math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 7.8s

Alternatives: 12

Speedup: 1.0×

• 25.1% 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: 92.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.9999998:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 2:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (exp re) 0.9999998)
   (exp re)
   (if (<= (exp re) 2.0)
     (* (cos im) (+ re 1.0))
     (* (exp re) (+ 1.0 (* -0.5 (* im im)))))))

C

double code(double re, double im) {
	double tmp;
	if (exp(re) <= 0.9999998) {
		tmp = exp(re);
	} else if (exp(re) <= 2.0) {
		tmp = cos(im) * (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 (exp(re) <= 0.9999998d0) then
        tmp = exp(re)
    else if (exp(re) <= 2.0d0) then
        tmp = cos(im) * (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 (Math.exp(re) <= 0.9999998) {
		tmp = Math.exp(re);
	} else if (Math.exp(re) <= 2.0) {
		tmp = Math.cos(im) * (re + 1.0);
	} else {
		tmp = Math.exp(re) * (1.0 + (-0.5 * (im * im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.exp(re) <= 0.9999998:
		tmp = math.exp(re)
	elif math.exp(re) <= 2.0:
		tmp = math.cos(im) * (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 (exp(re) <= 0.9999998)
		tmp = exp(re);
	elseif (exp(re) <= 2.0)
		tmp = Float64(cos(im) * 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 (exp(re) <= 0.9999998)
		tmp = exp(re);
	elseif (exp(re) <= 2.0)
		tmp = cos(im) * (re + 1.0);
	else
		tmp = exp(re) * (1.0 + (-0.5 * (im * im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Exp[re], $MachinePrecision], 0.9999998], N[Exp[re], $MachinePrecision], If[LessEqual[N[Exp[re], $MachinePrecision], 2.0], N[(N[Cos[im], $MachinePrecision] * N[(re + 1.0), $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 (exp.f64 re) < 0.999999799999999994

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

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

   if 0.999999799999999994 < (exp.f64 re) < 2

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. *-rgt-identity 100.0%

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

      3. distribute-lft-out 100.0%

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

   4. Simplified 100.0%

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

   if 2 < (exp.f64 re)

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 72.1%

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

      1. unpow2 72.1%

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

   4. Simplified 72.1%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.9999998:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 2:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 3: 91.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.15 \cdot 10^{-7}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 3.8 \cdot 10^{+23}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.15e-7)
   (exp re)
   (if (<= re 3.8e+23) (* (cos im) (+ re 1.0)) (exp re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.15e-7) {
		tmp = exp(re);
	} else if (re <= 3.8e+23) {
		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 <= (-2.15d-7)) then
        tmp = exp(re)
    else if (re <= 3.8d+23) 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 <= -2.15e-7) {
		tmp = Math.exp(re);
	} else if (re <= 3.8e+23) {
		tmp = Math.cos(im) * (re + 1.0);
	} else {
		tmp = Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2.15e-7:
		tmp = math.exp(re)
	elif re <= 3.8e+23:
		tmp = math.cos(im) * (re + 1.0)
	else:
		tmp = math.exp(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.15e-7)
		tmp = exp(re);
	elseif (re <= 3.8e+23)
		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 <= -2.15e-7)
		tmp = exp(re);
	elseif (re <= 3.8e+23)
		tmp = cos(im) * (re + 1.0);
	else
		tmp = exp(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.15e-7], N[Exp[re], $MachinePrecision], If[LessEqual[re, 3.8e+23], 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 < -2.1500000000000001e-7 or 3.79999999999999975e23 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 84.7%

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

   if -2.1500000000000001e-7 < re < 3.79999999999999975e23

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 98.5%

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

      1. +-commutative 98.5%

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

      2. *-rgt-identity 98.5%

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

      3. distribute-lft-out 98.5%

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

   4. Simplified 98.5%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.15 \cdot 10^{-7}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 3.8 \cdot 10^{+23}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]

Alternative 4: 67.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -6.8 \cdot 10^{+16}:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 620:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;re \leq 2.85 \cdot 10^{+146}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(\left(-0.5 + -0.25 \cdot \left(re \cdot re\right)\right) + re \cdot \left(-0.5 + \left(re \cdot re\right) \cdot -0.08333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - re \cdot re}{1 - re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -6.8e+16)
   (* -0.5 (* im im))
   (if (<= re 620.0)
     (cos im)
     (if (<= re 2.85e+146)
       (*
        (* im im)
        (+
         (+ -0.5 (* -0.25 (* re re)))
         (* re (+ -0.5 (* (* re re) -0.08333333333333333)))))
       (/ (- 1.0 (* re re)) (- 1.0 re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -6.8e+16) {
		tmp = -0.5 * (im * im);
	} else if (re <= 620.0) {
		tmp = cos(im);
	} else if (re <= 2.85e+146) {
		tmp = (im * im) * ((-0.5 + (-0.25 * (re * re))) + (re * (-0.5 + ((re * re) * -0.08333333333333333))));
	} else {
		tmp = (1.0 - (re * re)) / (1.0 - 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 <= (-6.8d+16)) then
        tmp = (-0.5d0) * (im * im)
    else if (re <= 620.0d0) then
        tmp = cos(im)
    else if (re <= 2.85d+146) then
        tmp = (im * im) * (((-0.5d0) + ((-0.25d0) * (re * re))) + (re * ((-0.5d0) + ((re * re) * (-0.08333333333333333d0)))))
    else
        tmp = (1.0d0 - (re * re)) / (1.0d0 - re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -6.8e+16) {
		tmp = -0.5 * (im * im);
	} else if (re <= 620.0) {
		tmp = Math.cos(im);
	} else if (re <= 2.85e+146) {
		tmp = (im * im) * ((-0.5 + (-0.25 * (re * re))) + (re * (-0.5 + ((re * re) * -0.08333333333333333))));
	} else {
		tmp = (1.0 - (re * re)) / (1.0 - re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -6.8e+16:
		tmp = -0.5 * (im * im)
	elif re <= 620.0:
		tmp = math.cos(im)
	elif re <= 2.85e+146:
		tmp = (im * im) * ((-0.5 + (-0.25 * (re * re))) + (re * (-0.5 + ((re * re) * -0.08333333333333333))))
	else:
		tmp = (1.0 - (re * re)) / (1.0 - re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -6.8e+16)
		tmp = Float64(-0.5 * Float64(im * im));
	elseif (re <= 620.0)
		tmp = cos(im);
	elseif (re <= 2.85e+146)
		tmp = Float64(Float64(im * im) * Float64(Float64(-0.5 + Float64(-0.25 * Float64(re * re))) + Float64(re * Float64(-0.5 + Float64(Float64(re * re) * -0.08333333333333333)))));
	else
		tmp = Float64(Float64(1.0 - Float64(re * re)) / Float64(1.0 - re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -6.8e+16)
		tmp = -0.5 * (im * im);
	elseif (re <= 620.0)
		tmp = cos(im);
	elseif (re <= 2.85e+146)
		tmp = (im * im) * ((-0.5 + (-0.25 * (re * re))) + (re * (-0.5 + ((re * re) * -0.08333333333333333))));
	else
		tmp = (1.0 - (re * re)) / (1.0 - re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -6.8e+16], N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 620.0], N[Cos[im], $MachinePrecision], If[LessEqual[re, 2.85e+146], N[(N[(im * im), $MachinePrecision] * N[(N[(-0.5 + N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(re * N[(-0.5 + N[(N[(re * re), $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(re * re), $MachinePrecision]), $MachinePrecision] / N[(1.0 - re), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -6.8e16

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 81.0%

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

      1. unpow2 81.0%

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

   4. Simplified 81.0%

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

   5. Taylor expanded in im around inf 81.0%

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

      1. *-commutative 81.0%

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

      2. associate-*r* 81.0%

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

      3. *-commutative 81.0%

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

      4. unpow2 81.0%

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

   7. Simplified 81.0%

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

   8. Taylor expanded in re around 0 29.9%

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

      1. unpow2 29.9%

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

   10. Simplified 29.9%

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

   if -6.8e16 < re < 620

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 95.6%

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

   if 620 < re < 2.85e146

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 73.1%

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

      1. unpow2 73.1%

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

   4. Simplified 73.1%

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

   5. Taylor expanded in im around inf 42.3%

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

      1. *-commutative 42.3%

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

      2. associate-*r* 42.3%

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

      3. *-commutative 42.3%

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

      4. unpow2 42.3%

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

   7. Simplified 42.3%

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

   8. Taylor expanded in re around 0 31.9%

      \[\leadsto \color{blue}{-0.08333333333333333 \cdot \left({re}^{3} \cdot {im}^{2}\right) + \left(-0.5 \cdot \left(re \cdot {im}^{2}\right) + \left(-0.5 \cdot {im}^{2} + -0.25 \cdot \left({re}^{2} \cdot {im}^{2}\right)\right)\right)} \]
   9. Step-by-step derivation

      1. associate-+r+ 31.9%

         \[\leadsto \color{blue}{\left(-0.08333333333333333 \cdot \left({re}^{3} \cdot {im}^{2}\right) + -0.5 \cdot \left(re \cdot {im}^{2}\right)\right) + \left(-0.5 \cdot {im}^{2} + -0.25 \cdot \left({re}^{2} \cdot {im}^{2}\right)\right)} \]

      2. +-commutative 31.9%

         \[\leadsto \color{blue}{\left(-0.5 \cdot {im}^{2} + -0.25 \cdot \left({re}^{2} \cdot {im}^{2}\right)\right) + \left(-0.08333333333333333 \cdot \left({re}^{3} \cdot {im}^{2}\right) + -0.5 \cdot \left(re \cdot {im}^{2}\right)\right)} \]

      3. +-commutative 31.9%

         \[\leadsto \color{blue}{\left(-0.25 \cdot \left({re}^{2} \cdot {im}^{2}\right) + -0.5 \cdot {im}^{2}\right)} + \left(-0.08333333333333333 \cdot \left({re}^{3} \cdot {im}^{2}\right) + -0.5 \cdot \left(re \cdot {im}^{2}\right)\right) \]

      4. associate-*r* 31.9%

         \[\leadsto \left(\color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot {im}^{2}} + -0.5 \cdot {im}^{2}\right) + \left(-0.08333333333333333 \cdot \left({re}^{3} \cdot {im}^{2}\right) + -0.5 \cdot \left(re \cdot {im}^{2}\right)\right) \]

      5. distribute-rgt-out 31.9%

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(-0.25 \cdot {re}^{2} + -0.5\right)} + \left(-0.08333333333333333 \cdot \left({re}^{3} \cdot {im}^{2}\right) + -0.5 \cdot \left(re \cdot {im}^{2}\right)\right) \]

      6. associate-*r* 31.9%

         \[\leadsto {im}^{2} \cdot \left(-0.25 \cdot {re}^{2} + -0.5\right) + \left(\color{blue}{\left(-0.08333333333333333 \cdot {re}^{3}\right) \cdot {im}^{2}} + -0.5 \cdot \left(re \cdot {im}^{2}\right)\right) \]

      7. associate-*r* 31.9%

         \[\leadsto {im}^{2} \cdot \left(-0.25 \cdot {re}^{2} + -0.5\right) + \left(\left(-0.08333333333333333 \cdot {re}^{3}\right) \cdot {im}^{2} + \color{blue}{\left(-0.5 \cdot re\right) \cdot {im}^{2}}\right) \]

      8. distribute-rgt-out 31.9%

         \[\leadsto {im}^{2} \cdot \left(-0.25 \cdot {re}^{2} + -0.5\right) + \color{blue}{{im}^{2} \cdot \left(-0.08333333333333333 \cdot {re}^{3} + -0.5 \cdot re\right)} \]

      9. distribute-lft-out 31.9%

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\left(-0.25 \cdot {re}^{2} + -0.5\right) + \left(-0.08333333333333333 \cdot {re}^{3} + -0.5 \cdot re\right)\right)} \]

      10. unpow2 31.9%

          \[\leadsto \color{blue}{\left(im \cdot im\right)} \cdot \left(\left(-0.25 \cdot {re}^{2} + -0.5\right) + \left(-0.08333333333333333 \cdot {re}^{3} + -0.5 \cdot re\right)\right) \]

      11. +-commutative 31.9%

          \[\leadsto \left(im \cdot im\right) \cdot \left(\color{blue}{\left(-0.5 + -0.25 \cdot {re}^{2}\right)} + \left(-0.08333333333333333 \cdot {re}^{3} + -0.5 \cdot re\right)\right) \]

      12. unpow2 31.9%

          \[\leadsto \left(im \cdot im\right) \cdot \left(\left(-0.5 + -0.25 \cdot \color{blue}{\left(re \cdot re\right)}\right) + \left(-0.08333333333333333 \cdot {re}^{3} + -0.5 \cdot re\right)\right) \]

   10. Simplified 31.9%

       \[\leadsto \color{blue}{\left(im \cdot im\right) \cdot \left(\left(-0.5 + -0.25 \cdot \left(re \cdot re\right)\right) + re \cdot \left(-0.08333333333333333 \cdot \left(re \cdot re\right) + -0.5\right)\right)} \]

   if 2.85e146 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 6.5%

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

      1. +-commutative 6.5%

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

      2. *-rgt-identity 6.5%

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

      3. distribute-lft-out 6.5%

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

   4. Simplified 6.5%

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

   5. Taylor expanded in im around 0 29.3%

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

      1. unpow2 71.4%

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

   7. Simplified 29.3%

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

      1. *-commutative 29.3%

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

      2. distribute-lft-in 29.3%

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

      3. *-commutative 29.3%

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

      4. *-un-lft-identity 29.3%

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

      5. flip-+ 48.6%

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

      6. div-inv 48.6%

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

      7. fma-def 48.6%

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

      8. metadata-eval 48.6%

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

      9. *-commutative 48.6%

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

      10. associate-*l* 48.6%

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

   9. Applied egg-rr 48.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 - re \cdot re, \frac{1}{1 - re}, \left(1 + re\right) \cdot \left(im \cdot \left(im \cdot -0.5\right)\right)\right)} \]
   10. Step-by-step derivation

       1. fma-udef 48.6%

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

       2. +-commutative 48.6%

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

       3. *-commutative 48.6%

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

       4. *-commutative 48.6%

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

       5. associate-*r* 48.6%

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

       6. associate-*r/ 48.6%

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

       7. *-rgt-identity 48.6%

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

   11. Simplified 48.6%

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

   12. Taylor expanded in im around 0 63.3%

       \[\leadsto \color{blue}{\frac{1}{1 - re} - \frac{{re}^{2}}{1 - re}} \]
   13. Step-by-step derivation

       1. unpow2 63.3%

          \[\leadsto \frac{1}{1 - re} - \frac{\color{blue}{re \cdot re}}{1 - re} \]

       2. div-sub 63.3%

          \[\leadsto \color{blue}{\frac{1 - re \cdot re}{1 - re}} \]

   14. Simplified 63.3%

       \[\leadsto \color{blue}{\frac{1 - re \cdot re}{1 - re}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -6.8 \cdot 10^{+16}:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 620:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;re \leq 2.85 \cdot 10^{+146}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(\left(-0.5 + -0.25 \cdot \left(re \cdot re\right)\right) + re \cdot \left(-0.5 + \left(re \cdot re\right) \cdot -0.08333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - re \cdot re}{1 - re}\\ \end{array} \]

Alternative 5: 91.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.9 \cdot 10^{-8}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 3.8 \cdot 10^{+23}:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4.9e-8) (exp re) (if (<= re 3.8e+23) (cos im) (exp re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4.9e-8) {
		tmp = exp(re);
	} else if (re <= 3.8e+23) {
		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 (re <= (-4.9d-8)) then
        tmp = exp(re)
    else if (re <= 3.8d+23) 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 (re <= -4.9e-8) {
		tmp = Math.exp(re);
	} else if (re <= 3.8e+23) {
		tmp = Math.cos(im);
	} else {
		tmp = Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -4.9e-8:
		tmp = math.exp(re)
	elif re <= 3.8e+23:
		tmp = math.cos(im)
	else:
		tmp = math.exp(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4.9e-8)
		tmp = exp(re);
	elseif (re <= 3.8e+23)
		tmp = cos(im);
	else
		tmp = exp(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -4.9e-8)
		tmp = exp(re);
	elseif (re <= 3.8e+23)
		tmp = cos(im);
	else
		tmp = exp(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4.9e-8], N[Exp[re], $MachinePrecision], If[LessEqual[re, 3.8e+23], N[Cos[im], $MachinePrecision], N[Exp[re], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if re < -4.9000000000000002e-8 or 3.79999999999999975e23 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 84.7%

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

   if -4.9000000000000002e-8 < re < 3.79999999999999975e23

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 98.2%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.9 \cdot 10^{-8}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 3.8 \cdot 10^{+23}:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]

Alternative 6: 45.6% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -82:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 1.05 \cdot 10^{-15} \lor \neg \left(re \leq 2.8 \cdot 10^{+146}\right):\\ \;\;\;\;\frac{1 - re \cdot re}{1 - re}\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(\left(-0.5 + -0.25 \cdot \left(re \cdot re\right)\right) + re \cdot \left(-0.5 + \left(re \cdot re\right) \cdot -0.08333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -82.0)
   (* -0.5 (* im im))
   (if (or (<= re 1.05e-15) (not (<= re 2.8e+146)))
     (/ (- 1.0 (* re re)) (- 1.0 re))
     (*
      (* im im)
      (+
       (+ -0.5 (* -0.25 (* re re)))
       (* re (+ -0.5 (* (* re re) -0.08333333333333333))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -82.0) {
		tmp = -0.5 * (im * im);
	} else if ((re <= 1.05e-15) || !(re <= 2.8e+146)) {
		tmp = (1.0 - (re * re)) / (1.0 - re);
	} else {
		tmp = (im * im) * ((-0.5 + (-0.25 * (re * re))) + (re * (-0.5 + ((re * re) * -0.08333333333333333))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-82.0d0)) then
        tmp = (-0.5d0) * (im * im)
    else if ((re <= 1.05d-15) .or. (.not. (re <= 2.8d+146))) then
        tmp = (1.0d0 - (re * re)) / (1.0d0 - re)
    else
        tmp = (im * im) * (((-0.5d0) + ((-0.25d0) * (re * re))) + (re * ((-0.5d0) + ((re * re) * (-0.08333333333333333d0)))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -82.0) {
		tmp = -0.5 * (im * im);
	} else if ((re <= 1.05e-15) || !(re <= 2.8e+146)) {
		tmp = (1.0 - (re * re)) / (1.0 - re);
	} else {
		tmp = (im * im) * ((-0.5 + (-0.25 * (re * re))) + (re * (-0.5 + ((re * re) * -0.08333333333333333))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -82.0:
		tmp = -0.5 * (im * im)
	elif (re <= 1.05e-15) or not (re <= 2.8e+146):
		tmp = (1.0 - (re * re)) / (1.0 - re)
	else:
		tmp = (im * im) * ((-0.5 + (-0.25 * (re * re))) + (re * (-0.5 + ((re * re) * -0.08333333333333333))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -82.0)
		tmp = Float64(-0.5 * Float64(im * im));
	elseif ((re <= 1.05e-15) || !(re <= 2.8e+146))
		tmp = Float64(Float64(1.0 - Float64(re * re)) / Float64(1.0 - re));
	else
		tmp = Float64(Float64(im * im) * Float64(Float64(-0.5 + Float64(-0.25 * Float64(re * re))) + Float64(re * Float64(-0.5 + Float64(Float64(re * re) * -0.08333333333333333)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -82.0)
		tmp = -0.5 * (im * im);
	elseif ((re <= 1.05e-15) || ~((re <= 2.8e+146)))
		tmp = (1.0 - (re * re)) / (1.0 - re);
	else
		tmp = (im * im) * ((-0.5 + (-0.25 * (re * re))) + (re * (-0.5 + ((re * re) * -0.08333333333333333))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -82.0], N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, 1.05e-15], N[Not[LessEqual[re, 2.8e+146]], $MachinePrecision]], N[(N[(1.0 - N[(re * re), $MachinePrecision]), $MachinePrecision] / N[(1.0 - re), $MachinePrecision]), $MachinePrecision], N[(N[(im * im), $MachinePrecision] * N[(N[(-0.5 + N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(re * N[(-0.5 + N[(N[(re * re), $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -82

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 82.3%

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

      1. unpow2 82.3%

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

   4. Simplified 82.3%

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

   5. Taylor expanded in im around inf 82.3%

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

      1. *-commutative 82.3%

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

      2. associate-*r* 82.3%

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

      3. *-commutative 82.3%

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

      4. unpow2 82.3%

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

   7. Simplified 82.3%

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

   8. Taylor expanded in re around 0 28.2%

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

      1. unpow2 28.2%

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

   10. Simplified 28.2%

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

   if -82 < re < 1.0499999999999999e-15 or 2.8000000000000001e146 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 79.9%

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

      1. +-commutative 79.9%

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

      2. *-rgt-identity 79.9%

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

      3. distribute-lft-out 79.9%

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

   4. Simplified 79.9%

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

   5. Taylor expanded in im around 0 48.1%

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

      1. unpow2 57.5%

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

   7. Simplified 48.1%

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

      1. *-commutative 48.1%

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

      2. distribute-lft-in 48.1%

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

      3. *-commutative 48.1%

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

      4. *-un-lft-identity 48.1%

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

      5. flip-+ 52.2%

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

      6. div-inv 52.2%

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

      7. fma-def 52.2%

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

      8. metadata-eval 52.2%

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

      9. *-commutative 52.2%

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

      10. associate-*l* 52.2%

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

   9. Applied egg-rr 52.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 - re \cdot re, \frac{1}{1 - re}, \left(1 + re\right) \cdot \left(im \cdot \left(im \cdot -0.5\right)\right)\right)} \]
   10. Step-by-step derivation

       1. fma-udef 52.2%

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

       2. +-commutative 52.2%

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

       3. *-commutative 52.2%

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

       4. *-commutative 52.2%

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

       5. associate-*r* 52.2%

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

       6. associate-*r/ 52.2%

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

       7. *-rgt-identity 52.2%

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

   11. Simplified 52.2%

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

   12. Taylor expanded in im around 0 57.5%

       \[\leadsto \color{blue}{\frac{1}{1 - re} - \frac{{re}^{2}}{1 - re}} \]
   13. Step-by-step derivation

       1. unpow2 57.5%

          \[\leadsto \frac{1}{1 - re} - \frac{\color{blue}{re \cdot re}}{1 - re} \]

       2. div-sub 57.5%

          \[\leadsto \color{blue}{\frac{1 - re \cdot re}{1 - re}} \]

   14. Simplified 57.5%

       \[\leadsto \color{blue}{\frac{1 - re \cdot re}{1 - re}} \]

   if 1.0499999999999999e-15 < re < 2.8000000000000001e146

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 70.5%

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

      1. unpow2 70.5%

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

   4. Simplified 70.5%

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

   5. Taylor expanded in im around inf 40.9%

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

      1. *-commutative 40.9%

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

      2. associate-*r* 40.9%

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

      3. *-commutative 40.9%

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

      4. unpow2 40.9%

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

   7. Simplified 40.9%

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

   8. Taylor expanded in re around 0 30.8%

      \[\leadsto \color{blue}{-0.08333333333333333 \cdot \left({re}^{3} \cdot {im}^{2}\right) + \left(-0.5 \cdot \left(re \cdot {im}^{2}\right) + \left(-0.5 \cdot {im}^{2} + -0.25 \cdot \left({re}^{2} \cdot {im}^{2}\right)\right)\right)} \]
   9. Step-by-step derivation

      1. associate-+r+ 30.8%

         \[\leadsto \color{blue}{\left(-0.08333333333333333 \cdot \left({re}^{3} \cdot {im}^{2}\right) + -0.5 \cdot \left(re \cdot {im}^{2}\right)\right) + \left(-0.5 \cdot {im}^{2} + -0.25 \cdot \left({re}^{2} \cdot {im}^{2}\right)\right)} \]

      2. +-commutative 30.8%

         \[\leadsto \color{blue}{\left(-0.5 \cdot {im}^{2} + -0.25 \cdot \left({re}^{2} \cdot {im}^{2}\right)\right) + \left(-0.08333333333333333 \cdot \left({re}^{3} \cdot {im}^{2}\right) + -0.5 \cdot \left(re \cdot {im}^{2}\right)\right)} \]

      3. +-commutative 30.8%

         \[\leadsto \color{blue}{\left(-0.25 \cdot \left({re}^{2} \cdot {im}^{2}\right) + -0.5 \cdot {im}^{2}\right)} + \left(-0.08333333333333333 \cdot \left({re}^{3} \cdot {im}^{2}\right) + -0.5 \cdot \left(re \cdot {im}^{2}\right)\right) \]

      4. associate-*r* 30.8%

         \[\leadsto \left(\color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot {im}^{2}} + -0.5 \cdot {im}^{2}\right) + \left(-0.08333333333333333 \cdot \left({re}^{3} \cdot {im}^{2}\right) + -0.5 \cdot \left(re \cdot {im}^{2}\right)\right) \]

      5. distribute-rgt-out 30.8%

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(-0.25 \cdot {re}^{2} + -0.5\right)} + \left(-0.08333333333333333 \cdot \left({re}^{3} \cdot {im}^{2}\right) + -0.5 \cdot \left(re \cdot {im}^{2}\right)\right) \]

      6. associate-*r* 30.8%

         \[\leadsto {im}^{2} \cdot \left(-0.25 \cdot {re}^{2} + -0.5\right) + \left(\color{blue}{\left(-0.08333333333333333 \cdot {re}^{3}\right) \cdot {im}^{2}} + -0.5 \cdot \left(re \cdot {im}^{2}\right)\right) \]

      7. associate-*r* 30.8%

         \[\leadsto {im}^{2} \cdot \left(-0.25 \cdot {re}^{2} + -0.5\right) + \left(\left(-0.08333333333333333 \cdot {re}^{3}\right) \cdot {im}^{2} + \color{blue}{\left(-0.5 \cdot re\right) \cdot {im}^{2}}\right) \]

      8. distribute-rgt-out 30.8%

         \[\leadsto {im}^{2} \cdot \left(-0.25 \cdot {re}^{2} + -0.5\right) + \color{blue}{{im}^{2} \cdot \left(-0.08333333333333333 \cdot {re}^{3} + -0.5 \cdot re\right)} \]

      9. distribute-lft-out 30.8%

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\left(-0.25 \cdot {re}^{2} + -0.5\right) + \left(-0.08333333333333333 \cdot {re}^{3} + -0.5 \cdot re\right)\right)} \]

      10. unpow2 30.8%

          \[\leadsto \color{blue}{\left(im \cdot im\right)} \cdot \left(\left(-0.25 \cdot {re}^{2} + -0.5\right) + \left(-0.08333333333333333 \cdot {re}^{3} + -0.5 \cdot re\right)\right) \]

      11. +-commutative 30.8%

          \[\leadsto \left(im \cdot im\right) \cdot \left(\color{blue}{\left(-0.5 + -0.25 \cdot {re}^{2}\right)} + \left(-0.08333333333333333 \cdot {re}^{3} + -0.5 \cdot re\right)\right) \]

      12. unpow2 30.8%

          \[\leadsto \left(im \cdot im\right) \cdot \left(\left(-0.5 + -0.25 \cdot \color{blue}{\left(re \cdot re\right)}\right) + \left(-0.08333333333333333 \cdot {re}^{3} + -0.5 \cdot re\right)\right) \]

   10. Simplified 30.8%

       \[\leadsto \color{blue}{\left(im \cdot im\right) \cdot \left(\left(-0.5 + -0.25 \cdot \left(re \cdot re\right)\right) + re \cdot \left(-0.08333333333333333 \cdot \left(re \cdot re\right) + -0.5\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -82:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 1.05 \cdot 10^{-15} \lor \neg \left(re \leq 2.8 \cdot 10^{+146}\right):\\ \;\;\;\;\frac{1 - re \cdot re}{1 - re}\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(\left(-0.5 + -0.25 \cdot \left(re \cdot re\right)\right) + re \cdot \left(-0.5 + \left(re \cdot re\right) \cdot -0.08333333333333333\right)\right)\\ \end{array} \]

Alternative 7: 39.0% accurate, 15.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -460:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 1.05 \cdot 10^{-15}:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\left(re \cdot im\right) \cdot \left(re \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -460.0)
   (* -0.5 (* im im))
   (if (<= re 1.05e-15) (+ re 1.0) (* -0.25 (* (* re im) (* re im))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -460.0) {
		tmp = -0.5 * (im * im);
	} else if (re <= 1.05e-15) {
		tmp = re + 1.0;
	} else {
		tmp = -0.25 * ((re * im) * (re * 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 <= (-460.0d0)) then
        tmp = (-0.5d0) * (im * im)
    else if (re <= 1.05d-15) then
        tmp = re + 1.0d0
    else
        tmp = (-0.25d0) * ((re * im) * (re * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -460.0) {
		tmp = -0.5 * (im * im);
	} else if (re <= 1.05e-15) {
		tmp = re + 1.0;
	} else {
		tmp = -0.25 * ((re * im) * (re * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -460.0:
		tmp = -0.5 * (im * im)
	elif re <= 1.05e-15:
		tmp = re + 1.0
	else:
		tmp = -0.25 * ((re * im) * (re * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -460.0)
		tmp = Float64(-0.5 * Float64(im * im));
	elseif (re <= 1.05e-15)
		tmp = Float64(re + 1.0);
	else
		tmp = Float64(-0.25 * Float64(Float64(re * im) * Float64(re * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -460.0)
		tmp = -0.5 * (im * im);
	elseif (re <= 1.05e-15)
		tmp = re + 1.0;
	else
		tmp = -0.25 * ((re * im) * (re * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -460.0], N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.05e-15], N[(re + 1.0), $MachinePrecision], N[(-0.25 * N[(N[(re * im), $MachinePrecision] * N[(re * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -460

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 82.3%

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

      1. unpow2 82.3%

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

   4. Simplified 82.3%

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

   5. Taylor expanded in im around inf 82.3%

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

      1. *-commutative 82.3%

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

      2. associate-*r* 82.3%

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

      3. *-commutative 82.3%

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

      4. unpow2 82.3%

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

   7. Simplified 82.3%

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

   8. Taylor expanded in re around 0 28.2%

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

      1. unpow2 28.2%

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

   10. Simplified 28.2%

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

   if -460 < re < 1.0499999999999999e-15

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.3%

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

      1. +-commutative 99.3%

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

      2. *-rgt-identity 99.3%

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

      3. distribute-lft-out 99.3%

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

   4. Simplified 99.3%

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

   5. Taylor expanded in im around 0 56.0%

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

   if 1.0499999999999999e-15 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 71.0%

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

      1. unpow2 71.0%

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

   4. Simplified 71.0%

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

   5. Taylor expanded in im around inf 33.9%

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

      1. *-commutative 33.9%

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

      2. associate-*r* 33.9%

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

      3. *-commutative 33.9%

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

      4. unpow2 33.9%

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

   7. Simplified 33.9%

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

   8. Taylor expanded in re around 0 28.1%

      \[\leadsto \color{blue}{-0.5 \cdot \left(re \cdot {im}^{2}\right) + \left(-0.5 \cdot {im}^{2} + -0.25 \cdot \left({re}^{2} \cdot {im}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 28.1%

         \[\leadsto \color{blue}{\left(-0.5 \cdot re\right) \cdot {im}^{2}} + \left(-0.5 \cdot {im}^{2} + -0.25 \cdot \left({re}^{2} \cdot {im}^{2}\right)\right) \]

      2. *-commutative 28.1%

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(-0.5 \cdot re\right)} + \left(-0.5 \cdot {im}^{2} + -0.25 \cdot \left({re}^{2} \cdot {im}^{2}\right)\right) \]

      3. +-commutative 28.1%

         \[\leadsto {im}^{2} \cdot \left(-0.5 \cdot re\right) + \color{blue}{\left(-0.25 \cdot \left({re}^{2} \cdot {im}^{2}\right) + -0.5 \cdot {im}^{2}\right)} \]

      4. associate-*r* 28.1%

         \[\leadsto {im}^{2} \cdot \left(-0.5 \cdot re\right) + \left(\color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot {im}^{2}} + -0.5 \cdot {im}^{2}\right) \]

      5. distribute-rgt-out 28.1%

         \[\leadsto {im}^{2} \cdot \left(-0.5 \cdot re\right) + \color{blue}{{im}^{2} \cdot \left(-0.25 \cdot {re}^{2} + -0.5\right)} \]

      6. distribute-lft-out 28.1%

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(-0.5 \cdot re + \left(-0.25 \cdot {re}^{2} + -0.5\right)\right)} \]

      7. unpow2 28.1%

         \[\leadsto \color{blue}{\left(im \cdot im\right)} \cdot \left(-0.5 \cdot re + \left(-0.25 \cdot {re}^{2} + -0.5\right)\right) \]

      8. *-commutative 28.1%

         \[\leadsto \left(im \cdot im\right) \cdot \left(\color{blue}{re \cdot -0.5} + \left(-0.25 \cdot {re}^{2} + -0.5\right)\right) \]

      9. +-commutative 28.1%

         \[\leadsto \left(im \cdot im\right) \cdot \left(re \cdot -0.5 + \color{blue}{\left(-0.5 + -0.25 \cdot {re}^{2}\right)}\right) \]

      10. unpow2 28.1%

          \[\leadsto \left(im \cdot im\right) \cdot \left(re \cdot -0.5 + \left(-0.5 + -0.25 \cdot \color{blue}{\left(re \cdot re\right)}\right)\right) \]

   10. Simplified 28.1%

       \[\leadsto \color{blue}{\left(im \cdot im\right) \cdot \left(re \cdot -0.5 + \left(-0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\right)} \]

   11. Taylor expanded in re around inf 28.1%

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

       1. unpow2 28.1%

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

       2. unpow2 28.1%

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

       3. *-commutative 28.1%

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

       4. unswap-sqr 28.2%

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

   13. Simplified 28.2%

       \[\leadsto \color{blue}{-0.25 \cdot \left(\left(im \cdot re\right) \cdot \left(im \cdot re\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -460:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 1.05 \cdot 10^{-15}:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\left(re \cdot im\right) \cdot \left(re \cdot im\right)\right)\\ \end{array} \]

Alternative 8: 38.3% accurate, 18.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -310:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 1.05 \cdot 10^{-15}:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(re \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -310.0)
   (* -0.5 (* im im))
   (if (<= re 1.05e-15) (+ re 1.0) (* -0.5 (* re (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -310.0) {
		tmp = -0.5 * (im * im);
	} else if (re <= 1.05e-15) {
		tmp = re + 1.0;
	} else {
		tmp = -0.5 * (re * (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 <= (-310.0d0)) then
        tmp = (-0.5d0) * (im * im)
    else if (re <= 1.05d-15) then
        tmp = re + 1.0d0
    else
        tmp = (-0.5d0) * (re * (im * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -310.0) {
		tmp = -0.5 * (im * im);
	} else if (re <= 1.05e-15) {
		tmp = re + 1.0;
	} else {
		tmp = -0.5 * (re * (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -310.0:
		tmp = -0.5 * (im * im)
	elif re <= 1.05e-15:
		tmp = re + 1.0
	else:
		tmp = -0.5 * (re * (im * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -310.0)
		tmp = Float64(-0.5 * Float64(im * im));
	elseif (re <= 1.05e-15)
		tmp = Float64(re + 1.0);
	else
		tmp = Float64(-0.5 * Float64(re * Float64(im * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -310.0)
		tmp = -0.5 * (im * im);
	elseif (re <= 1.05e-15)
		tmp = re + 1.0;
	else
		tmp = -0.5 * (re * (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -310.0], N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.05e-15], N[(re + 1.0), $MachinePrecision], N[(-0.5 * N[(re * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -310

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 82.3%

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

      1. unpow2 82.3%

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

   4. Simplified 82.3%

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

   5. Taylor expanded in im around inf 82.3%

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

      1. *-commutative 82.3%

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

      2. associate-*r* 82.3%

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

      3. *-commutative 82.3%

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

      4. unpow2 82.3%

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

   7. Simplified 82.3%

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

   8. Taylor expanded in re around 0 28.2%

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

      1. unpow2 28.2%

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

   10. Simplified 28.2%

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

   if -310 < re < 1.0499999999999999e-15

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.3%

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

      1. +-commutative 99.3%

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

      2. *-rgt-identity 99.3%

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

      3. distribute-lft-out 99.3%

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

   4. Simplified 99.3%

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

   5. Taylor expanded in im around 0 56.0%

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

   if 1.0499999999999999e-15 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 6.9%

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

      1. +-commutative 6.9%

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

      2. *-rgt-identity 6.9%

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

      3. distribute-lft-out 6.9%

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

   4. Simplified 6.9%

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

   5. Taylor expanded in im around 0 28.7%

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

      1. unpow2 71.0%

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

   7. Simplified 28.7%

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

   8. Taylor expanded in im around inf 26.8%

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

      1. *-commutative 26.8%

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

      2. associate-*r* 26.8%

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

      3. *-commutative 26.8%

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

      4. unpow2 26.8%

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

   10. Simplified 26.8%

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

   11. Taylor expanded in re around inf 26.8%

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

       1. unpow2 26.8%

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

   13. Simplified 26.8%

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

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -310:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 1.05 \cdot 10^{-15}:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(re \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 9: 44.2% accurate, 18.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -125:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - re \cdot re}{1 - re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -125.0) (* -0.5 (* im im)) (/ (- 1.0 (* re re)) (- 1.0 re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -125.0) {
		tmp = -0.5 * (im * im);
	} else {
		tmp = (1.0 - (re * re)) / (1.0 - 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 <= (-125.0d0)) then
        tmp = (-0.5d0) * (im * im)
    else
        tmp = (1.0d0 - (re * re)) / (1.0d0 - re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -125.0) {
		tmp = -0.5 * (im * im);
	} else {
		tmp = (1.0 - (re * re)) / (1.0 - re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -125.0:
		tmp = -0.5 * (im * im)
	else:
		tmp = (1.0 - (re * re)) / (1.0 - re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -125.0)
		tmp = Float64(-0.5 * Float64(im * im));
	else
		tmp = Float64(Float64(1.0 - Float64(re * re)) / Float64(1.0 - re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -125.0)
		tmp = -0.5 * (im * im);
	else
		tmp = (1.0 - (re * re)) / (1.0 - re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -125.0], N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(re * re), $MachinePrecision]), $MachinePrecision] / N[(1.0 - re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -125

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 82.3%

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

      1. unpow2 82.3%

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

   4. Simplified 82.3%

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

   5. Taylor expanded in im around inf 82.3%

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

      1. *-commutative 82.3%

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

      2. associate-*r* 82.3%

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

      3. *-commutative 82.3%

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

      4. unpow2 82.3%

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

   7. Simplified 82.3%

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

   8. Taylor expanded in re around 0 28.2%

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

      1. unpow2 28.2%

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

   10. Simplified 28.2%

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

   if -125 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 69.8%

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

      1. +-commutative 69.8%

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

      2. *-rgt-identity 69.8%

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

      3. distribute-lft-out 69.8%

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

   4. Simplified 69.8%

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

   5. Taylor expanded in im around 0 45.3%

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

      1. unpow2 59.3%

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

   7. Simplified 45.3%

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

      1. *-commutative 45.3%

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

      2. distribute-lft-in 45.3%

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

      3. *-commutative 45.3%

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

      4. *-un-lft-identity 45.3%

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

      5. flip-+ 48.8%

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

      6. div-inv 48.8%

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

      7. fma-def 48.8%

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

      8. metadata-eval 48.8%

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

      9. *-commutative 48.8%

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

      10. associate-*l* 48.8%

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

   9. Applied egg-rr 48.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 - re \cdot re, \frac{1}{1 - re}, \left(1 + re\right) \cdot \left(im \cdot \left(im \cdot -0.5\right)\right)\right)} \]
   10. Step-by-step derivation

       1. fma-udef 48.8%

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

       2. +-commutative 48.8%

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

       3. *-commutative 48.8%

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

       4. *-commutative 48.8%

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

       5. associate-*r* 48.8%

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

       6. associate-*r/ 48.8%

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

       7. *-rgt-identity 48.8%

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

   11. Simplified 48.8%

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

   12. Taylor expanded in im around 0 49.9%

       \[\leadsto \color{blue}{\frac{1}{1 - re} - \frac{{re}^{2}}{1 - re}} \]
   13. Step-by-step derivation

       1. unpow2 49.9%

          \[\leadsto \frac{1}{1 - re} - \frac{\color{blue}{re \cdot re}}{1 - re} \]

       2. div-sub 49.9%

          \[\leadsto \color{blue}{\frac{1 - re \cdot re}{1 - re}} \]

   14. Simplified 49.9%

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

4. Final simplification 44.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -125:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - re \cdot re}{1 - re}\\ \end{array} \]

Alternative 10: 36.8% accurate, 22.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -360 \lor \neg \left(re \leq 1.05 \cdot 10^{-15}\right):\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;re + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -360.0) (not (<= re 1.05e-15))) (* -0.5 (* im im)) (+ re 1.0)))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -360.0) || !(re <= 1.05e-15)) {
		tmp = -0.5 * (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 <= (-360.0d0)) .or. (.not. (re <= 1.05d-15))) then
        tmp = (-0.5d0) * (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 <= -360.0) || !(re <= 1.05e-15)) {
		tmp = -0.5 * (im * im);
	} else {
		tmp = re + 1.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -360.0) or not (re <= 1.05e-15):
		tmp = -0.5 * (im * im)
	else:
		tmp = re + 1.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -360.0) || !(re <= 1.05e-15))
		tmp = Float64(-0.5 * 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 <= -360.0) || ~((re <= 1.05e-15)))
		tmp = -0.5 * (im * im);
	else
		tmp = re + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -360.0], N[Not[LessEqual[re, 1.05e-15]], $MachinePrecision]], N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision], N[(re + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -360 or 1.0499999999999999e-15 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 76.6%

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

      1. unpow2 76.6%

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

   4. Simplified 76.6%

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

   5. Taylor expanded in im around inf 58.1%

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

      1. *-commutative 58.1%

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

      2. associate-*r* 58.1%

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

      3. *-commutative 58.1%

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

      4. unpow2 58.1%

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

   7. Simplified 58.1%

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

   8. Taylor expanded in re around 0 21.5%

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

      1. unpow2 21.5%

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

   10. Simplified 21.5%

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

   if -360 < re < 1.0499999999999999e-15

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.3%

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

      1. +-commutative 99.3%

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

      2. *-rgt-identity 99.3%

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

      3. distribute-lft-out 99.3%

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

   4. Simplified 99.3%

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

   5. Taylor expanded in im around 0 56.0%

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

4. Final simplification 39.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -360 \lor \neg \left(re \leq 1.05 \cdot 10^{-15}\right):\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;re + 1\\ \end{array} \]

Alternative 11: 29.6% 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 53.4%

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

   1. +-commutative 53.4%

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

   2. *-rgt-identity 53.4%

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

   3. distribute-lft-out 53.4%

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

4. Simplified 53.4%

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

5. Taylor expanded in im around 0 30.4%

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

6. Final simplification 30.4%

   \[\leadsto re + 1 \]

Alternative 12: 29.1% 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 29.8%

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

4. Final simplification 29.8%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023200 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))