdot \left(re + 1\right)\right)\\ \mathbf{elif}\;re \leq 4.3 \cdot 10^{+19}:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;re \leq 1.1 \cdot 10^{+134}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(0.5 \cdot \left(re \cdot re\right) + \left(re + 1\right)\right)\\ \mathbf{elif}\;re \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{re \cdot re + \left(1 + re \cdot \left(re \cdot 0.5\right)\right) \cdot \left(-1 - re \cdot \left(re \cdot 0.5\right)\right)}{re + \left(-1 - re \cdot \left(re \cdot 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \]

Alternative 7: 46.1% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left(re \cdot 0.5\right)\\ t_1 := -1 - t_0\\ t_2 := \frac{re \cdot re + \left(1 + t_0\right) \cdot t_1}{re + t_1}\\ \mathbf{if}\;re \leq -3400:\\ \;\;\;\;-0.5 \cdot \left(\left(im \cdot im\right) \cdot \left(re + 1\right)\right)\\ \mathbf{elif}\;re \leq 3 \cdot 10^{+83}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;re \leq 1.1 \cdot 10^{+134}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot t_0\\ \mathbf{elif}\;re \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (* re 0.5)))
        (t_1 (- -1.0 t_0))
        (t_2 (/ (+ (* re re) (* (+ 1.0 t_0) t_1)) (+ re t_1))))
   (if (<= re -3400.0)
     (* -0.5 (* (* im im) (+ re 1.0)))
     (if (<= re 3e+83)
       t_2
       (if (<= re 1.1e+134)
         (* (+ 1.0 (* -0.5 (* im im))) t_0)
         (if (<= re 1.4e+154) t_2 t_0))))))

C

double code(double re, double im) {
	double t_0 = re * (re * 0.5);
	double t_1 = -1.0 - t_0;
	double t_2 = ((re * re) + ((1.0 + t_0) * t_1)) / (re + t_1);
	double tmp;
	if (re <= -3400.0) {
		tmp = -0.5 * ((im * im) * (re + 1.0));
	} else if (re <= 3e+83) {
		tmp = t_2;
	} else if (re <= 1.1e+134) {
		tmp = (1.0 + (-0.5 * (im * im))) * t_0;
	} else if (re <= 1.4e+154) {
		tmp = t_2;
	} 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = re * (re * 0.5d0)
    t_1 = (-1.0d0) - t_0
    t_2 = ((re * re) + ((1.0d0 + t_0) * t_1)) / (re + t_1)
    if (re <= (-3400.0d0)) then
        tmp = (-0.5d0) * ((im * im) * (re + 1.0d0))
    else if (re <= 3d+83) then
        tmp = t_2
    else if (re <= 1.1d+134) then
        tmp = (1.0d0 + ((-0.5d0) * (im * im))) * t_0
    else if (re <= 1.4d+154) then
        tmp = t_2
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re * (re * 0.5);
	double t_1 = -1.0 - t_0;
	double t_2 = ((re * re) + ((1.0 + t_0) * t_1)) / (re + t_1);
	double tmp;
	if (re <= -3400.0) {
		tmp = -0.5 * ((im * im) * (re + 1.0));
	} else if (re <= 3e+83) {
		tmp = t_2;
	} else if (re <= 1.1e+134) {
		tmp = (1.0 + (-0.5 * (im * im))) * t_0;
	} else if (re <= 1.4e+154) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * (re * 0.5)
	t_1 = -1.0 - t_0
	t_2 = ((re * re) + ((1.0 + t_0) * t_1)) / (re + t_1)
	tmp = 0
	if re <= -3400.0:
		tmp = -0.5 * ((im * im) * (re + 1.0))
	elif re <= 3e+83:
		tmp = t_2
	elif re <= 1.1e+134:
		tmp = (1.0 + (-0.5 * (im * im))) * t_0
	elif re <= 1.4e+154:
		tmp = t_2
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(re * 0.5))
	t_1 = Float64(-1.0 - t_0)
	t_2 = Float64(Float64(Float64(re * re) + Float64(Float64(1.0 + t_0) * t_1)) / Float64(re + t_1))
	tmp = 0.0
	if (re <= -3400.0)
		tmp = Float64(-0.5 * Float64(Float64(im * im) * Float64(re + 1.0)));
	elseif (re <= 3e+83)
		tmp = t_2;
	elseif (re <= 1.1e+134)
		tmp = Float64(Float64(1.0 + Float64(-0.5 * Float64(im * im))) * t_0);
	elseif (re <= 1.4e+154)
		tmp = t_2;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * (re * 0.5);
	t_1 = -1.0 - t_0;
	t_2 = ((re * re) + ((1.0 + t_0) * t_1)) / (re + t_1);
	tmp = 0.0;
	if (re <= -3400.0)
		tmp = -0.5 * ((im * im) * (re + 1.0));
	elseif (re <= 3e+83)
		tmp = t_2;
	elseif (re <= 1.1e+134)
		tmp = (1.0 + (-0.5 * (im * im))) * t_0;
	elseif (re <= 1.4e+154)
		tmp = t_2;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-1.0 - t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(re * re), $MachinePrecision] + N[(N[(1.0 + t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(re + t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -3400.0], N[(-0.5 * N[(N[(im * im), $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 3e+83], t$95$2, If[LessEqual[re, 1.1e+134], N[(N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[re, 1.4e+154], t$95$2, t$95$0]]]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -3400

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 2.2%

      \[\leadsto \color{blue}{\cos im \cdot re + \cos im} \]
   3. Step-by-step derivation

      1. *-rgt-identity 2.2%

         \[\leadsto \cos im \cdot re + \color{blue}{\cos im \cdot 1} \]

      2. distribute-lft-out 2.2%

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

   4. Simplified 2.2%

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

   5. Taylor expanded in im around 0 2.0%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \cdot \left(re + 1\right) \]
   6. Step-by-step derivation

      1. unpow2 1.7%

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

   7. Simplified 2.0%

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

   8. Taylor expanded in im around inf 33.6%

      \[\leadsto \color{blue}{-0.5 \cdot \left(\left(1 + re\right) \cdot {im}^{2}\right)} \]
   9. Step-by-step derivation

      1. unpow2 33.6%

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

      2. +-commutative 33.6%

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

      3. *-commutative 33.6%

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

      4. +-commutative 33.6%

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

   10. Simplified 33.6%

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

   if -3400 < re < 3e83 or 1.1e134 < re < 1.4e154

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 82.5%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos im \cdot {re}^{2}\right) + \left(\cos im \cdot re + \cos im\right)} \]
   3. Step-by-step derivation

      1. +-commutative 82.5%

         \[\leadsto \color{blue}{\left(\cos im \cdot re + \cos im\right) + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right)} \]

      2. +-commutative 82.5%

         \[\leadsto \color{blue}{\left(\cos im + \cos im \cdot re\right)} + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right) \]

      3. *-rgt-identity 82.5%

         \[\leadsto \left(\color{blue}{\cos im \cdot 1} + \cos im \cdot re\right) + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right) \]

      4. distribute-lft-out 82.5%

         \[\leadsto \color{blue}{\cos im \cdot \left(1 + re\right)} + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right) \]

      5. *-commutative 82.5%

         \[\leadsto \cos im \cdot \left(1 + re\right) + \color{blue}{\left(\cos im \cdot {re}^{2}\right) \cdot 0.5} \]

      6. associate-*l* 82.5%

         \[\leadsto \cos im \cdot \left(1 + re\right) + \color{blue}{\cos im \cdot \left({re}^{2} \cdot 0.5\right)} \]

      7. distribute-lft-out 82.5%

         \[\leadsto \color{blue}{\cos im \cdot \left(\left(1 + re\right) + {re}^{2} \cdot 0.5\right)} \]

      8. +-commutative 82.5%

         \[\leadsto \cos im \cdot \left(\color{blue}{\left(re + 1\right)} + {re}^{2} \cdot 0.5\right) \]

      9. *-commutative 82.5%

         \[\leadsto \cos im \cdot \left(\left(re + 1\right) + \color{blue}{0.5 \cdot {re}^{2}}\right) \]

      10. unpow2 82.5%

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

   4. Simplified 82.5%

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

   5. Taylor expanded in im around 0 49.0%

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

      1. associate-+l+ 49.0%

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

      2. flip-+ 52.3%

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

      3. *-commutative 52.3%

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

      4. associate-*l* 52.3%

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

      5. *-commutative 52.3%

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

      6. associate-*l* 52.3%

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

      7. *-commutative 52.3%

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

      8. associate-*l* 52.3%

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

   7. Applied egg-rr 52.3%

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

   if 3e83 < re < 1.1e134

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 5.8%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos im \cdot {re}^{2}\right) + \left(\cos im \cdot re + \cos im\right)} \]
   3. Step-by-step derivation

      1. +-commutative 5.8%

         \[\leadsto \color{blue}{\left(\cos im \cdot re + \cos im\right) + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right)} \]

      2. +-commutative 5.8%

         \[\leadsto \color{blue}{\left(\cos im + \cos im \cdot re\right)} + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right) \]

      3. *-rgt-identity 5.8%

         \[\leadsto \left(\color{blue}{\cos im \cdot 1} + \cos im \cdot re\right) + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right) \]

      4. distribute-lft-out 5.8%

         \[\leadsto \color{blue}{\cos im \cdot \left(1 + re\right)} + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right) \]

      5. *-commutative 5.8%

         \[\leadsto \cos im \cdot \left(1 + re\right) + \color{blue}{\left(\cos im \cdot {re}^{2}\right) \cdot 0.5} \]

      6. associate-*l* 5.8%

         \[\leadsto \cos im \cdot \left(1 + re\right) + \color{blue}{\cos im \cdot \left({re}^{2} \cdot 0.5\right)} \]

      7. distribute-lft-out 5.8%

         \[\leadsto \color{blue}{\cos im \cdot \left(\left(1 + re\right) + {re}^{2} \cdot 0.5\right)} \]

      8. +-commutative 5.8%

         \[\leadsto \cos im \cdot \left(\color{blue}{\left(re + 1\right)} + {re}^{2} \cdot 0.5\right) \]

      9. *-commutative 5.8%

         \[\leadsto \cos im \cdot \left(\left(re + 1\right) + \color{blue}{0.5 \cdot {re}^{2}}\right) \]

      10. unpow2 5.8%

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

   4. Simplified 5.8%

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

   5. Taylor expanded in im around 0 70.2%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 70.2%

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

   7. Simplified 70.2%

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

   8. Taylor expanded in re around inf 70.2%

      \[\leadsto \left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left(0.5 \cdot {re}^{2}\right)} \]
   9. Step-by-step derivation

      1. unpow2 70.2%

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

      2. *-commutative 70.2%

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

      3. associate-*r* 70.2%

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

   10. Simplified 70.2%

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

   if 1.4e154 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos im \cdot {re}^{2}\right) + \left(\cos im \cdot re + \cos im\right)} \]
   3. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{\left(\cos im \cdot re + \cos im\right) + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right)} \]

      2. +-commutative 100.0%

         \[\leadsto \color{blue}{\left(\cos im + \cos im \cdot re\right)} + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right) \]

      3. *-rgt-identity 100.0%

         \[\leadsto \left(\color{blue}{\cos im \cdot 1} + \cos im \cdot re\right) + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right) \]

      4. distribute-lft-out 100.0%

         \[\leadsto \color{blue}{\cos im \cdot \left(1 + re\right)} + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right) \]

      5. *-commutative 100.0%

         \[\leadsto \cos im \cdot \left(1 + re\right) + \color{blue}{\left(\cos im \cdot {re}^{2}\right) \cdot 0.5} \]

      6. associate-*l* 100.0%

         \[\leadsto \cos im \cdot \left(1 + re\right) + \color{blue}{\cos im \cdot \left({re}^{2} \cdot 0.5\right)} \]

      7. distribute-lft-out 100.0%

         \[\leadsto \color{blue}{\cos im \cdot \left(\left(1 + re\right) + {re}^{2} \cdot 0.5\right)} \]

      8. +-commutative 100.0%

         \[\leadsto \cos im \cdot \left(\color{blue}{\left(re + 1\right)} + {re}^{2} \cdot 0.5\right) \]

      9. *-commutative 100.0%

         \[\leadsto \cos im \cdot \left(\left(re + 1\right) + \color{blue}{0.5 \cdot {re}^{2}}\right) \]

      10. unpow2 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around 0 76.9%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 76.9%

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

   7. Simplified 76.9%

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

   8. Taylor expanded in re around inf 76.9%

      \[\leadsto \left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left(0.5 \cdot {re}^{2}\right)} \]
   9. Step-by-step derivation

      1. unpow2 76.9%

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

      2. *-commutative 76.9%

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

      3. associate-*r* 76.9%

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

   10. Simplified 76.9%

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

   11. Taylor expanded in im around 0 76.9%

       \[\leadsto \color{blue}{0.5 \cdot {re}^{2}} \]
   12. Step-by-step derivation

       1. unpow2 76.9%

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

       2. *-commutative 76.9%

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

       3. associate-*r* 76.9%

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

   13. Simplified 76.9%

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

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3400:\\ \;\;\;\;-0.5 \cdot \left(\left(im \cdot im\right) \cdot \left(re + 1\right)\right)\\ \mathbf{elif}\;re \leq 3 \cdot 10^{+83}:\\ \;\;\;\;\frac{re \cdot re + \left(1 + re \cdot \left(re \cdot 0.5\right)\right) \cdot \left(-1 - re \cdot \left(re \cdot 0.5\right)\right)}{re + \left(-1 - re \cdot \left(re \cdot 0.5\right)\right)}\\ \mathbf{elif}\;re \leq 1.1 \cdot 10^{+134}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \mathbf{elif}\;re \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{re \cdot re + \left(1 + re \cdot \left(re \cdot 0.5\right)\right) \cdot \left(-1 - re \cdot \left(re \cdot 0.5\right)\right)}{re + \left(-1 - re \cdot \left(re \cdot 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \]

Alternative 8: 45.7% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(re \cdot re\right) + \left(re + 1\right)\\ \mathbf{if}\;re \leq -3400:\\ \;\;\;\;-0.5 \cdot \left(\left(im \cdot im\right) \cdot \left(re + 1\right)\right)\\ \mathbf{elif}\;re \leq 2.5 \cdot 10^{+19}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ (* 0.5 (* re re)) (+ re 1.0))))
   (if (<= re -3400.0)
     (* -0.5 (* (* im im) (+ re 1.0)))
     (if (<= re 2.5e+19) t_0 (* (+ 1.0 (* -0.5 (* im im))) t_0)))))

C

double code(double re, double im) {
	double t_0 = (0.5 * (re * re)) + (re + 1.0);
	double tmp;
	if (re <= -3400.0) {
		tmp = -0.5 * ((im * im) * (re + 1.0));
	} else if (re <= 2.5e+19) {
		tmp = t_0;
	} else {
		tmp = (1.0 + (-0.5 * (im * im))) * 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 = (0.5d0 * (re * re)) + (re + 1.0d0)
    if (re <= (-3400.0d0)) then
        tmp = (-0.5d0) * ((im * im) * (re + 1.0d0))
    else if (re <= 2.5d+19) then
        tmp = t_0
    else
        tmp = (1.0d0 + ((-0.5d0) * (im * im))) * t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (0.5 * (re * re)) + (re + 1.0);
	double tmp;
	if (re <= -3400.0) {
		tmp = -0.5 * ((im * im) * (re + 1.0));
	} else if (re <= 2.5e+19) {
		tmp = t_0;
	} else {
		tmp = (1.0 + (-0.5 * (im * im))) * t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (0.5 * (re * re)) + (re + 1.0)
	tmp = 0
	if re <= -3400.0:
		tmp = -0.5 * ((im * im) * (re + 1.0))
	elif re <= 2.5e+19:
		tmp = t_0
	else:
		tmp = (1.0 + (-0.5 * (im * im))) * t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * Float64(re * re)) + Float64(re + 1.0))
	tmp = 0.0
	if (re <= -3400.0)
		tmp = Float64(-0.5 * Float64(Float64(im * im) * Float64(re + 1.0)));
	elseif (re <= 2.5e+19)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 + Float64(-0.5 * Float64(im * im))) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (0.5 * (re * re)) + (re + 1.0);
	tmp = 0.0;
	if (re <= -3400.0)
		tmp = -0.5 * ((im * im) * (re + 1.0));
	elseif (re <= 2.5e+19)
		tmp = t_0;
	else
		tmp = (1.0 + (-0.5 * (im * im))) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision] + N[(re + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -3400.0], N[(-0.5 * N[(N[(im * im), $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.5e+19], t$95$0, N[(N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -3400

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 2.2%

      \[\leadsto \color{blue}{\cos im \cdot re + \cos im} \]
   3. Step-by-step derivation

      1. *-rgt-identity 2.2%

         \[\leadsto \cos im \cdot re + \color{blue}{\cos im \cdot 1} \]

      2. distribute-lft-out 2.2%

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

   4. Simplified 2.2%

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

   5. Taylor expanded in im around 0 2.0%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \cdot \left(re + 1\right) \]
   6. Step-by-step derivation

      1. unpow2 1.7%

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

   7. Simplified 2.0%

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

   8. Taylor expanded in im around inf 33.6%

      \[\leadsto \color{blue}{-0.5 \cdot \left(\left(1 + re\right) \cdot {im}^{2}\right)} \]
   9. Step-by-step derivation

      1. unpow2 33.6%

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

      2. +-commutative 33.6%

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

      3. *-commutative 33.6%

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

      4. +-commutative 33.6%

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

   10. Simplified 33.6%

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

   if -3400 < re < 2.5e19

   1. Initial program 99.9%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 94.7%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos im \cdot {re}^{2}\right) + \left(\cos im \cdot re + \cos im\right)} \]
   3. Step-by-step derivation

      1. +-commutative 94.7%

         \[\leadsto \color{blue}{\left(\cos im \cdot re + \cos im\right) + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right)} \]

      2. +-commutative 94.7%

         \[\leadsto \color{blue}{\left(\cos im + \cos im \cdot re\right)} + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right) \]

      3. *-rgt-identity 94.7%

         \[\leadsto \left(\color{blue}{\cos im \cdot 1} + \cos im \cdot re\right) + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right) \]

      4. distribute-lft-out 94.7%

         \[\leadsto \color{blue}{\cos im \cdot \left(1 + re\right)} + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right) \]

      5. *-commutative 94.7%

         \[\leadsto \cos im \cdot \left(1 + re\right) + \color{blue}{\left(\cos im \cdot {re}^{2}\right) \cdot 0.5} \]

      6. associate-*l* 94.7%

         \[\leadsto \cos im \cdot \left(1 + re\right) + \color{blue}{\cos im \cdot \left({re}^{2} \cdot 0.5\right)} \]

      7. distribute-lft-out 94.7%

         \[\leadsto \color{blue}{\cos im \cdot \left(\left(1 + re\right) + {re}^{2} \cdot 0.5\right)} \]

      8. +-commutative 94.7%

         \[\leadsto \cos im \cdot \left(\color{blue}{\left(re + 1\right)} + {re}^{2} \cdot 0.5\right) \]

      9. *-commutative 94.7%

         \[\leadsto \cos im \cdot \left(\left(re + 1\right) + \color{blue}{0.5 \cdot {re}^{2}}\right) \]

      10. unpow2 94.7%

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

   4. Simplified 94.7%

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

   5. Taylor expanded in im around 0 55.9%

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

   if 2.5e19 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 45.8%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos im \cdot {re}^{2}\right) + \left(\cos im \cdot re + \cos im\right)} \]
   3. Step-by-step derivation

      1. +-commutative 45.8%

         \[\leadsto \color{blue}{\left(\cos im \cdot re + \cos im\right) + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right)} \]

      2. +-commutative 45.8%

         \[\leadsto \color{blue}{\left(\cos im + \cos im \cdot re\right)} + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right) \]

      3. *-rgt-identity 45.8%

         \[\leadsto \left(\color{blue}{\cos im \cdot 1} + \cos im \cdot re\right) + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right) \]

      4. distribute-lft-out 45.8%

         \[\leadsto \color{blue}{\cos im \cdot \left(1 + re\right)} + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right) \]

      5. *-commutative 45.8%

         \[\leadsto \cos im \cdot \left(1 + re\right) + \color{blue}{\left(\cos im \cdot {re}^{2}\right) \cdot 0.5} \]

      6. associate-*l* 45.8%

         \[\leadsto \cos im \cdot \left(1 + re\right) + \color{blue}{\cos im \cdot \left({re}^{2} \cdot 0.5\right)} \]

      7. distribute-lft-out 45.8%

         \[\leadsto \color{blue}{\cos im \cdot \left(\left(1 + re\right) + {re}^{2} \cdot 0.5\right)} \]

      8. +-commutative 45.8%

         \[\leadsto \cos im \cdot \left(\color{blue}{\left(re + 1\right)} + {re}^{2} \cdot 0.5\right) \]

      9. *-commutative 45.8%

         \[\leadsto \cos im \cdot \left(\left(re + 1\right) + \color{blue}{0.5 \cdot {re}^{2}}\right) \]

      10. unpow2 45.8%

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

   4. Simplified 45.8%

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

   5. Taylor expanded in im around 0 50.6%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 50.6%

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

   7. Simplified 50.6%

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

4. Final simplification 49.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3400:\\ \;\;\;\;-0.5 \cdot \left(\left(im \cdot im\right) \cdot \left(re + 1\right)\right)\\ \mathbf{elif}\;re \leq 2.5 \cdot 10^{+19}:\\ \;\;\;\;0.5 \cdot \left(re \cdot re\right) + \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(0.5 \cdot \left(re \cdot re\right) + \left(re + 1\right)\right)\\ \end{array} \]

Alternative 9: 45.7% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3400:\\ \;\;\;\;-0.5 \cdot \left(\left(im \cdot im\right) \cdot \left(re + 1\right)\right)\\ \mathbf{elif}\;re \leq 2.6 \cdot 10^{+19}:\\ \;\;\;\;0.5 \cdot \left(re \cdot re\right) + \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3400.0)
   (* -0.5 (* (* im im) (+ re 1.0)))
   (if (<= re 2.6e+19)
     (+ (* 0.5 (* re re)) (+ re 1.0))
     (* (+ 1.0 (* -0.5 (* im im))) (* re (* re 0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3400.0) {
		tmp = -0.5 * ((im * im) * (re + 1.0));
	} else if (re <= 2.6e+19) {
		tmp = (0.5 * (re * re)) + (re + 1.0);
	} else {
		tmp = (1.0 + (-0.5 * (im * im))) * (re * (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 <= (-3400.0d0)) then
        tmp = (-0.5d0) * ((im * im) * (re + 1.0d0))
    else if (re <= 2.6d+19) then
        tmp = (0.5d0 * (re * re)) + (re + 1.0d0)
    else
        tmp = (1.0d0 + ((-0.5d0) * (im * im))) * (re * (re * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -3400.0) {
		tmp = -0.5 * ((im * im) * (re + 1.0));
	} else if (re <= 2.6e+19) {
		tmp = (0.5 * (re * re)) + (re + 1.0);
	} else {
		tmp = (1.0 + (-0.5 * (im * im))) * (re * (re * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3400.0:
		tmp = -0.5 * ((im * im) * (re + 1.0))
	elif re <= 2.6e+19:
		tmp = (0.5 * (re * re)) + (re + 1.0)
	else:
		tmp = (1.0 + (-0.5 * (im * im))) * (re * (re * 0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3400.0)
		tmp = Float64(-0.5 * Float64(Float64(im * im) * Float64(re + 1.0)));
	elseif (re <= 2.6e+19)
		tmp = Float64(Float64(0.5 * Float64(re * re)) + Float64(re + 1.0));
	else
		tmp = Float64(Float64(1.0 + Float64(-0.5 * Float64(im * im))) * Float64(re * Float64(re * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3400.0)
		tmp = -0.5 * ((im * im) * (re + 1.0));
	elseif (re <= 2.6e+19)
		tmp = (0.5 * (re * re)) + (re + 1.0);
	else
		tmp = (1.0 + (-0.5 * (im * im))) * (re * (re * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3400.0], N[(-0.5 * N[(N[(im * im), $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.6e+19], N[(N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision] + N[(re + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -3400

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 2.2%

      \[\leadsto \color{blue}{\cos im \cdot re + \cos im} \]
   3. Step-by-step derivation

      1. *-rgt-identity 2.2%

         \[\leadsto \cos im \cdot re + \color{blue}{\cos im \cdot 1} \]

      2. distribute-lft-out 2.2%

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

   4. Simplified 2.2%

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

   5. Taylor expanded in im around 0 2.0%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \cdot \left(re + 1\right) \]
   6. Step-by-step derivation

      1. unpow2 1.7%

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

   7. Simplified 2.0%

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

   8. Taylor expanded in im around inf 33.6%

      \[\leadsto \color{blue}{-0.5 \cdot \left(\left(1 + re\right) \cdot {im}^{2}\right)} \]
   9. Step-by-step derivation

      1. unpow2 33.6%

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

      2. +-commutative 33.6%

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

      3. *-commutative 33.6%

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

      4. +-commutative 33.6%

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

   10. Simplified 33.6%

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

   if -3400 < re < 2.6e19

   1. Initial program 99.9%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 94.7%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos im \cdot {re}^{2}\right) + \left(\cos im \cdot re + \cos im\right)} \]
   3. Step-by-step derivation

      1. +-commutative 94.7%

         \[\leadsto \color{blue}{\left(\cos im \cdot re + \cos im\right) + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right)} \]

      2. +-commutative 94.7%

         \[\leadsto \color{blue}{\left(\cos im + \cos im \cdot re\right)} + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right) \]

      3. *-rgt-identity 94.7%

         \[\leadsto \left(\color{blue}{\cos im \cdot 1} + \cos im \cdot re\right) + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right) \]

      4. distribute-lft-out 94.7%

         \[\leadsto \color{blue}{\cos im \cdot \left(1 + re\right)} + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right) \]

      5. *-commutative 94.7%

         \[\leadsto \cos im \cdot \left(1 + re\right) + \color{blue}{\left(\cos im \cdot {re}^{2}\right) \cdot 0.5} \]

      6. associate-*l* 94.7%

         \[\leadsto \cos im \cdot \left(1 + re\right) + \color{blue}{\cos im \cdot \left({re}^{2} \cdot 0.5\right)} \]

      7. distribute-lft-out 94.7%

         \[\leadsto \color{blue}{\cos im \cdot \left(\left(1 + re\right) + {re}^{2} \cdot 0.5\right)} \]

      8. +-commutative 94.7%

         \[\leadsto \cos im \cdot \left(\color{blue}{\left(re + 1\right)} + {re}^{2} \cdot 0.5\right) \]

      9. *-commutative 94.7%

         \[\leadsto \cos im \cdot \left(\left(re + 1\right) + \color{blue}{0.5 \cdot {re}^{2}}\right) \]

      10. unpow2 94.7%

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

   4. Simplified 94.7%

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

   5. Taylor expanded in im around 0 55.9%

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

   if 2.6e19 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 45.8%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos im \cdot {re}^{2}\right) + \left(\cos im \cdot re + \cos im\right)} \]
   3. Step-by-step derivation

      1. +-commutative 45.8%

         \[\leadsto \color{blue}{\left(\cos im \cdot re + \cos im\right) + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right)} \]

      2. +-commutative 45.8%

         \[\leadsto \color{blue}{\left(\cos im + \cos im \cdot re\right)} + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right) \]

      3. *-rgt-identity 45.8%

         \[\leadsto \left(\color{blue}{\cos im \cdot 1} + \cos im \cdot re\right) + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right) \]

      4. distribute-lft-out 45.8%

         \[\leadsto \color{blue}{\cos im \cdot \left(1 + re\right)} + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right) \]

      5. *-commutative 45.8%

         \[\leadsto \cos im \cdot \left(1 + re\right) + \color{blue}{\left(\cos im \cdot {re}^{2}\right) \cdot 0.5} \]

      6. associate-*l* 45.8%

         \[\leadsto \cos im \cdot \left(1 + re\right) + \color{blue}{\cos im \cdot \left({re}^{2} \cdot 0.5\right)} \]

      7. distribute-lft-out 45.8%

         \[\leadsto \color{blue}{\cos im \cdot \left(\left(1 + re\right) + {re}^{2} \cdot 0.5\right)} \]

      8. +-commutative 45.8%

         \[\leadsto \cos im \cdot \left(\color{blue}{\left(re + 1\right)} + {re}^{2} \cdot 0.5\right) \]

      9. *-commutative 45.8%

         \[\leadsto \cos im \cdot \left(\left(re + 1\right) + \color{blue}{0.5 \cdot {re}^{2}}\right) \]

      10. unpow2 45.8%

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

   4. Simplified 45.8%

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

   5. Taylor expanded in im around 0 50.6%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 50.6%

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

   7. Simplified 50.6%

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

   8. Taylor expanded in re around inf 50.6%

      \[\leadsto \left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left(0.5 \cdot {re}^{2}\right)} \]
   9. Step-by-step derivation

      1. unpow2 50.6%

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

      2. *-commutative 50.6%

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

      3. associate-*r* 50.6%

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

   10. Simplified 50.6%

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

4. Final simplification 49.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3400:\\ \;\;\;\;-0.5 \cdot \left(\left(im \cdot im\right) \cdot \left(re + 1\right)\right)\\ \mathbf{elif}\;re \leq 2.6 \cdot 10^{+19}:\\ \;\;\;\;0.5 \cdot \left(re \cdot re\right) + \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 10: 45.0% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.5 \cdot \left(\left(im \cdot im\right) \cdot \left(re + 1\right)\right)\\ \mathbf{if}\;re \leq -3400:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 1.85 \cdot 10^{+25}:\\ \;\;\;\;re + 1\\ \mathbf{elif}\;re \leq 6.8 \cdot 10^{+136}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* -0.5 (* (* im im) (+ re 1.0)))))
   (if (<= re -3400.0)
     t_0
     (if (<= re 1.85e+25)
       (+ re 1.0)
       (if (<= re 6.8e+136) t_0 (* re (* re 0.5)))))))

C

double code(double re, double im) {
	double t_0 = -0.5 * ((im * im) * (re + 1.0));
	double tmp;
	if (re <= -3400.0) {
		tmp = t_0;
	} else if (re <= 1.85e+25) {
		tmp = re + 1.0;
	} else if (re <= 6.8e+136) {
		tmp = t_0;
	} else {
		tmp = re * (re * 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 = (-0.5d0) * ((im * im) * (re + 1.0d0))
    if (re <= (-3400.0d0)) then
        tmp = t_0
    else if (re <= 1.85d+25) then
        tmp = re + 1.0d0
    else if (re <= 6.8d+136) then
        tmp = t_0
    else
        tmp = re * (re * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = -0.5 * ((im * im) * (re + 1.0));
	double tmp;
	if (re <= -3400.0) {
		tmp = t_0;
	} else if (re <= 1.85e+25) {
		tmp = re + 1.0;
	} else if (re <= 6.8e+136) {
		tmp = t_0;
	} else {
		tmp = re * (re * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = -0.5 * ((im * im) * (re + 1.0))
	tmp = 0
	if re <= -3400.0:
		tmp = t_0
	elif re <= 1.85e+25:
		tmp = re + 1.0
	elif re <= 6.8e+136:
		tmp = t_0
	else:
		tmp = re * (re * 0.5)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(-0.5 * Float64(Float64(im * im) * Float64(re + 1.0)))
	tmp = 0.0
	if (re <= -3400.0)
		tmp = t_0;
	elseif (re <= 1.85e+25)
		tmp = Float64(re + 1.0);
	elseif (re <= 6.8e+136)
		tmp = t_0;
	else
		tmp = Float64(re * Float64(re * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = -0.5 * ((im * im) * (re + 1.0));
	tmp = 0.0;
	if (re <= -3400.0)
		tmp = t_0;
	elseif (re <= 1.85e+25)
		tmp = re + 1.0;
	elseif (re <= 6.8e+136)
		tmp = t_0;
	else
		tmp = re * (re * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(-0.5 * N[(N[(im * im), $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -3400.0], t$95$0, If[LessEqual[re, 1.85e+25], N[(re + 1.0), $MachinePrecision], If[LessEqual[re, 6.8e+136], t$95$0, N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -3400 or 1.8499999999999999e25 < re < 6.79999999999999993e136

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 2.7%

      \[\leadsto \color{blue}{\cos im \cdot re + \cos im} \]
   3. Step-by-step derivation

      1. *-rgt-identity 2.7%

         \[\leadsto \cos im \cdot re + \color{blue}{\cos im \cdot 1} \]

      2. distribute-lft-out 2.7%

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

   4. Simplified 2.7%

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

   5. Taylor expanded in im around 0 12.3%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \cdot \left(re + 1\right) \]
   6. Step-by-step derivation

      1. unpow2 13.4%

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

   7. Simplified 12.3%

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

   8. Taylor expanded in im around inf 32.7%

      \[\leadsto \color{blue}{-0.5 \cdot \left(\left(1 + re\right) \cdot {im}^{2}\right)} \]
   9. Step-by-step derivation

      1. unpow2 32.7%

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

      2. +-commutative 32.7%

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

      3. *-commutative 32.7%

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

      4. +-commutative 32.7%

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

   10. Simplified 32.7%

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

   if -3400 < re < 1.8499999999999999e25

   1. Initial program 99.9%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 94.6%

      \[\leadsto \color{blue}{\cos im \cdot re + \cos im} \]
   3. Step-by-step derivation

      1. *-rgt-identity 94.6%

         \[\leadsto \cos im \cdot re + \color{blue}{\cos im \cdot 1} \]

      2. distribute-lft-out 94.6%

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

   4. Simplified 94.6%

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

   5. Taylor expanded in im around 0 55.9%

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

   if 6.79999999999999993e136 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 85.4%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos im \cdot {re}^{2}\right) + \left(\cos im \cdot re + \cos im\right)} \]
   3. Step-by-step derivation

      1. +-commutative 85.4%

         \[\leadsto \color{blue}{\left(\cos im \cdot re + \cos im\right) + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right)} \]

      2. +-commutative 85.4%

         \[\leadsto \color{blue}{\left(\cos im + \cos im \cdot re\right)} + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right) \]

      3. *-rgt-identity 85.4%

         \[\leadsto \left(\color{blue}{\cos im \cdot 1} + \cos im \cdot re\right) + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right) \]

      4. distribute-lft-out 85.4%

         \[\leadsto \color{blue}{\cos im \cdot \left(1 + re\right)} + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right) \]

      5. *-commutative 85.4%

         \[\leadsto \cos im \cdot \left(1 + re\right) + \color{blue}{\left(\cos im \cdot {re}^{2}\right) \cdot 0.5} \]

      6. associate-*l* 85.4%

         \[\leadsto \cos im \cdot \left(1 + re\right) + \color{blue}{\cos im \cdot \left({re}^{2} \cdot 0.5\right)} \]

      7. distribute-lft-out 85.4%

         \[\leadsto \color{blue}{\cos im \cdot \left(\left(1 + re\right) + {re}^{2} \cdot 0.5\right)} \]

      8. +-commutative 85.4%

         \[\leadsto \cos im \cdot \left(\color{blue}{\left(re + 1\right)} + {re}^{2} \cdot 0.5\right) \]

      9. *-commutative 85.4%

         \[\leadsto \cos im \cdot \left(\left(re + 1\right) + \color{blue}{0.5 \cdot {re}^{2}}\right) \]

      10. unpow2 85.4%

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

   4. Simplified 85.4%

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

   5. Taylor expanded in im around 0 65.4%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 65.4%

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

   7. Simplified 65.4%

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

   8. Taylor expanded in re around inf 65.4%

      \[\leadsto \left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left(0.5 \cdot {re}^{2}\right)} \]
   9. Step-by-step derivation

      1. unpow2 65.4%

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

      2. *-commutative 65.4%

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

      3. associate-*r* 65.4%

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

   10. Simplified 65.4%

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

   11. Taylor expanded in im around 0 66.1%

       \[\leadsto \color{blue}{0.5 \cdot {re}^{2}} \]
   12. Step-by-step derivation

       1. unpow2 66.1%

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

       2. *-commutative 66.1%

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

       3. associate-*r* 66.1%

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

   13. Simplified 66.1%

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

4. Final simplification 49.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3400:\\ \;\;\;\;-0.5 \cdot \left(\left(im \cdot im\right) \cdot \left(re + 1\right)\right)\\ \mathbf{elif}\;re \leq 1.85 \cdot 10^{+25}:\\ \;\;\;\;re + 1\\ \mathbf{elif}\;re \leq 6.8 \cdot 10^{+136}:\\ \;\;\;\;-0.5 \cdot \left(\left(im \cdot im\right) \cdot \left(re + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \]

Alternative 11: 45.4% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3400:\\ \;\;\;\;-0.5 \cdot \left(\left(im \cdot im\right) \cdot \left(re + 1\right)\right)\\ \mathbf{elif}\;re \leq 2.5 \cdot 10^{+19}:\\ \;\;\;\;re + 1\\ \mathbf{elif}\;re \leq 2.7 \cdot 10^{+140}:\\ \;\;\;\;-0.25 \cdot \left(re \cdot \left(re \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3400.0)
   (* -0.5 (* (* im im) (+ re 1.0)))
   (if (<= re 2.5e+19)
     (+ re 1.0)
     (if (<= re 2.7e+140)
       (* -0.25 (* re (* re (* im im))))
       (* re (* re 0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3400.0) {
		tmp = -0.5 * ((im * im) * (re + 1.0));
	} else if (re <= 2.5e+19) {
		tmp = re + 1.0;
	} else if (re <= 2.7e+140) {
		tmp = -0.25 * (re * (re * (im * im)));
	} else {
		tmp = re * (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 <= (-3400.0d0)) then
        tmp = (-0.5d0) * ((im * im) * (re + 1.0d0))
    else if (re <= 2.5d+19) then
        tmp = re + 1.0d0
    else if (re <= 2.7d+140) then
        tmp = (-0.25d0) * (re * (re * (im * im)))
    else
        tmp = re * (re * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -3400.0) {
		tmp = -0.5 * ((im * im) * (re + 1.0));
	} else if (re <= 2.5e+19) {
		tmp = re + 1.0;
	} else if (re <= 2.7e+140) {
		tmp = -0.25 * (re * (re * (im * im)));
	} else {
		tmp = re * (re * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3400.0:
		tmp = -0.5 * ((im * im) * (re + 1.0))
	elif re <= 2.5e+19:
		tmp = re + 1.0
	elif re <= 2.7e+140:
		tmp = -0.25 * (re * (re * (im * im)))
	else:
		tmp = re * (re * 0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3400.0)
		tmp = Float64(-0.5 * Float64(Float64(im * im) * Float64(re + 1.0)));
	elseif (re <= 2.5e+19)
		tmp = Float64(re + 1.0);
	elseif (re <= 2.7e+140)
		tmp = Float64(-0.25 * Float64(re * Float64(re * Float64(im * im))));
	else
		tmp = Float64(re * Float64(re * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3400.0)
		tmp = -0.5 * ((im * im) * (re + 1.0));
	elseif (re <= 2.5e+19)
		tmp = re + 1.0;
	elseif (re <= 2.7e+140)
		tmp = -0.25 * (re * (re * (im * im)));
	else
		tmp = re * (re * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3400.0], N[(-0.5 * N[(N[(im * im), $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.5e+19], N[(re + 1.0), $MachinePrecision], If[LessEqual[re, 2.7e+140], N[(-0.25 * N[(re * N[(re * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -3400

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 2.2%

      \[\leadsto \color{blue}{\cos im \cdot re + \cos im} \]
   3. Step-by-step derivation

      1. *-rgt-identity 2.2%

         \[\leadsto \cos im \cdot re + \color{blue}{\cos im \cdot 1} \]

      2. distribute-lft-out 2.2%

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

   4. Simplified 2.2%

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

   5. Taylor expanded in im around 0 2.0%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \cdot \left(re + 1\right) \]
   6. Step-by-step derivation

      1. unpow2 1.7%

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

   7. Simplified 2.0%

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

   8. Taylor expanded in im around inf 33.6%

      \[\leadsto \color{blue}{-0.5 \cdot \left(\left(1 + re\right) \cdot {im}^{2}\right)} \]
   9. Step-by-step derivation

      1. unpow2 33.6%

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

      2. +-commutative 33.6%

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

      3. *-commutative 33.6%

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

      4. +-commutative 33.6%

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

   10. Simplified 33.6%

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

   if -3400 < re < 2.5e19

   1. Initial program 99.9%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 94.6%

      \[\leadsto \color{blue}{\cos im \cdot re + \cos im} \]
   3. Step-by-step derivation

      1. *-rgt-identity 94.6%

         \[\leadsto \cos im \cdot re + \color{blue}{\cos im \cdot 1} \]

      2. distribute-lft-out 94.6%

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

   4. Simplified 94.6%

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

   5. Taylor expanded in im around 0 55.9%

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

   if 2.5e19 < re < 2.70000000000000018e140

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 4.8%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos im \cdot {re}^{2}\right) + \left(\cos im \cdot re + \cos im\right)} \]
   3. Step-by-step derivation

      1. +-commutative 4.8%

         \[\leadsto \color{blue}{\left(\cos im \cdot re + \cos im\right) + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right)} \]

      2. +-commutative 4.8%

         \[\leadsto \color{blue}{\left(\cos im + \cos im \cdot re\right)} + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right) \]

      3. *-rgt-identity 4.8%

         \[\leadsto \left(\color{blue}{\cos im \cdot 1} + \cos im \cdot re\right) + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right) \]

      4. distribute-lft-out 4.8%

         \[\leadsto \color{blue}{\cos im \cdot \left(1 + re\right)} + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right) \]

      5. *-commutative 4.8%

         \[\leadsto \cos im \cdot \left(1 + re\right) + \color{blue}{\left(\cos im \cdot {re}^{2}\right) \cdot 0.5} \]

      6. associate-*l* 4.8%

         \[\leadsto \cos im \cdot \left(1 + re\right) + \color{blue}{\cos im \cdot \left({re}^{2} \cdot 0.5\right)} \]

      7. distribute-lft-out 4.8%

         \[\leadsto \color{blue}{\cos im \cdot \left(\left(1 + re\right) + {re}^{2} \cdot 0.5\right)} \]

      8. +-commutative 4.8%

         \[\leadsto \cos im \cdot \left(\color{blue}{\left(re + 1\right)} + {re}^{2} \cdot 0.5\right) \]

      9. *-commutative 4.8%

         \[\leadsto \cos im \cdot \left(\left(re + 1\right) + \color{blue}{0.5 \cdot {re}^{2}}\right) \]

      10. unpow2 4.8%

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

   4. Simplified 4.8%

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

   5. Taylor expanded in im around 0 35.3%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 35.3%

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

   7. Simplified 35.3%

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

   8. Taylor expanded in re around inf 35.3%

      \[\leadsto \left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left(0.5 \cdot {re}^{2}\right)} \]
   9. Step-by-step derivation

      1. unpow2 35.3%

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

      2. *-commutative 35.3%

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

      3. associate-*r* 35.3%

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

   10. Simplified 35.3%

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

   11. Taylor expanded in im around inf 34.3%

       \[\leadsto \color{blue}{-0.25 \cdot \left({re}^{2} \cdot {im}^{2}\right)} \]
   12. Step-by-step derivation

       1. unpow2 34.3%

          \[\leadsto -0.25 \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot {im}^{2}\right) \]

       2. unpow2 34.3%

          \[\leadsto -0.25 \cdot \left(\left(re \cdot re\right) \cdot \color{blue}{\left(im \cdot im\right)}\right) \]

       3. associate-*l* 34.3%

          \[\leadsto -0.25 \cdot \color{blue}{\left(re \cdot \left(re \cdot \left(im \cdot im\right)\right)\right)} \]

   13. Simplified 34.3%

       \[\leadsto \color{blue}{-0.25 \cdot \left(re \cdot \left(re \cdot \left(im \cdot im\right)\right)\right)} \]

   if 2.70000000000000018e140 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 85.4%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos im \cdot {re}^{2}\right) + \left(\cos im \cdot re + \cos im\right)} \]
   3. Step-by-step derivation

      1. +-commutative 85.4%

         \[\leadsto \color{blue}{\left(\cos im \cdot re + \cos im\right) + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right)} \]

      2. +-commutative 85.4%

         \[\leadsto \color{blue}{\left(\cos im + \cos im \cdot re\right)} + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right) \]

      3. *-rgt-identity 85.4%

         \[\leadsto \left(\color{blue}{\cos im \cdot 1} + \cos im \cdot re\right) + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right) \]

      4. distribute-lft-out 85.4%

         \[\leadsto \color{blue}{\cos im \cdot \left(1 + re\right)} + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right) \]

      5. *-commutative 85.4%

         \[\leadsto \cos im \cdot \left(1 + re\right) + \color{blue}{\left(\cos im \cdot {re}^{2}\right) \cdot 0.5} \]

      6. associate-*l* 85.4%

         \[\leadsto \cos im \cdot \left(1 + re\right) + \color{blue}{\cos im \cdot \left({re}^{2} \cdot 0.5\right)} \]

      7. distribute-lft-out 85.4%

         \[\leadsto \color{blue}{\cos im \cdot \left(\left(1 + re\right) + {re}^{2} \cdot 0.5\right)} \]

      8. +-commutative 85.4%

         \[\leadsto \cos im \cdot \left(\color{blue}{\left(re + 1\right)} + {re}^{2} \cdot 0.5\right) \]

      9. *-commutative 85.4%

         \[\leadsto \cos im \cdot \left(\left(re + 1\right) + \color{blue}{0.5 \cdot {re}^{2}}\right) \]

      10. unpow2 85.4%

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

   4. Simplified 85.4%

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

   5. Taylor expanded in im around 0 65.4%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 65.4%

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

   7. Simplified 65.4%

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

   8. Taylor expanded in re around inf 65.4%

      \[\leadsto \left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left(0.5 \cdot {re}^{2}\right)} \]
   9. Step-by-step derivation

      1. unpow2 65.4%

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

      2. *-commutative 65.4%

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

      3. associate-*r* 65.4%

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

   10. Simplified 65.4%

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

   11. Taylor expanded in im around 0 66.1%

       \[\leadsto \color{blue}{0.5 \cdot {re}^{2}} \]
   12. Step-by-step derivation

       1. unpow2 66.1%

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

       2. *-commutative 66.1%

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

       3. associate-*r* 66.1%

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

   13. Simplified 66.1%

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

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3400:\\ \;\;\;\;-0.5 \cdot \left(\left(im \cdot im\right) \cdot \left(re + 1\right)\right)\\ \mathbf{elif}\;re \leq 2.5 \cdot 10^{+19}:\\ \;\;\;\;re + 1\\ \mathbf{elif}\;re \leq 2.7 \cdot 10^{+140}:\\ \;\;\;\;-0.25 \cdot \left(re \cdot \left(re \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \]

Alternative 12: 45.5% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3400:\\ \;\;\;\;-0.5 \cdot \left(\left(im \cdot im\right) \cdot \left(re + 1\right)\right)\\ \mathbf{elif}\;re \leq 2.35 \cdot 10^{+20}:\\ \;\;\;\;0.5 \cdot \left(re \cdot re\right) + \left(re + 1\right)\\ \mathbf{elif}\;re \leq 2.7 \cdot 10^{+140}:\\ \;\;\;\;-0.25 \cdot \left(re \cdot \left(re \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3400.0)
   (* -0.5 (* (* im im) (+ re 1.0)))
   (if (<= re 2.35e+20)
     (+ (* 0.5 (* re re)) (+ re 1.0))
     (if (<= re 2.7e+140)
       (* -0.25 (* re (* re (* im im))))
       (* re (* re 0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3400.0) {
		tmp = -0.5 * ((im * im) * (re + 1.0));
	} else if (re <= 2.35e+20) {
		tmp = (0.5 * (re * re)) + (re + 1.0);
	} else if (re <= 2.7e+140) {
		tmp = -0.25 * (re * (re * (im * im)));
	} else {
		tmp = re * (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 <= (-3400.0d0)) then
        tmp = (-0.5d0) * ((im * im) * (re + 1.0d0))
    else if (re <= 2.35d+20) then
        tmp = (0.5d0 * (re * re)) + (re + 1.0d0)
    else if (re <= 2.7d+140) then
        tmp = (-0.25d0) * (re * (re * (im * im)))
    else
        tmp = re * (re * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -3400.0) {
		tmp = -0.5 * ((im * im) * (re + 1.0));
	} else if (re <= 2.35e+20) {
		tmp = (0.5 * (re * re)) + (re + 1.0);
	} else if (re <= 2.7e+140) {
		tmp = -0.25 * (re * (re * (im * im)));
	} else {
		tmp = re * (re * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3400.0:
		tmp = -0.5 * ((im * im) * (re + 1.0))
	elif re <= 2.35e+20:
		tmp = (0.5 * (re * re)) + (re + 1.0)
	elif re <= 2.7e+140:
		tmp = -0.25 * (re * (re * (im * im)))
	else:
		tmp = re * (re * 0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3400.0)
		tmp = Float64(-0.5 * Float64(Float64(im * im) * Float64(re + 1.0)));
	elseif (re <= 2.35e+20)
		tmp = Float64(Float64(0.5 * Float64(re * re)) + Float64(re + 1.0));
	elseif (re <= 2.7e+140)
		tmp = Float64(-0.25 * Float64(re * Float64(re * Float64(im * im))));
	else
		tmp = Float64(re * Float64(re * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3400.0)
		tmp = -0.5 * ((im * im) * (re + 1.0));
	elseif (re <= 2.35e+20)
		tmp = (0.5 * (re * re)) + (re + 1.0);
	elseif (re <= 2.7e+140)
		tmp = -0.25 * (re * (re * (im * im)));
	else
		tmp = re * (re * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3400.0], N[(-0.5 * N[(N[(im * im), $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.35e+20], N[(N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision] + N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.7e+140], N[(-0.25 * N[(re * N[(re * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -3400

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 2.2%

      \[\leadsto \color{blue}{\cos im \cdot re + \cos im} \]
   3. Step-by-step derivation

      1. *-rgt-identity 2.2%

         \[\leadsto \cos im \cdot re + \color{blue}{\cos im \cdot 1} \]

      2. distribute-lft-out 2.2%

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

   4. Simplified 2.2%

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

   5. Taylor expanded in im around 0 2.0%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \cdot \left(re + 1\right) \]
   6. Step-by-step derivation

      1. unpow2 1.7%

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

   7. Simplified 2.0%

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

   8. Taylor expanded in im around inf 33.6%

      \[\leadsto \color{blue}{-0.5 \cdot \left(\left(1 + re\right) \cdot {im}^{2}\right)} \]
   9. Step-by-step derivation

      1. unpow2 33.6%

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

      2. +-commutative 33.6%

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

      3. *-commutative 33.6%

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

      4. +-commutative 33.6%

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

   10. Simplified 33.6%

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

   if -3400 < re < 2.35e20

   1. Initial program 99.9%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 94.7%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos im \cdot {re}^{2}\right) + \left(\cos im \cdot re + \cos im\right)} \]
   3. Step-by-step derivation

      1. +-commutative 94.7%

         \[\leadsto \color{blue}{\left(\cos im \cdot re + \cos im\right) + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right)} \]

      2. +-commutative 94.7%

         \[\leadsto \color{blue}{\left(\cos im + \cos im \cdot re\right)} + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right) \]

      3. *-rgt-identity 94.7%

         \[\leadsto \left(\color{blue}{\cos im \cdot 1} + \cos im \cdot re\right) + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right) \]

      4. distribute-lft-out 94.7%

         \[\leadsto \color{blue}{\cos im \cdot \left(1 + re\right)} + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right) \]

      5. *-commutative 94.7%

         \[\leadsto \cos im \cdot \left(1 + re\right) + \color{blue}{\left(\cos im \cdot {re}^{2}\right) \cdot 0.5} \]

      6. associate-*l* 94.7%

         \[\leadsto \cos im \cdot \left(1 + re\right) + \color{blue}{\cos im \cdot \left({re}^{2} \cdot 0.5\right)} \]

      7. distribute-lft-out 94.7%

         \[\leadsto \color{blue}{\cos im \cdot \left(\left(1 + re\right) + {re}^{2} \cdot 0.5\right)} \]

      8. +-commutative 94.7%

         \[\leadsto \cos im \cdot \left(\color{blue}{\left(re + 1\right)} + {re}^{2} \cdot 0.5\right) \]

      9. *-commutative 94.7%

         \[\leadsto \cos im \cdot \left(\left(re + 1\right) + \color{blue}{0.5 \cdot {re}^{2}}\right) \]

      10. unpow2 94.7%

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

   4. Simplified 94.7%

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

   5. Taylor expanded in im around 0 55.9%

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

   if 2.35e20 < re < 2.70000000000000018e140

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 4.8%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos im \cdot {re}^{2}\right) + \left(\cos im \cdot re + \cos im\right)} \]
   3. Step-by-step derivation

      1. +-commutative 4.8%

         \[\leadsto \color{blue}{\left(\cos im \cdot re + \cos im\right) + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right)} \]

      2. +-commutative 4.8%

         \[\leadsto \color{blue}{\left(\cos im + \cos im \cdot re\right)} + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right) \]

      3. *-rgt-identity 4.8%

         \[\leadsto \left(\color{blue}{\cos im \cdot 1} + \cos im \cdot re\right) + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right) \]

      4. distribute-lft-out 4.8%

         \[\leadsto \color{blue}{\cos im \cdot \left(1 + re\right)} + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right) \]

      5. *-commutative 4.8%

         \[\leadsto \cos im \cdot \left(1 + re\right) + \color{blue}{\left(\cos im \cdot {re}^{2}\right) \cdot 0.5} \]

      6. associate-*l* 4.8%

         \[\leadsto \cos im \cdot \left(1 + re\right) + \color{blue}{\cos im \cdot \left({re}^{2} \cdot 0.5\right)} \]

      7. distribute-lft-out 4.8%

         \[\leadsto \color{blue}{\cos im \cdot \left(\left(1 + re\right) + {re}^{2} \cdot 0.5\right)} \]

      8. +-commutative 4.8%

         \[\leadsto \cos im \cdot \left(\color{blue}{\left(re + 1\right)} + {re}^{2} \cdot 0.5\right) \]

      9. *-commutative 4.8%

         \[\leadsto \cos im \cdot \left(\left(re + 1\right) + \color{blue}{0.5 \cdot {re}^{2}}\right) \]

      10. unpow2 4.8%

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

   4. Simplified 4.8%

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

   5. Taylor expanded in im around 0 35.3%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 35.3%

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

   7. Simplified 35.3%

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

   8. Taylor expanded in re around inf 35.3%

      \[\leadsto \left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left(0.5 \cdot {re}^{2}\right)} \]
   9. Step-by-step derivation

      1. unpow2 35.3%

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

      2. *-commutative 35.3%

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

      3. associate-*r* 35.3%

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

   10. Simplified 35.3%

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

   11. Taylor expanded in im around inf 34.3%

       \[\leadsto \color{blue}{-0.25 \cdot \left({re}^{2} \cdot {im}^{2}\right)} \]
   12. Step-by-step derivation

       1. unpow2 34.3%

          \[\leadsto -0.25 \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot {im}^{2}\right) \]

       2. unpow2 34.3%

          \[\leadsto -0.25 \cdot \left(\left(re \cdot re\right) \cdot \color{blue}{\left(im \cdot im\right)}\right) \]

       3. associate-*l* 34.3%

          \[\leadsto -0.25 \cdot \color{blue}{\left(re \cdot \left(re \cdot \left(im \cdot im\right)\right)\right)} \]

   13. Simplified 34.3%

       \[\leadsto \color{blue}{-0.25 \cdot \left(re \cdot \left(re \cdot \left(im \cdot im\right)\right)\right)} \]

   if 2.70000000000000018e140 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 85.4%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos im \cdot {re}^{2}\right) + \left(\cos im \cdot re + \cos im\right)} \]
   3. Step-by-step derivation

      1. +-commutative 85.4%

         \[\leadsto \color{blue}{\left(\cos im \cdot re + \cos im\right) + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right)} \]

      2. +-commutative 85.4%

         \[\leadsto \color{blue}{\left(\cos im + \cos im \cdot re\right)} + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right) \]

      3. *-rgt-identity 85.4%

         \[\leadsto \left(\color{blue}{\cos im \cdot 1} + \cos im \cdot re\right) + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right) \]

      4. distribute-lft-out 85.4%

         \[\leadsto \color{blue}{\cos im \cdot \left(1 + re\right)} + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right) \]

      5. *-commutative 85.4%

         \[\leadsto \cos im \cdot \left(1 + re\right) + \color{blue}{\left(\cos im \cdot {re}^{2}\right) \cdot 0.5} \]

      6. associate-*l* 85.4%

         \[\leadsto \cos im \cdot \left(1 + re\right) + \color{blue}{\cos im \cdot \left({re}^{2} \cdot 0.5\right)} \]

      7. distribute-lft-out 85.4%

         \[\leadsto \color{blue}{\cos im \cdot \left(\left(1 + re\right) + {re}^{2} \cdot 0.5\right)} \]

      8. +-commutative 85.4%

         \[\leadsto \cos im \cdot \left(\color{blue}{\left(re + 1\right)} + {re}^{2} \cdot 0.5\right) \]

      9. *-commutative 85.4%

         \[\leadsto \cos im \cdot \left(\left(re + 1\right) + \color{blue}{0.5 \cdot {re}^{2}}\right) \]

      10. unpow2 85.4%

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

   4. Simplified 85.4%

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

   5. Taylor expanded in im around 0 65.4%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 65.4%

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

   7. Simplified 65.4%

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

   8. Taylor expanded in re around inf 65.4%

      \[\leadsto \left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left(0.5 \cdot {re}^{2}\right)} \]
   9. Step-by-step derivation

      1. unpow2 65.4%

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

      2. *-commutative 65.4%

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

      3. associate-*r* 65.4%

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

   10. Simplified 65.4%

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

   11. Taylor expanded in im around 0 66.1%

       \[\leadsto \color{blue}{0.5 \cdot {re}^{2}} \]
   12. Step-by-step derivation

       1. unpow2 66.1%

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

       2. *-commutative 66.1%

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

       3. associate-*r* 66.1%

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

   13. Simplified 66.1%

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

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3400:\\ \;\;\;\;-0.5 \cdot \left(\left(im \cdot im\right) \cdot \left(re + 1\right)\right)\\ \mathbf{elif}\;re \leq 2.35 \cdot 10^{+20}:\\ \;\;\;\;0.5 \cdot \left(re \cdot re\right) + \left(re + 1\right)\\ \mathbf{elif}\;re \leq 2.7 \cdot 10^{+140}:\\ \;\;\;\;-0.25 \cdot \left(re \cdot \left(re \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \]

Alternative 13: 39.3% accurate, 18.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 2.5 \cdot 10^{+19}:\\ \;\;\;\;1\\ \mathbf{elif}\;re \leq 2.7 \cdot 10^{+140}:\\ \;\;\;\;1 + -0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 2.5e+19)
   1.0
   (if (<= re 2.7e+140) (+ 1.0 (* -0.5 (* im im))) (* re (* re 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 2.5e+19) {
		tmp = 1.0;
	} else if (re <= 2.7e+140) {
		tmp = 1.0 + (-0.5 * (im * im));
	} else {
		tmp = re * (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 <= 2.5d+19) then
        tmp = 1.0d0
    else if (re <= 2.7d+140) then
        tmp = 1.0d0 + ((-0.5d0) * (im * im))
    else
        tmp = re * (re * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 2.5e+19) {
		tmp = 1.0;
	} else if (re <= 2.7e+140) {
		tmp = 1.0 + (-0.5 * (im * im));
	} else {
		tmp = re * (re * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 2.5e+19:
		tmp = 1.0
	elif re <= 2.7e+140:
		tmp = 1.0 + (-0.5 * (im * im))
	else:
		tmp = re * (re * 0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 2.5e+19)
		tmp = 1.0;
	elseif (re <= 2.7e+140)
		tmp = Float64(1.0 + Float64(-0.5 * Float64(im * im)));
	else
		tmp = Float64(re * Float64(re * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 2.5e+19)
		tmp = 1.0;
	elseif (re <= 2.7e+140)
		tmp = 1.0 + (-0.5 * (im * im));
	else
		tmp = re * (re * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 2.5e+19], 1.0, If[LessEqual[re, 2.7e+140], N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < 2.5e19

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 68.1%

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

   3. Taylor expanded in im around 0 40.6%

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

   if 2.5e19 < re < 2.70000000000000018e140

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 3.1%

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

   3. Taylor expanded in im around 0 25.3%

      \[\leadsto \color{blue}{1 + -0.5 \cdot {im}^{2}} \]
   4. Step-by-step derivation

      1. unpow2 35.3%

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

   5. Simplified 25.3%

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

   if 2.70000000000000018e140 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 85.4%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos im \cdot {re}^{2}\right) + \left(\cos im \cdot re + \cos im\right)} \]
   3. Step-by-step derivation

      1. +-commutative 85.4%

         \[\leadsto \color{blue}{\left(\cos im \cdot re + \cos im\right) + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right)} \]

      2. +-commutative 85.4%

         \[\leadsto \color{blue}{\left(\cos im + \cos im \cdot re\right)} + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right) \]

      3. *-rgt-identity 85.4%

         \[\leadsto \left(\color{blue}{\cos im \cdot 1} + \cos im \cdot re\right) + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right) \]

      4. distribute-lft-out 85.4%

         \[\leadsto \color{blue}{\cos im \cdot \left(1 + re\right)} + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right) \]

      5. *-commutative 85.4%

         \[\leadsto \cos im \cdot \left(1 + re\right) + \color{blue}{\left(\cos im \cdot {re}^{2}\right) \cdot 0.5} \]

      6. associate-*l* 85.4%

         \[\leadsto \cos im \cdot \left(1 + re\right) + \color{blue}{\cos im \cdot \left({re}^{2} \cdot 0.5\right)} \]

      7. distribute-lft-out 85.4%

         \[\leadsto \color{blue}{\cos im \cdot \left(\left(1 + re\right) + {re}^{2} \cdot 0.5\right)} \]

      8. +-commutative 85.4%

         \[\leadsto \cos im \cdot \left(\color{blue}{\left(re + 1\right)} + {re}^{2} \cdot 0.5\right) \]

      9. *-commutative 85.4%

         \[\leadsto \cos im \cdot \left(\left(re + 1\right) + \color{blue}{0.5 \cdot {re}^{2}}\right) \]

      10. unpow2 85.4%

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

   4. Simplified 85.4%

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

   5. Taylor expanded in im around 0 65.4%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 65.4%

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

   7. Simplified 65.4%

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

   8. Taylor expanded in re around inf 65.4%

      \[\leadsto \left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left(0.5 \cdot {re}^{2}\right)} \]
   9. Step-by-step derivation

      1. unpow2 65.4%

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

      2. *-commutative 65.4%

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

      3. associate-*r* 65.4%

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

   10. Simplified 65.4%

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

   11. Taylor expanded in im around 0 66.1%

       \[\leadsto \color{blue}{0.5 \cdot {re}^{2}} \]
   12. Step-by-step derivation

       1. unpow2 66.1%

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

       2. *-commutative 66.1%

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

       3. associate-*r* 66.1%

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

   13. Simplified 66.1%

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

4. Final simplification 41.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 2.5 \cdot 10^{+19}:\\ \;\;\;\;1\\ \mathbf{elif}\;re \leq 2.7 \cdot 10^{+140}:\\ \;\;\;\;1 + -0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \]

Alternative 14: 38.1% accurate, 28.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.42:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= re 1.42) 1.0 (* re (* re 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.42) {
		tmp = 1.0;
	} else {
		tmp = re * (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 <= 1.42d0) then
        tmp = 1.0d0
    else
        tmp = re * (re * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.42) {
		tmp = 1.0;
	} else {
		tmp = re * (re * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.42:
		tmp = 1.0
	else:
		tmp = re * (re * 0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.42)
		tmp = 1.0;
	else
		tmp = Float64(re * Float64(re * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.42)
		tmp = 1.0;
	else
		tmp = re * (re * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.42], 1.0, N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.4199999999999999

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 70.2%

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

   3. Taylor expanded in im around 0 41.8%

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

   if 1.4199999999999999 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0 42.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos im \cdot {re}^{2}\right) + \left(\cos im \cdot re + \cos im\right)} \]
   3. Step-by-step derivation

      1. +-commutative 42.0%

         \[\leadsto \color{blue}{\left(\cos im \cdot re + \cos im\right) + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right)} \]

      2. +-commutative 42.0%

         \[\leadsto \color{blue}{\left(\cos im + \cos im \cdot re\right)} + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right) \]

      3. *-rgt-identity 42.0%

         \[\leadsto \left(\color{blue}{\cos im \cdot 1} + \cos im \cdot re\right) + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right) \]

      4. distribute-lft-out 42.0%

         \[\leadsto \color{blue}{\cos im \cdot \left(1 + re\right)} + 0.5 \cdot \left(\cos im \cdot {re}^{2}\right) \]

      5. *-commutative 42.0%

         \[\leadsto \cos im \cdot \left(1 + re\right) + \color{blue}{\left(\cos im \cdot {re}^{2}\right) \cdot 0.5} \]

      6. associate-*l* 42.0%

         \[\leadsto \cos im \cdot \left(1 + re\right) + \color{blue}{\cos im \cdot \left({re}^{2} \cdot 0.5\right)} \]

      7. distribute-lft-out 42.0%

         \[\leadsto \color{blue}{\cos im \cdot \left(\left(1 + re\right) + {re}^{2} \cdot 0.5\right)} \]

      8. +-commutative 42.0%

         \[\leadsto \cos im \cdot \left(\color{blue}{\left(re + 1\right)} + {re}^{2} \cdot 0.5\right) \]

      9. *-commutative 42.0%

         \[\leadsto \cos im \cdot \left(\left(re + 1\right) + \color{blue}{0.5 \cdot {re}^{2}}\right) \]

      10. unpow2 42.0%

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

   4. Simplified 42.0%

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

   5. Taylor expanded in im around 0 46.2%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 46.2%

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

   7. Simplified 46.2%

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

   8. Taylor expanded in re around inf 46.2%

      \[\leadsto \left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left(0.5 \cdot {re}^{2}\right)} \]
   9. Step-by-step derivation

      1. unpow2 46.2%

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

      2. *-commutative 46.2%

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

      3. associate-*r* 46.2%

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

   10. Simplified 46.2%

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

   11. Taylor expanded in im around 0 32.1%

       \[\leadsto \color{blue}{0.5 \cdot {re}^{2}} \]
   12. Step-by-step derivation

       1. unpow2 32.1%

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

       2. *-commutative 32.1%

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

       3. associate-*r* 32.1%

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

   13. Simplified 32.1%

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

4. Final simplification 39.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.42:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \]

Alternative 15: 28.9% 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. Taylor expanded in re around 0 53.1%

   \[\leadsto \color{blue}{\cos im \cdot re + \cos im} \]
3. Step-by-step derivation

   1. *-rgt-identity 53.1%

      \[\leadsto \cos im \cdot re + \color{blue}{\cos im \cdot 1} \]

   2. distribute-lft-out 53.0%

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

4. Simplified 53.0%

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

5. Taylor expanded in im around 0 31.7%

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

6. Final simplification 31.7%

   \[\leadsto re + 1 \]

Alternative 16: 28.5% 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. Taylor expanded in re around 0 52.6%

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

3. Taylor expanded in im around 0 31.5%

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

4. Final simplification 31.5%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023188 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))