math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 7.5s

Alternatives: 14

Speedup: 1.0×

• 25.2% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 1.00000005:\\ \;\;\;\;re + \cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (exp re) 2e-10)
   (exp re)
   (if (<= (exp re) 1.00000005) (+ re (cos im)) (exp re))))

C

double code(double re, double im) {
	double tmp;
	if (exp(re) <= 2e-10) {
		tmp = exp(re);
	} else if (exp(re) <= 1.00000005) {
		tmp = re + 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) <= 2d-10) then
        tmp = exp(re)
    else if (exp(re) <= 1.00000005d0) then
        tmp = re + 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) <= 2e-10) {
		tmp = Math.exp(re);
	} else if (Math.exp(re) <= 1.00000005) {
		tmp = re + Math.cos(im);
	} else {
		tmp = Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.exp(re) <= 2e-10:
		tmp = math.exp(re)
	elif math.exp(re) <= 1.00000005:
		tmp = re + math.cos(im)
	else:
		tmp = math.exp(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (exp(re) <= 2e-10)
		tmp = exp(re);
	elseif (exp(re) <= 1.00000005)
		tmp = Float64(re + cos(im));
	else
		tmp = exp(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (exp(re) <= 2e-10)
		tmp = exp(re);
	elseif (exp(re) <= 1.00000005)
		tmp = re + cos(im);
	else
		tmp = exp(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Exp[re], $MachinePrecision], 2e-10], N[Exp[re], $MachinePrecision], If[LessEqual[N[Exp[re], $MachinePrecision], 1.00000005], N[(re + N[Cos[im], $MachinePrecision]), $MachinePrecision], N[Exp[re], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 2.00000000000000007e-10 or 1.00000004999999992 < (exp.f64 re)

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 84.5%

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

   if 2.00000000000000007e-10 < (exp.f64 re) < 1.00000004999999992

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.7%

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

      1. distribute-rgt1-in 99.7%

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

   4. Simplified 99.7%

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

      1. *-commutative 99.7%

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

      2. distribute-lft-in 99.7%

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

      3. *-commutative 99.7%

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

      4. *-un-lft-identity 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in im around 0 99.0%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 1.00000005:\\ \;\;\;\;re + \cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]

Alternative 3: 92.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 1.00000005:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (exp re) 2e-10)
   (exp re)
   (if (<= (exp re) 1.00000005) (cos im) (exp re))))

C

double code(double re, double im) {
	double tmp;
	if (exp(re) <= 2e-10) {
		tmp = exp(re);
	} else if (exp(re) <= 1.00000005) {
		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) <= 2d-10) then
        tmp = exp(re)
    else if (exp(re) <= 1.00000005d0) 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) <= 2e-10) {
		tmp = Math.exp(re);
	} else if (Math.exp(re) <= 1.00000005) {
		tmp = Math.cos(im);
	} else {
		tmp = Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.exp(re) <= 2e-10:
		tmp = math.exp(re)
	elif math.exp(re) <= 1.00000005:
		tmp = math.cos(im)
	else:
		tmp = math.exp(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (exp(re) <= 2e-10)
		tmp = exp(re);
	elseif (exp(re) <= 1.00000005)
		tmp = cos(im);
	else
		tmp = exp(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (exp(re) <= 2e-10)
		tmp = exp(re);
	elseif (exp(re) <= 1.00000005)
		tmp = cos(im);
	else
		tmp = exp(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Exp[re], $MachinePrecision], 2e-10], N[Exp[re], $MachinePrecision], If[LessEqual[N[Exp[re], $MachinePrecision], 1.00000005], N[Cos[im], $MachinePrecision], N[Exp[re], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 2.00000000000000007e-10 or 1.00000004999999992 < (exp.f64 re)

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 84.5%

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

   if 2.00000000000000007e-10 < (exp.f64 re) < 1.00000004999999992

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 98.8%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 1.00000005:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]

Alternative 4: 97.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.0205:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 1700000 \lor \neg \left(re \leq 1.05 \cdot 10^{+103}\right):\\ \;\;\;\;\cos im \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot 0.16666666666666666 + 0.5\right) + \left(re + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.0205)
   (exp re)
   (if (or (<= re 1700000.0) (not (<= re 1.05e+103)))
     (*
      (cos im)
      (+ (* (* re re) (+ (* re 0.16666666666666666) 0.5)) (+ re 1.0)))
     (* (exp re) (+ 1.0 (* -0.5 (* im im)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.0205) {
		tmp = exp(re);
	} else if ((re <= 1700000.0) || !(re <= 1.05e+103)) {
		tmp = cos(im) * (((re * re) * ((re * 0.16666666666666666) + 0.5)) + (re + 1.0));
	} 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 <= (-0.0205d0)) then
        tmp = exp(re)
    else if ((re <= 1700000.0d0) .or. (.not. (re <= 1.05d+103))) then
        tmp = cos(im) * (((re * re) * ((re * 0.16666666666666666d0) + 0.5d0)) + (re + 1.0d0))
    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 <= -0.0205) {
		tmp = Math.exp(re);
	} else if ((re <= 1700000.0) || !(re <= 1.05e+103)) {
		tmp = Math.cos(im) * (((re * re) * ((re * 0.16666666666666666) + 0.5)) + (re + 1.0));
	} else {
		tmp = Math.exp(re) * (1.0 + (-0.5 * (im * im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.0205:
		tmp = math.exp(re)
	elif (re <= 1700000.0) or not (re <= 1.05e+103):
		tmp = math.cos(im) * (((re * re) * ((re * 0.16666666666666666) + 0.5)) + (re + 1.0))
	else:
		tmp = math.exp(re) * (1.0 + (-0.5 * (im * im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.0205)
		tmp = exp(re);
	elseif ((re <= 1700000.0) || !(re <= 1.05e+103))
		tmp = Float64(cos(im) * Float64(Float64(Float64(re * re) * Float64(Float64(re * 0.16666666666666666) + 0.5)) + Float64(re + 1.0)));
	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 <= -0.0205)
		tmp = exp(re);
	elseif ((re <= 1700000.0) || ~((re <= 1.05e+103)))
		tmp = cos(im) * (((re * re) * ((re * 0.16666666666666666) + 0.5)) + (re + 1.0));
	else
		tmp = exp(re) * (1.0 + (-0.5 * (im * im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.0205], N[Exp[re], $MachinePrecision], If[Or[LessEqual[re, 1700000.0], N[Not[LessEqual[re, 1.05e+103]], $MachinePrecision]], N[(N[Cos[im], $MachinePrecision] * N[(N[(N[(re * re), $MachinePrecision] * N[(N[(re * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + N[(re + 1.0), $MachinePrecision]), $MachinePrecision]), $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 < -0.0205000000000000009

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 96.5%

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

   if -0.0205000000000000009 < re < 1.7e6 or 1.0500000000000001e103 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Step-by-step derivation

      1. add-cube-cbrt 99.1%

         \[\leadsto e^{re} \cdot \color{blue}{\left(\left(\sqrt[3]{\cos im} \cdot \sqrt[3]{\cos im}\right) \cdot \sqrt[3]{\cos im}\right)} \]

      2. pow2 99.1%

         \[\leadsto e^{re} \cdot \left(\color{blue}{{\left(\sqrt[3]{\cos im}\right)}^{2}} \cdot \sqrt[3]{\cos im}\right) \]

   3. Applied egg-rr 99.1%

      \[\leadsto e^{re} \cdot \color{blue}{\left({\left(\sqrt[3]{\cos im}\right)}^{2} \cdot \sqrt[3]{\cos im}\right)} \]

   4. Taylor expanded in re around 0 99.4%

      \[\leadsto \color{blue}{\cos im + \left(0.16666666666666666 \cdot \left({re}^{3} \cdot \cos im\right) + \left(0.5 \cdot \left({re}^{2} \cdot \cos im\right) + re \cdot \cos im\right)\right)} \]
   5. Step-by-step derivation

      1. +-commutative 99.4%

         \[\leadsto \color{blue}{\left(0.16666666666666666 \cdot \left({re}^{3} \cdot \cos im\right) + \left(0.5 \cdot \left({re}^{2} \cdot \cos im\right) + re \cdot \cos im\right)\right) + \cos im} \]

      2. associate-+r+ 99.4%

         \[\leadsto \color{blue}{\left(\left(0.16666666666666666 \cdot \left({re}^{3} \cdot \cos im\right) + 0.5 \cdot \left({re}^{2} \cdot \cos im\right)\right) + re \cdot \cos im\right)} + \cos im \]

      3. associate-+l+ 99.4%

         \[\leadsto \color{blue}{\left(0.16666666666666666 \cdot \left({re}^{3} \cdot \cos im\right) + 0.5 \cdot \left({re}^{2} \cdot \cos im\right)\right) + \left(re \cdot \cos im + \cos im\right)} \]

      4. associate-*r* 99.4%

         \[\leadsto \left(\color{blue}{\left(0.16666666666666666 \cdot {re}^{3}\right) \cdot \cos im} + 0.5 \cdot \left({re}^{2} \cdot \cos im\right)\right) + \left(re \cdot \cos im + \cos im\right) \]

      5. associate-*r* 99.4%

         \[\leadsto \left(\left(0.16666666666666666 \cdot {re}^{3}\right) \cdot \cos im + \color{blue}{\left(0.5 \cdot {re}^{2}\right) \cdot \cos im}\right) + \left(re \cdot \cos im + \cos im\right) \]

      6. distribute-rgt-out 99.4%

         \[\leadsto \color{blue}{\cos im \cdot \left(0.16666666666666666 \cdot {re}^{3} + 0.5 \cdot {re}^{2}\right)} + \left(re \cdot \cos im + \cos im\right) \]

      7. distribute-lft1-in 99.4%

         \[\leadsto \cos im \cdot \left(0.16666666666666666 \cdot {re}^{3} + 0.5 \cdot {re}^{2}\right) + \color{blue}{\left(re + 1\right) \cdot \cos im} \]

      8. *-commutative 99.4%

         \[\leadsto \cos im \cdot \left(0.16666666666666666 \cdot {re}^{3} + 0.5 \cdot {re}^{2}\right) + \color{blue}{\cos im \cdot \left(re + 1\right)} \]

      9. distribute-lft-out 99.4%

         \[\leadsto \color{blue}{\cos im \cdot \left(\left(0.16666666666666666 \cdot {re}^{3} + 0.5 \cdot {re}^{2}\right) + \left(re + 1\right)\right)} \]

   6. Simplified 99.4%

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

   if 1.7e6 < re < 1.0500000000000001e103

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 77.3%

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

      1. unpow2 20.3%

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

   4. Simplified 77.3%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0205:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 1700000 \lor \neg \left(re \leq 1.05 \cdot 10^{+103}\right):\\ \;\;\;\;\cos im \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot 0.16666666666666666 + 0.5\right) + \left(re + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 5: 96.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.0063:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 1700000 \lor \neg \left(re \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;\cos im \cdot \left(1 + \left(re + \left(re \cdot re\right) \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.0063)
   (exp re)
   (if (or (<= re 1700000.0) (not (<= re 1.35e+154)))
     (* (cos im) (+ 1.0 (+ re (* (* re re) 0.5))))
     (* (exp re) (+ 1.0 (* -0.5 (* im im)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.0063) {
		tmp = exp(re);
	} else if ((re <= 1700000.0) || !(re <= 1.35e+154)) {
		tmp = cos(im) * (1.0 + (re + ((re * re) * 0.5)));
	} 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 <= (-0.0063d0)) then
        tmp = exp(re)
    else if ((re <= 1700000.0d0) .or. (.not. (re <= 1.35d+154))) then
        tmp = cos(im) * (1.0d0 + (re + ((re * re) * 0.5d0)))
    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 <= -0.0063) {
		tmp = Math.exp(re);
	} else if ((re <= 1700000.0) || !(re <= 1.35e+154)) {
		tmp = Math.cos(im) * (1.0 + (re + ((re * re) * 0.5)));
	} else {
		tmp = Math.exp(re) * (1.0 + (-0.5 * (im * im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.0063:
		tmp = math.exp(re)
	elif (re <= 1700000.0) or not (re <= 1.35e+154):
		tmp = math.cos(im) * (1.0 + (re + ((re * re) * 0.5)))
	else:
		tmp = math.exp(re) * (1.0 + (-0.5 * (im * im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.0063)
		tmp = exp(re);
	elseif ((re <= 1700000.0) || !(re <= 1.35e+154))
		tmp = Float64(cos(im) * Float64(1.0 + Float64(re + Float64(Float64(re * re) * 0.5))));
	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 <= -0.0063)
		tmp = exp(re);
	elseif ((re <= 1700000.0) || ~((re <= 1.35e+154)))
		tmp = cos(im) * (1.0 + (re + ((re * re) * 0.5)));
	else
		tmp = exp(re) * (1.0 + (-0.5 * (im * im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.0063], N[Exp[re], $MachinePrecision], If[Or[LessEqual[re, 1700000.0], N[Not[LessEqual[re, 1.35e+154]], $MachinePrecision]], N[(N[Cos[im], $MachinePrecision] * N[(1.0 + N[(re + N[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 < -0.0063

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 96.5%

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

   if -0.0063 < re < 1.7e6 or 1.35000000000000003e154 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Step-by-step derivation

      1. add-cube-cbrt 99.1%

         \[\leadsto e^{re} \cdot \color{blue}{\left(\left(\sqrt[3]{\cos im} \cdot \sqrt[3]{\cos im}\right) \cdot \sqrt[3]{\cos im}\right)} \]

      2. pow2 99.1%

         \[\leadsto e^{re} \cdot \left(\color{blue}{{\left(\sqrt[3]{\cos im}\right)}^{2}} \cdot \sqrt[3]{\cos im}\right) \]

   3. Applied egg-rr 99.1%

      \[\leadsto e^{re} \cdot \color{blue}{\left({\left(\sqrt[3]{\cos im}\right)}^{2} \cdot \sqrt[3]{\cos im}\right)} \]

   4. Taylor expanded in re around 0 99.4%

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

      1. *-rgt-identity 99.4%

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

      2. +-commutative 99.4%

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

      3. associate-*r* 99.4%

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

      4. distribute-rgt-out 99.4%

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

      5. distribute-lft-out 99.3%

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

      6. unpow2 99.3%

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

   6. Simplified 99.3%

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

   if 1.7e6 < re < 1.35000000000000003e154

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 78.6%

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

      1. unpow2 16.6%

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

   4. Simplified 78.6%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0063:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 1700000 \lor \neg \left(re \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;\cos im \cdot \left(1 + \left(re + \left(re \cdot re\right) \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 6: 93.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.0305:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 5.5 \cdot 10^{-8}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.0305)
   (exp re)
   (if (<= re 5.5e-8) (* (cos im) (+ re 1.0)) (exp re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.0305) {
		tmp = exp(re);
	} else if (re <= 5.5e-8) {
		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 (re <= (-0.0305d0)) then
        tmp = exp(re)
    else if (re <= 5.5d-8) 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 (re <= -0.0305) {
		tmp = Math.exp(re);
	} else if (re <= 5.5e-8) {
		tmp = Math.cos(im) * (re + 1.0);
	} else {
		tmp = Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.0305:
		tmp = math.exp(re)
	elif re <= 5.5e-8:
		tmp = math.cos(im) * (re + 1.0)
	else:
		tmp = math.exp(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.0305)
		tmp = exp(re);
	elseif (re <= 5.5e-8)
		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 (re <= -0.0305)
		tmp = exp(re);
	elseif (re <= 5.5e-8)
		tmp = cos(im) * (re + 1.0);
	else
		tmp = exp(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.0305], N[Exp[re], $MachinePrecision], If[LessEqual[re, 5.5e-8], N[(N[Cos[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], N[Exp[re], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if re < -0.030499999999999999 or 5.5000000000000003e-8 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 84.5%

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

   if -0.030499999999999999 < re < 5.5000000000000003e-8

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.7%

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

      1. distribute-rgt1-in 99.7%

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

   4. Simplified 99.7%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0305:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 5.5 \cdot 10^{-8}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]

Alternative 7: 69.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.8 \cdot 10^{+27}:\\ \;\;\;\;-0.5 \cdot \left(\left(re + 1\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 2.95 \cdot 10^{-8}:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot \left(re \cdot 0.16666666666666666 + 0.5\right) + \left(re + 1\right)\right) \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.8e+27)
   (* -0.5 (* (+ re 1.0) (* im im)))
   (if (<= re 2.95e-8)
     (cos im)
     (*
      (+ (* (* re re) (+ (* re 0.16666666666666666) 0.5)) (+ re 1.0))
      (+ 1.0 (* -0.5 (* im im)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.8e+27) {
		tmp = -0.5 * ((re + 1.0) * (im * im));
	} else if (re <= 2.95e-8) {
		tmp = cos(im);
	} else {
		tmp = (((re * re) * ((re * 0.16666666666666666) + 0.5)) + (re + 1.0)) * (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 <= (-2.8d+27)) then
        tmp = (-0.5d0) * ((re + 1.0d0) * (im * im))
    else if (re <= 2.95d-8) then
        tmp = cos(im)
    else
        tmp = (((re * re) * ((re * 0.16666666666666666d0) + 0.5d0)) + (re + 1.0d0)) * (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 <= -2.8e+27) {
		tmp = -0.5 * ((re + 1.0) * (im * im));
	} else if (re <= 2.95e-8) {
		tmp = Math.cos(im);
	} else {
		tmp = (((re * re) * ((re * 0.16666666666666666) + 0.5)) + (re + 1.0)) * (1.0 + (-0.5 * (im * im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2.8e+27:
		tmp = -0.5 * ((re + 1.0) * (im * im))
	elif re <= 2.95e-8:
		tmp = math.cos(im)
	else:
		tmp = (((re * re) * ((re * 0.16666666666666666) + 0.5)) + (re + 1.0)) * (1.0 + (-0.5 * (im * im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.8e+27)
		tmp = Float64(-0.5 * Float64(Float64(re + 1.0) * Float64(im * im)));
	elseif (re <= 2.95e-8)
		tmp = cos(im);
	else
		tmp = Float64(Float64(Float64(Float64(re * re) * Float64(Float64(re * 0.16666666666666666) + 0.5)) + Float64(re + 1.0)) * 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 <= -2.8e+27)
		tmp = -0.5 * ((re + 1.0) * (im * im));
	elseif (re <= 2.95e-8)
		tmp = cos(im);
	else
		tmp = (((re * re) * ((re * 0.16666666666666666) + 0.5)) + (re + 1.0)) * (1.0 + (-0.5 * (im * im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.8e+27], N[(-0.5 * N[(N[(re + 1.0), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.95e-8], N[Cos[im], $MachinePrecision], N[(N[(N[(N[(re * re), $MachinePrecision] * N[(N[(re * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + N[(re + 1.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.7999999999999999e27

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 2.1%

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

      1. distribute-rgt1-in 2.1%

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

   4. Simplified 2.1%

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

   5. Taylor expanded in im around 0 1.9%

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

      1. unpow2 2.7%

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

   7. Simplified 1.9%

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

   8. Taylor expanded in im around inf 28.2%

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

      1. +-commutative 28.2%

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

      2. unpow2 28.2%

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

   10. Simplified 28.2%

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

   if -2.7999999999999999e27 < re < 2.9499999999999999e-8

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 94.4%

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

   if 2.9499999999999999e-8 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Step-by-step derivation

      1. add-cube-cbrt 100.0%

         \[\leadsto e^{re} \cdot \color{blue}{\left(\left(\sqrt[3]{\cos im} \cdot \sqrt[3]{\cos im}\right) \cdot \sqrt[3]{\cos im}\right)} \]

      2. pow2 100.0%

         \[\leadsto e^{re} \cdot \left(\color{blue}{{\left(\sqrt[3]{\cos im}\right)}^{2}} \cdot \sqrt[3]{\cos im}\right) \]

   3. Applied egg-rr 100.0%

      \[\leadsto e^{re} \cdot \color{blue}{\left({\left(\sqrt[3]{\cos im}\right)}^{2} \cdot \sqrt[3]{\cos im}\right)} \]

   4. Taylor expanded in re around 0 63.6%

      \[\leadsto \color{blue}{\cos im + \left(0.16666666666666666 \cdot \left({re}^{3} \cdot \cos im\right) + \left(0.5 \cdot \left({re}^{2} \cdot \cos im\right) + re \cdot \cos im\right)\right)} \]
   5. Step-by-step derivation

      1. +-commutative 63.6%

         \[\leadsto \color{blue}{\left(0.16666666666666666 \cdot \left({re}^{3} \cdot \cos im\right) + \left(0.5 \cdot \left({re}^{2} \cdot \cos im\right) + re \cdot \cos im\right)\right) + \cos im} \]

      2. associate-+r+ 63.6%

         \[\leadsto \color{blue}{\left(\left(0.16666666666666666 \cdot \left({re}^{3} \cdot \cos im\right) + 0.5 \cdot \left({re}^{2} \cdot \cos im\right)\right) + re \cdot \cos im\right)} + \cos im \]

      3. associate-+l+ 63.6%

         \[\leadsto \color{blue}{\left(0.16666666666666666 \cdot \left({re}^{3} \cdot \cos im\right) + 0.5 \cdot \left({re}^{2} \cdot \cos im\right)\right) + \left(re \cdot \cos im + \cos im\right)} \]

      4. associate-*r* 63.6%

         \[\leadsto \left(\color{blue}{\left(0.16666666666666666 \cdot {re}^{3}\right) \cdot \cos im} + 0.5 \cdot \left({re}^{2} \cdot \cos im\right)\right) + \left(re \cdot \cos im + \cos im\right) \]

      5. associate-*r* 63.6%

         \[\leadsto \left(\left(0.16666666666666666 \cdot {re}^{3}\right) \cdot \cos im + \color{blue}{\left(0.5 \cdot {re}^{2}\right) \cdot \cos im}\right) + \left(re \cdot \cos im + \cos im\right) \]

      6. distribute-rgt-out 63.6%

         \[\leadsto \color{blue}{\cos im \cdot \left(0.16666666666666666 \cdot {re}^{3} + 0.5 \cdot {re}^{2}\right)} + \left(re \cdot \cos im + \cos im\right) \]

      7. distribute-lft1-in 63.6%

         \[\leadsto \cos im \cdot \left(0.16666666666666666 \cdot {re}^{3} + 0.5 \cdot {re}^{2}\right) + \color{blue}{\left(re + 1\right) \cdot \cos im} \]

      8. *-commutative 63.6%

         \[\leadsto \cos im \cdot \left(0.16666666666666666 \cdot {re}^{3} + 0.5 \cdot {re}^{2}\right) + \color{blue}{\cos im \cdot \left(re + 1\right)} \]

      9. distribute-lft-out 63.6%

         \[\leadsto \color{blue}{\cos im \cdot \left(\left(0.16666666666666666 \cdot {re}^{3} + 0.5 \cdot {re}^{2}\right) + \left(re + 1\right)\right)} \]

   6. Simplified 63.6%

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

   7. Taylor expanded in im around 0 49.7%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \cdot \left(\left(re \cdot re\right) \cdot \left(0.16666666666666666 \cdot re + 0.5\right) + \left(1 + re\right)\right) \]
   8. Step-by-step derivation

      1. unpow2 14.7%

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

   9. Simplified 49.7%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.8 \cdot 10^{+27}:\\ \;\;\;\;-0.5 \cdot \left(\left(re + 1\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 2.95 \cdot 10^{-8}:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot \left(re \cdot 0.16666666666666666 + 0.5\right) + \left(re + 1\right)\right) \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 8: 47.7% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(re \cdot re\right) \cdot \left(re \cdot 0.16666666666666666 + 0.5\right) + \left(re + 1\right)\\ \mathbf{if}\;re \leq -1.15 \cdot 10^{+24}:\\ \;\;\;\;-0.5 \cdot \left(\left(re + 1\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 1700000:\\ \;\;\;\;\left(re + 1\right) + \left(\left(re \cdot re\right) \cdot \left(re \cdot 0.16666666666666666\right) + \left(re \cdot re\right) \cdot 0.5\right)\\ \mathbf{elif}\;re \leq 1.75 \cdot 10^{+133}:\\ \;\;\;\;t_0 \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ (* (* re re) (+ (* re 0.16666666666666666) 0.5)) (+ re 1.0))))
   (if (<= re -1.15e+24)
     (* -0.5 (* (+ re 1.0) (* im im)))
     (if (<= re 1700000.0)
       (+
        (+ re 1.0)
        (+ (* (* re re) (* re 0.16666666666666666)) (* (* re re) 0.5)))
       (if (<= re 1.75e+133) (* t_0 (+ 1.0 (* -0.5 (* im im)))) t_0)))))

C

double code(double re, double im) {
	double t_0 = ((re * re) * ((re * 0.16666666666666666) + 0.5)) + (re + 1.0);
	double tmp;
	if (re <= -1.15e+24) {
		tmp = -0.5 * ((re + 1.0) * (im * im));
	} else if (re <= 1700000.0) {
		tmp = (re + 1.0) + (((re * re) * (re * 0.16666666666666666)) + ((re * re) * 0.5));
	} else if (re <= 1.75e+133) {
		tmp = t_0 * (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 = ((re * re) * ((re * 0.16666666666666666d0) + 0.5d0)) + (re + 1.0d0)
    if (re <= (-1.15d+24)) then
        tmp = (-0.5d0) * ((re + 1.0d0) * (im * im))
    else if (re <= 1700000.0d0) then
        tmp = (re + 1.0d0) + (((re * re) * (re * 0.16666666666666666d0)) + ((re * re) * 0.5d0))
    else if (re <= 1.75d+133) then
        tmp = t_0 * (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 = ((re * re) * ((re * 0.16666666666666666) + 0.5)) + (re + 1.0);
	double tmp;
	if (re <= -1.15e+24) {
		tmp = -0.5 * ((re + 1.0) * (im * im));
	} else if (re <= 1700000.0) {
		tmp = (re + 1.0) + (((re * re) * (re * 0.16666666666666666)) + ((re * re) * 0.5));
	} else if (re <= 1.75e+133) {
		tmp = t_0 * (1.0 + (-0.5 * (im * im)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = ((re * re) * ((re * 0.16666666666666666) + 0.5)) + (re + 1.0)
	tmp = 0
	if re <= -1.15e+24:
		tmp = -0.5 * ((re + 1.0) * (im * im))
	elif re <= 1700000.0:
		tmp = (re + 1.0) + (((re * re) * (re * 0.16666666666666666)) + ((re * re) * 0.5))
	elif re <= 1.75e+133:
		tmp = t_0 * (1.0 + (-0.5 * (im * im)))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(Float64(re * re) * Float64(Float64(re * 0.16666666666666666) + 0.5)) + Float64(re + 1.0))
	tmp = 0.0
	if (re <= -1.15e+24)
		tmp = Float64(-0.5 * Float64(Float64(re + 1.0) * Float64(im * im)));
	elseif (re <= 1700000.0)
		tmp = Float64(Float64(re + 1.0) + Float64(Float64(Float64(re * re) * Float64(re * 0.16666666666666666)) + Float64(Float64(re * re) * 0.5)));
	elseif (re <= 1.75e+133)
		tmp = Float64(t_0 * 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 = ((re * re) * ((re * 0.16666666666666666) + 0.5)) + (re + 1.0);
	tmp = 0.0;
	if (re <= -1.15e+24)
		tmp = -0.5 * ((re + 1.0) * (im * im));
	elseif (re <= 1700000.0)
		tmp = (re + 1.0) + (((re * re) * (re * 0.16666666666666666)) + ((re * re) * 0.5));
	elseif (re <= 1.75e+133)
		tmp = t_0 * (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[(N[(re * re), $MachinePrecision] * N[(N[(re * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + N[(re + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -1.15e+24], N[(-0.5 * N[(N[(re + 1.0), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1700000.0], N[(N[(re + 1.0), $MachinePrecision] + N[(N[(N[(re * re), $MachinePrecision] * N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + N[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.75e+133], N[(t$95$0 * N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 4 regimes

2. if re < -1.15e24

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 2.1%

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

      1. distribute-rgt1-in 2.1%

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

   4. Simplified 2.1%

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

   5. Taylor expanded in im around 0 2.0%

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

      1. unpow2 2.7%

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

   7. Simplified 2.0%

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

   8. Taylor expanded in im around inf 27.7%

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

      1. +-commutative 27.7%

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

      2. unpow2 27.7%

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

   10. Simplified 27.7%

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

   if -1.15e24 < re < 1.7e6

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Step-by-step derivation

      1. add-cube-cbrt 98.9%

         \[\leadsto e^{re} \cdot \color{blue}{\left(\left(\sqrt[3]{\cos im} \cdot \sqrt[3]{\cos im}\right) \cdot \sqrt[3]{\cos im}\right)} \]

      2. pow2 98.9%

         \[\leadsto e^{re} \cdot \left(\color{blue}{{\left(\sqrt[3]{\cos im}\right)}^{2}} \cdot \sqrt[3]{\cos im}\right) \]

   3. Applied egg-rr 98.9%

      \[\leadsto e^{re} \cdot \color{blue}{\left({\left(\sqrt[3]{\cos im}\right)}^{2} \cdot \sqrt[3]{\cos im}\right)} \]

   4. Taylor expanded in re around 0 95.4%

      \[\leadsto \color{blue}{\cos im + \left(0.16666666666666666 \cdot \left({re}^{3} \cdot \cos im\right) + \left(0.5 \cdot \left({re}^{2} \cdot \cos im\right) + re \cdot \cos im\right)\right)} \]
   5. Step-by-step derivation

      1. +-commutative 95.4%

         \[\leadsto \color{blue}{\left(0.16666666666666666 \cdot \left({re}^{3} \cdot \cos im\right) + \left(0.5 \cdot \left({re}^{2} \cdot \cos im\right) + re \cdot \cos im\right)\right) + \cos im} \]

      2. associate-+r+ 95.4%

         \[\leadsto \color{blue}{\left(\left(0.16666666666666666 \cdot \left({re}^{3} \cdot \cos im\right) + 0.5 \cdot \left({re}^{2} \cdot \cos im\right)\right) + re \cdot \cos im\right)} + \cos im \]

      3. associate-+l+ 95.4%

         \[\leadsto \color{blue}{\left(0.16666666666666666 \cdot \left({re}^{3} \cdot \cos im\right) + 0.5 \cdot \left({re}^{2} \cdot \cos im\right)\right) + \left(re \cdot \cos im + \cos im\right)} \]

      4. associate-*r* 95.4%

         \[\leadsto \left(\color{blue}{\left(0.16666666666666666 \cdot {re}^{3}\right) \cdot \cos im} + 0.5 \cdot \left({re}^{2} \cdot \cos im\right)\right) + \left(re \cdot \cos im + \cos im\right) \]

      5. associate-*r* 95.4%

         \[\leadsto \left(\left(0.16666666666666666 \cdot {re}^{3}\right) \cdot \cos im + \color{blue}{\left(0.5 \cdot {re}^{2}\right) \cdot \cos im}\right) + \left(re \cdot \cos im + \cos im\right) \]

      6. distribute-rgt-out 95.4%

         \[\leadsto \color{blue}{\cos im \cdot \left(0.16666666666666666 \cdot {re}^{3} + 0.5 \cdot {re}^{2}\right)} + \left(re \cdot \cos im + \cos im\right) \]

      7. distribute-lft1-in 95.3%

         \[\leadsto \cos im \cdot \left(0.16666666666666666 \cdot {re}^{3} + 0.5 \cdot {re}^{2}\right) + \color{blue}{\left(re + 1\right) \cdot \cos im} \]

      8. *-commutative 95.3%

         \[\leadsto \cos im \cdot \left(0.16666666666666666 \cdot {re}^{3} + 0.5 \cdot {re}^{2}\right) + \color{blue}{\cos im \cdot \left(re + 1\right)} \]

      9. distribute-lft-out 95.3%

         \[\leadsto \color{blue}{\cos im \cdot \left(\left(0.16666666666666666 \cdot {re}^{3} + 0.5 \cdot {re}^{2}\right) + \left(re + 1\right)\right)} \]

   6. Simplified 95.3%

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

   7. Taylor expanded in im around 0 50.5%

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

      1. distribute-lft-in 50.5%

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

   9. Applied egg-rr 50.5%

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

   if 1.7e6 < re < 1.7499999999999999e133

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Step-by-step derivation

      1. add-cube-cbrt 100.0%

         \[\leadsto e^{re} \cdot \color{blue}{\left(\left(\sqrt[3]{\cos im} \cdot \sqrt[3]{\cos im}\right) \cdot \sqrt[3]{\cos im}\right)} \]

      2. pow2 100.0%

         \[\leadsto e^{re} \cdot \left(\color{blue}{{\left(\sqrt[3]{\cos im}\right)}^{2}} \cdot \sqrt[3]{\cos im}\right) \]

   3. Applied egg-rr 100.0%

      \[\leadsto e^{re} \cdot \color{blue}{\left({\left(\sqrt[3]{\cos im}\right)}^{2} \cdot \sqrt[3]{\cos im}\right)} \]

   4. Taylor expanded in re around 0 16.6%

      \[\leadsto \color{blue}{\cos im + \left(0.16666666666666666 \cdot \left({re}^{3} \cdot \cos im\right) + \left(0.5 \cdot \left({re}^{2} \cdot \cos im\right) + re \cdot \cos im\right)\right)} \]
   5. Step-by-step derivation

      1. +-commutative 16.6%

         \[\leadsto \color{blue}{\left(0.16666666666666666 \cdot \left({re}^{3} \cdot \cos im\right) + \left(0.5 \cdot \left({re}^{2} \cdot \cos im\right) + re \cdot \cos im\right)\right) + \cos im} \]

      2. associate-+r+ 16.6%

         \[\leadsto \color{blue}{\left(\left(0.16666666666666666 \cdot \left({re}^{3} \cdot \cos im\right) + 0.5 \cdot \left({re}^{2} \cdot \cos im\right)\right) + re \cdot \cos im\right)} + \cos im \]

      3. associate-+l+ 16.6%

         \[\leadsto \color{blue}{\left(0.16666666666666666 \cdot \left({re}^{3} \cdot \cos im\right) + 0.5 \cdot \left({re}^{2} \cdot \cos im\right)\right) + \left(re \cdot \cos im + \cos im\right)} \]

      4. associate-*r* 16.6%

         \[\leadsto \left(\color{blue}{\left(0.16666666666666666 \cdot {re}^{3}\right) \cdot \cos im} + 0.5 \cdot \left({re}^{2} \cdot \cos im\right)\right) + \left(re \cdot \cos im + \cos im\right) \]

      5. associate-*r* 16.6%

         \[\leadsto \left(\left(0.16666666666666666 \cdot {re}^{3}\right) \cdot \cos im + \color{blue}{\left(0.5 \cdot {re}^{2}\right) \cdot \cos im}\right) + \left(re \cdot \cos im + \cos im\right) \]

      6. distribute-rgt-out 16.6%

         \[\leadsto \color{blue}{\cos im \cdot \left(0.16666666666666666 \cdot {re}^{3} + 0.5 \cdot {re}^{2}\right)} + \left(re \cdot \cos im + \cos im\right) \]

      7. distribute-lft1-in 16.6%

         \[\leadsto \cos im \cdot \left(0.16666666666666666 \cdot {re}^{3} + 0.5 \cdot {re}^{2}\right) + \color{blue}{\left(re + 1\right) \cdot \cos im} \]

      8. *-commutative 16.6%

         \[\leadsto \cos im \cdot \left(0.16666666666666666 \cdot {re}^{3} + 0.5 \cdot {re}^{2}\right) + \color{blue}{\cos im \cdot \left(re + 1\right)} \]

      9. distribute-lft-out 16.6%

         \[\leadsto \color{blue}{\cos im \cdot \left(\left(0.16666666666666666 \cdot {re}^{3} + 0.5 \cdot {re}^{2}\right) + \left(re + 1\right)\right)} \]

   6. Simplified 16.6%

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

   7. Taylor expanded in im around 0 31.2%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \cdot \left(\left(re \cdot re\right) \cdot \left(0.16666666666666666 \cdot re + 0.5\right) + \left(1 + re\right)\right) \]
   8. Step-by-step derivation

      1. unpow2 18.3%

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

   9. Simplified 31.2%

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

   if 1.7499999999999999e133 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Step-by-step derivation

      1. add-cube-cbrt 100.0%

         \[\leadsto e^{re} \cdot \color{blue}{\left(\left(\sqrt[3]{\cos im} \cdot \sqrt[3]{\cos im}\right) \cdot \sqrt[3]{\cos im}\right)} \]

      2. pow2 100.0%

         \[\leadsto e^{re} \cdot \left(\color{blue}{{\left(\sqrt[3]{\cos im}\right)}^{2}} \cdot \sqrt[3]{\cos im}\right) \]

   3. Applied egg-rr 100.0%

      \[\leadsto e^{re} \cdot \color{blue}{\left({\left(\sqrt[3]{\cos im}\right)}^{2} \cdot \sqrt[3]{\cos im}\right)} \]

   4. Taylor expanded in re around 0 100.0%

      \[\leadsto \color{blue}{\cos im + \left(0.16666666666666666 \cdot \left({re}^{3} \cdot \cos im\right) + \left(0.5 \cdot \left({re}^{2} \cdot \cos im\right) + re \cdot \cos im\right)\right)} \]
   5. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{\left(0.16666666666666666 \cdot \left({re}^{3} \cdot \cos im\right) + \left(0.5 \cdot \left({re}^{2} \cdot \cos im\right) + re \cdot \cos im\right)\right) + \cos im} \]

      2. associate-+r+ 100.0%

         \[\leadsto \color{blue}{\left(\left(0.16666666666666666 \cdot \left({re}^{3} \cdot \cos im\right) + 0.5 \cdot \left({re}^{2} \cdot \cos im\right)\right) + re \cdot \cos im\right)} + \cos im \]

      3. associate-+l+ 100.0%

         \[\leadsto \color{blue}{\left(0.16666666666666666 \cdot \left({re}^{3} \cdot \cos im\right) + 0.5 \cdot \left({re}^{2} \cdot \cos im\right)\right) + \left(re \cdot \cos im + \cos im\right)} \]

      4. associate-*r* 100.0%

         \[\leadsto \left(\color{blue}{\left(0.16666666666666666 \cdot {re}^{3}\right) \cdot \cos im} + 0.5 \cdot \left({re}^{2} \cdot \cos im\right)\right) + \left(re \cdot \cos im + \cos im\right) \]

      5. associate-*r* 100.0%

         \[\leadsto \left(\left(0.16666666666666666 \cdot {re}^{3}\right) \cdot \cos im + \color{blue}{\left(0.5 \cdot {re}^{2}\right) \cdot \cos im}\right) + \left(re \cdot \cos im + \cos im\right) \]

      6. distribute-rgt-out 100.0%

         \[\leadsto \color{blue}{\cos im \cdot \left(0.16666666666666666 \cdot {re}^{3} + 0.5 \cdot {re}^{2}\right)} + \left(re \cdot \cos im + \cos im\right) \]

      7. distribute-lft1-in 100.0%

         \[\leadsto \cos im \cdot \left(0.16666666666666666 \cdot {re}^{3} + 0.5 \cdot {re}^{2}\right) + \color{blue}{\left(re + 1\right) \cdot \cos im} \]

      8. *-commutative 100.0%

         \[\leadsto \cos im \cdot \left(0.16666666666666666 \cdot {re}^{3} + 0.5 \cdot {re}^{2}\right) + \color{blue}{\cos im \cdot \left(re + 1\right)} \]

      9. distribute-lft-out 100.0%

         \[\leadsto \color{blue}{\cos im \cdot \left(\left(0.16666666666666666 \cdot {re}^{3} + 0.5 \cdot {re}^{2}\right) + \left(re + 1\right)\right)} \]

   6. Simplified 100.0%

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

   7. Taylor expanded in im around 0 72.7%

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

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.15 \cdot 10^{+24}:\\ \;\;\;\;-0.5 \cdot \left(\left(re + 1\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 1700000:\\ \;\;\;\;\left(re + 1\right) + \left(\left(re \cdot re\right) \cdot \left(re \cdot 0.16666666666666666\right) + \left(re \cdot re\right) \cdot 0.5\right)\\ \mathbf{elif}\;re \leq 1.75 \cdot 10^{+133}:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot \left(re \cdot 0.16666666666666666 + 0.5\right) + \left(re + 1\right)\right) \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(re \cdot 0.16666666666666666 + 0.5\right) + \left(re + 1\right)\\ \end{array} \]

Alternative 9: 46.2% accurate, 13.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.15 \cdot 10^{+24}:\\ \;\;\;\;-0.5 \cdot \left(\left(re + 1\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(re \cdot 0.16666666666666666 + 0.5\right) + \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.15e+24)
   (* -0.5 (* (+ re 1.0) (* im im)))
   (+ (* (* re re) (+ (* re 0.16666666666666666) 0.5)) (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.15e+24) {
		tmp = -0.5 * ((re + 1.0) * (im * im));
	} else {
		tmp = ((re * re) * ((re * 0.16666666666666666) + 0.5)) + (re + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.15d+24)) then
        tmp = (-0.5d0) * ((re + 1.0d0) * (im * im))
    else
        tmp = ((re * re) * ((re * 0.16666666666666666d0) + 0.5d0)) + (re + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.15e+24) {
		tmp = -0.5 * ((re + 1.0) * (im * im));
	} else {
		tmp = ((re * re) * ((re * 0.16666666666666666) + 0.5)) + (re + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.15e+24:
		tmp = -0.5 * ((re + 1.0) * (im * im))
	else:
		tmp = ((re * re) * ((re * 0.16666666666666666) + 0.5)) + (re + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.15e+24)
		tmp = Float64(-0.5 * Float64(Float64(re + 1.0) * Float64(im * im)));
	else
		tmp = Float64(Float64(Float64(re * re) * Float64(Float64(re * 0.16666666666666666) + 0.5)) + Float64(re + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.15e+24)
		tmp = -0.5 * ((re + 1.0) * (im * im));
	else
		tmp = ((re * re) * ((re * 0.16666666666666666) + 0.5)) + (re + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.15e+24], N[(-0.5 * N[(N[(re + 1.0), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] * N[(N[(re * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + N[(re + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.15e24

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 2.1%

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

      1. distribute-rgt1-in 2.1%

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

   4. Simplified 2.1%

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

   5. Taylor expanded in im around 0 2.0%

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

      1. unpow2 2.7%

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

   7. Simplified 2.0%

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

   8. Taylor expanded in im around inf 27.7%

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

      1. +-commutative 27.7%

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

      2. unpow2 27.7%

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

   10. Simplified 27.7%

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

   if -1.15e24 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Step-by-step derivation

      1. add-cube-cbrt 99.2%

         \[\leadsto e^{re} \cdot \color{blue}{\left(\left(\sqrt[3]{\cos im} \cdot \sqrt[3]{\cos im}\right) \cdot \sqrt[3]{\cos im}\right)} \]

      2. pow2 99.2%

         \[\leadsto e^{re} \cdot \left(\color{blue}{{\left(\sqrt[3]{\cos im}\right)}^{2}} \cdot \sqrt[3]{\cos im}\right) \]

   3. Applied egg-rr 99.2%

      \[\leadsto e^{re} \cdot \color{blue}{\left({\left(\sqrt[3]{\cos im}\right)}^{2} \cdot \sqrt[3]{\cos im}\right)} \]

   4. Taylor expanded in re around 0 86.5%

      \[\leadsto \color{blue}{\cos im + \left(0.16666666666666666 \cdot \left({re}^{3} \cdot \cos im\right) + \left(0.5 \cdot \left({re}^{2} \cdot \cos im\right) + re \cdot \cos im\right)\right)} \]
   5. Step-by-step derivation

      1. +-commutative 86.5%

         \[\leadsto \color{blue}{\left(0.16666666666666666 \cdot \left({re}^{3} \cdot \cos im\right) + \left(0.5 \cdot \left({re}^{2} \cdot \cos im\right) + re \cdot \cos im\right)\right) + \cos im} \]

      2. associate-+r+ 86.5%

         \[\leadsto \color{blue}{\left(\left(0.16666666666666666 \cdot \left({re}^{3} \cdot \cos im\right) + 0.5 \cdot \left({re}^{2} \cdot \cos im\right)\right) + re \cdot \cos im\right)} + \cos im \]

      3. associate-+l+ 86.5%

         \[\leadsto \color{blue}{\left(0.16666666666666666 \cdot \left({re}^{3} \cdot \cos im\right) + 0.5 \cdot \left({re}^{2} \cdot \cos im\right)\right) + \left(re \cdot \cos im + \cos im\right)} \]

      4. associate-*r* 86.5%

         \[\leadsto \left(\color{blue}{\left(0.16666666666666666 \cdot {re}^{3}\right) \cdot \cos im} + 0.5 \cdot \left({re}^{2} \cdot \cos im\right)\right) + \left(re \cdot \cos im + \cos im\right) \]

      5. associate-*r* 86.5%

         \[\leadsto \left(\left(0.16666666666666666 \cdot {re}^{3}\right) \cdot \cos im + \color{blue}{\left(0.5 \cdot {re}^{2}\right) \cdot \cos im}\right) + \left(re \cdot \cos im + \cos im\right) \]

      6. distribute-rgt-out 86.5%

         \[\leadsto \color{blue}{\cos im \cdot \left(0.16666666666666666 \cdot {re}^{3} + 0.5 \cdot {re}^{2}\right)} + \left(re \cdot \cos im + \cos im\right) \]

      7. distribute-lft1-in 86.5%

         \[\leadsto \cos im \cdot \left(0.16666666666666666 \cdot {re}^{3} + 0.5 \cdot {re}^{2}\right) + \color{blue}{\left(re + 1\right) \cdot \cos im} \]

      8. *-commutative 86.5%

         \[\leadsto \cos im \cdot \left(0.16666666666666666 \cdot {re}^{3} + 0.5 \cdot {re}^{2}\right) + \color{blue}{\cos im \cdot \left(re + 1\right)} \]

      9. distribute-lft-out 86.5%

         \[\leadsto \color{blue}{\cos im \cdot \left(\left(0.16666666666666666 \cdot {re}^{3} + 0.5 \cdot {re}^{2}\right) + \left(re + 1\right)\right)} \]

   6. Simplified 86.5%

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

   7. Taylor expanded in im around 0 49.4%

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

4. Final simplification 45.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.15 \cdot 10^{+24}:\\ \;\;\;\;-0.5 \cdot \left(\left(re + 1\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(re \cdot 0.16666666666666666 + 0.5\right) + \left(re + 1\right)\\ \end{array} \]

Alternative 10: 37.8% accurate, 15.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.15 \cdot 10^{+24} \lor \neg \left(re \leq 1700000\right):\\ \;\;\;\;-0.5 \cdot \left(\left(re + 1\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -1.15e+24) (not (<= re 1700000.0)))
   (* -0.5 (* (+ re 1.0) (* im im)))
   (+ re 1.0)))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -1.15e+24) || !(re <= 1700000.0)) {
		tmp = -0.5 * ((re + 1.0) * (im * im));
	} else {
		tmp = re + 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((re <= (-1.15d+24)) .or. (.not. (re <= 1700000.0d0))) then
        tmp = (-0.5d0) * ((re + 1.0d0) * (im * im))
    else
        tmp = re + 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -1.15e+24) || !(re <= 1700000.0)) {
		tmp = -0.5 * ((re + 1.0) * (im * im));
	} else {
		tmp = re + 1.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -1.15e+24) or not (re <= 1700000.0):
		tmp = -0.5 * ((re + 1.0) * (im * im))
	else:
		tmp = re + 1.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -1.15e+24) || !(re <= 1700000.0))
		tmp = Float64(-0.5 * Float64(Float64(re + 1.0) * Float64(im * im)));
	else
		tmp = Float64(re + 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -1.15e+24) || ~((re <= 1700000.0)))
		tmp = -0.5 * ((re + 1.0) * (im * im));
	else
		tmp = re + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -1.15e+24], N[Not[LessEqual[re, 1700000.0]], $MachinePrecision]], N[(-0.5 * N[(N[(re + 1.0), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.15e24 or 1.7e6 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 3.9%

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

      1. distribute-rgt1-in 3.9%

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

   4. Simplified 3.9%

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

   5. Taylor expanded in im around 0 12.8%

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

      1. unpow2 8.9%

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

   7. Simplified 12.8%

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

   8. Taylor expanded in im around inf 23.6%

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

      1. +-commutative 23.6%

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

      2. unpow2 23.6%

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

   10. Simplified 23.6%

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

   if -1.15e24 < re < 1.7e6

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 95.1%

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

      1. distribute-rgt1-in 95.1%

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

   4. Simplified 95.1%

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

   5. Taylor expanded in im around 0 50.5%

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

4. Final simplification 39.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.15 \cdot 10^{+24} \lor \neg \left(re \leq 1700000\right):\\ \;\;\;\;-0.5 \cdot \left(\left(re + 1\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re + 1\\ \end{array} \]

Alternative 11: 43.5% accurate, 18.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.15 \cdot 10^{+24}:\\ \;\;\;\;-0.5 \cdot \left(\left(re + 1\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(re + \left(re \cdot re\right) \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.15e+24)
   (* -0.5 (* (+ re 1.0) (* im im)))
   (+ 1.0 (+ re (* (* re re) 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.15e+24) {
		tmp = -0.5 * ((re + 1.0) * (im * im));
	} else {
		tmp = 1.0 + (re + ((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 <= (-1.15d+24)) then
        tmp = (-0.5d0) * ((re + 1.0d0) * (im * im))
    else
        tmp = 1.0d0 + (re + ((re * re) * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.15e+24) {
		tmp = -0.5 * ((re + 1.0) * (im * im));
	} else {
		tmp = 1.0 + (re + ((re * re) * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.15e+24:
		tmp = -0.5 * ((re + 1.0) * (im * im))
	else:
		tmp = 1.0 + (re + ((re * re) * 0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.15e+24)
		tmp = Float64(-0.5 * Float64(Float64(re + 1.0) * Float64(im * im)));
	else
		tmp = Float64(1.0 + Float64(re + Float64(Float64(re * re) * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.15e+24)
		tmp = -0.5 * ((re + 1.0) * (im * im));
	else
		tmp = 1.0 + (re + ((re * re) * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.15e+24], N[(-0.5 * N[(N[(re + 1.0), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(re + N[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.15e24

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 2.1%

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

      1. distribute-rgt1-in 2.1%

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

   4. Simplified 2.1%

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

   5. Taylor expanded in im around 0 2.0%

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

      1. unpow2 2.7%

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

   7. Simplified 2.0%

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

   8. Taylor expanded in im around inf 27.7%

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

      1. +-commutative 27.7%

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

      2. unpow2 27.7%

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

   10. Simplified 27.7%

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

   if -1.15e24 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Step-by-step derivation

      1. add-cube-cbrt 99.2%

         \[\leadsto e^{re} \cdot \color{blue}{\left(\left(\sqrt[3]{\cos im} \cdot \sqrt[3]{\cos im}\right) \cdot \sqrt[3]{\cos im}\right)} \]

      2. pow2 99.2%

         \[\leadsto e^{re} \cdot \left(\color{blue}{{\left(\sqrt[3]{\cos im}\right)}^{2}} \cdot \sqrt[3]{\cos im}\right) \]

   3. Applied egg-rr 99.2%

      \[\leadsto e^{re} \cdot \color{blue}{\left({\left(\sqrt[3]{\cos im}\right)}^{2} \cdot \sqrt[3]{\cos im}\right)} \]

   4. Taylor expanded in re around 0 86.5%

      \[\leadsto \color{blue}{\cos im + \left(0.16666666666666666 \cdot \left({re}^{3} \cdot \cos im\right) + \left(0.5 \cdot \left({re}^{2} \cdot \cos im\right) + re \cdot \cos im\right)\right)} \]
   5. Step-by-step derivation

      1. +-commutative 86.5%

         \[\leadsto \color{blue}{\left(0.16666666666666666 \cdot \left({re}^{3} \cdot \cos im\right) + \left(0.5 \cdot \left({re}^{2} \cdot \cos im\right) + re \cdot \cos im\right)\right) + \cos im} \]

      2. associate-+r+ 86.5%

         \[\leadsto \color{blue}{\left(\left(0.16666666666666666 \cdot \left({re}^{3} \cdot \cos im\right) + 0.5 \cdot \left({re}^{2} \cdot \cos im\right)\right) + re \cdot \cos im\right)} + \cos im \]

      3. associate-+l+ 86.5%

         \[\leadsto \color{blue}{\left(0.16666666666666666 \cdot \left({re}^{3} \cdot \cos im\right) + 0.5 \cdot \left({re}^{2} \cdot \cos im\right)\right) + \left(re \cdot \cos im + \cos im\right)} \]

      4. associate-*r* 86.5%

         \[\leadsto \left(\color{blue}{\left(0.16666666666666666 \cdot {re}^{3}\right) \cdot \cos im} + 0.5 \cdot \left({re}^{2} \cdot \cos im\right)\right) + \left(re \cdot \cos im + \cos im\right) \]

      5. associate-*r* 86.5%

         \[\leadsto \left(\left(0.16666666666666666 \cdot {re}^{3}\right) \cdot \cos im + \color{blue}{\left(0.5 \cdot {re}^{2}\right) \cdot \cos im}\right) + \left(re \cdot \cos im + \cos im\right) \]

      6. distribute-rgt-out 86.5%

         \[\leadsto \color{blue}{\cos im \cdot \left(0.16666666666666666 \cdot {re}^{3} + 0.5 \cdot {re}^{2}\right)} + \left(re \cdot \cos im + \cos im\right) \]

      7. distribute-lft1-in 86.5%

         \[\leadsto \cos im \cdot \left(0.16666666666666666 \cdot {re}^{3} + 0.5 \cdot {re}^{2}\right) + \color{blue}{\left(re + 1\right) \cdot \cos im} \]

      8. *-commutative 86.5%

         \[\leadsto \cos im \cdot \left(0.16666666666666666 \cdot {re}^{3} + 0.5 \cdot {re}^{2}\right) + \color{blue}{\cos im \cdot \left(re + 1\right)} \]

      9. distribute-lft-out 86.5%

         \[\leadsto \color{blue}{\cos im \cdot \left(\left(0.16666666666666666 \cdot {re}^{3} + 0.5 \cdot {re}^{2}\right) + \left(re + 1\right)\right)} \]

   6. Simplified 86.5%

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

   7. Taylor expanded in re around 0 83.8%

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

      1. unpow2 83.8%

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

      2. associate-*r* 83.8%

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

   9. Simplified 83.8%

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

   10. Taylor expanded in im around 0 47.5%

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

       1. unpow2 47.5%

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

   12. Simplified 47.5%

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

4. Final simplification 43.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.15 \cdot 10^{+24}:\\ \;\;\;\;-0.5 \cdot \left(\left(re + 1\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(re + \left(re \cdot re\right) \cdot 0.5\right)\\ \end{array} \]

Alternative 12: 31.1% accurate, 22.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1700000:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1700000.0) (+ re 1.0) (+ 1.0 (* -0.5 (* im im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1700000.0) {
		tmp = re + 1.0;
	} else {
		tmp = 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 <= 1700000.0d0) then
        tmp = re + 1.0d0
    else
        tmp = 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 <= 1700000.0) {
		tmp = re + 1.0;
	} else {
		tmp = 1.0 + (-0.5 * (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1700000.0:
		tmp = re + 1.0
	else:
		tmp = 1.0 + (-0.5 * (im * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1700000.0)
		tmp = Float64(re + 1.0);
	else
		tmp = 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 <= 1700000.0)
		tmp = re + 1.0;
	else
		tmp = 1.0 + (-0.5 * (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1700000.0], N[(re + 1.0), $MachinePrecision], N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.7e6

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 71.6%

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

      1. distribute-rgt1-in 71.6%

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

   4. Simplified 71.6%

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

   5. Taylor expanded in im around 0 38.3%

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

   if 1.7e6 < re

   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 14.2%

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

      1. unpow2 14.2%

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

   5. Simplified 14.2%

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

4. Final simplification 32.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1700000:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \left(im \cdot im\right)\\ \end{array} \]

Alternative 13: 28.4% 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 56.6%

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

   1. distribute-rgt1-in 56.6%

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

4. Simplified 56.6%

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

5. Taylor expanded in im around 0 30.6%

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

6. Final simplification 30.6%

   \[\leadsto re + 1 \]

Alternative 14: 27.9% 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 55.8%

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

3. Taylor expanded in im around 0 30.2%

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

4. Final simplification 30.2%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023293 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))