math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 6.3s

Alternatives: 10

Speedup: 1.0×

• 24.3% 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: 71.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.5 \cdot 10^{+78}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.5e+78) (exp re) (* (exp re) (+ 1.0 (* -0.5 (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.5e+78) {
		tmp = exp(re);
	} else {
		tmp = exp(re) * (1.0 + (-0.5 * (im * im)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 1.5d+78) then
        tmp = exp(re)
    else
        tmp = exp(re) * (1.0d0 + ((-0.5d0) * (im * im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.5e+78) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.exp(re) * (1.0 + (-0.5 * (im * im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.5e+78:
		tmp = math.exp(re)
	else:
		tmp = math.exp(re) * (1.0 + (-0.5 * (im * im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.5e+78)
		tmp = exp(re);
	else
		tmp = Float64(exp(re) * Float64(1.0 + Float64(-0.5 * Float64(im * im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.5e+78)
		tmp = exp(re);
	else
		tmp = exp(re) * (1.0 + (-0.5 * (im * im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.5e+78], N[Exp[re], $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.49999999999999991e78

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 57.0%

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

      1. unpow2 57.0%

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

   4. Simplified 57.0%

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

   5. Taylor expanded in im around 0 67.4%

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

   if 1.49999999999999991e78 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 73.8%

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

      1. unpow2 73.8%

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

   4. Simplified 73.8%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.5 \cdot 10^{+78}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot \left(1 + -0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 3: 71.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(re, im)
	return exp(re)
end

MATLAB

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

Wolfram

code[re_, im_] := N[Exp[re], $MachinePrecision]

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in im around 0 59.7%

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

   1. unpow2 59.7%

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

4. Simplified 59.7%

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

5. Taylor expanded in im around 0 66.1%

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

6. Final simplification 66.1%

   \[\leadsto e^{re} \]

Alternative 4: 45.4% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.056:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 1400000:\\ \;\;\;\;re + 1\\ \mathbf{elif}\;re \leq 3.3 \cdot 10^{+165}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(\left(-0.5 + re \cdot \left(re \cdot -0.25\right)\right) + re \cdot \left(-0.5 + \left(re \cdot re\right) \cdot -0.08333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - re \cdot re}{1 - re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.056)
   (* -0.5 (* im im))
   (if (<= re 1400000.0)
     (+ re 1.0)
     (if (<= re 3.3e+165)
       (*
        (* im im)
        (+
         (+ -0.5 (* re (* re -0.25)))
         (* 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 <= -0.056) {
		tmp = -0.5 * (im * im);
	} else if (re <= 1400000.0) {
		tmp = re + 1.0;
	} else if (re <= 3.3e+165) {
		tmp = (im * im) * ((-0.5 + (re * (re * -0.25))) + (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 <= (-0.056d0)) then
        tmp = (-0.5d0) * (im * im)
    else if (re <= 1400000.0d0) then
        tmp = re + 1.0d0
    else if (re <= 3.3d+165) then
        tmp = (im * im) * (((-0.5d0) + (re * (re * (-0.25d0)))) + (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 <= -0.056) {
		tmp = -0.5 * (im * im);
	} else if (re <= 1400000.0) {
		tmp = re + 1.0;
	} else if (re <= 3.3e+165) {
		tmp = (im * im) * ((-0.5 + (re * (re * -0.25))) + (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 <= -0.056:
		tmp = -0.5 * (im * im)
	elif re <= 1400000.0:
		tmp = re + 1.0
	elif re <= 3.3e+165:
		tmp = (im * im) * ((-0.5 + (re * (re * -0.25))) + (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 <= -0.056)
		tmp = Float64(-0.5 * Float64(im * im));
	elseif (re <= 1400000.0)
		tmp = Float64(re + 1.0);
	elseif (re <= 3.3e+165)
		tmp = Float64(Float64(im * im) * Float64(Float64(-0.5 + Float64(re * Float64(re * -0.25))) + 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 <= -0.056)
		tmp = -0.5 * (im * im);
	elseif (re <= 1400000.0)
		tmp = re + 1.0;
	elseif (re <= 3.3e+165)
		tmp = (im * im) * ((-0.5 + (re * (re * -0.25))) + (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, -0.056], N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1400000.0], N[(re + 1.0), $MachinePrecision], If[LessEqual[re, 3.3e+165], N[(N[(im * im), $MachinePrecision] * N[(N[(-0.5 + N[(re * N[(re * -0.25), $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 < -0.0560000000000000012

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 80.9%

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

      1. unpow2 80.9%

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

   4. Simplified 80.9%

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

   5. Taylor expanded in im around inf 78.2%

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

      1. unpow2 78.2%

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

      2. associate-*r* 78.2%

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

      3. *-commutative 78.2%

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

      4. associate-*r* 78.2%

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

   7. Simplified 78.2%

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

   8. Taylor expanded in re around 0 29.1%

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

      1. unpow2 29.1%

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

   10. Simplified 29.1%

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

   if -0.0560000000000000012 < re < 1.4e6

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 41.9%

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

      1. unpow2 41.9%

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

   4. Simplified 41.9%

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

   5. Taylor expanded in re around 0 40.2%

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

      1. associate-+r+ 40.2%

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

      2. *-commutative 40.2%

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

      3. distribute-rgt1-in 40.4%

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

      4. *-commutative 40.4%

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

      5. +-commutative 40.4%

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

      6. unpow2 40.4%

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

      7. fma-udef 40.4%

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

      8. +-commutative 40.4%

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

   7. Simplified 40.4%

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

   8. Taylor expanded in im around 0 45.4%

      \[\leadsto \color{blue}{1 + re} \]
   9. Step-by-step derivation

      1. +-commutative 45.4%

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

   10. Simplified 45.4%

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

   if 1.4e6 < re < 3.2999999999999999e165

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 69.7%

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

      1. unpow2 69.7%

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

   4. Simplified 69.7%

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

   5. Taylor expanded in im around inf 33.3%

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

      1. unpow2 33.3%

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

      2. associate-*r* 33.3%

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

      3. *-commutative 33.3%

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

      4. associate-*r* 33.3%

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

   7. Simplified 33.3%

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

   8. Taylor expanded in re around 0 33.5%

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

      1. associate-+r+ 33.5%

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

      2. +-commutative 33.5%

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

      3. associate-*r* 33.5%

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

      4. distribute-rgt-out 33.5%

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

      5. associate-*r* 33.5%

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

      6. associate-*r* 33.5%

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

      7. distribute-rgt-out 33.5%

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

      8. distribute-lft-out 33.5%

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

      9. unpow2 33.5%

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

      10. unpow2 33.5%

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

      11. associate-*r* 33.5%

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

      12. *-commutative 33.5%

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

      13. *-commutative 33.5%

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

   10. Simplified 33.5%

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

   if 3.2999999999999999e165 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 75.0%

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

      1. unpow2 75.0%

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

   4. Simplified 75.0%

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

   5. Taylor expanded in re around 0 36.0%

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

      1. associate-+r+ 36.0%

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

      2. *-commutative 36.0%

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

      3. distribute-rgt1-in 36.0%

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

      4. *-commutative 36.0%

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

      5. +-commutative 36.0%

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

      6. unpow2 36.0%

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

      7. fma-udef 36.0%

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

      8. +-commutative 36.0%

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

   7. Simplified 36.0%

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

      1. flip-+ 75.0%

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

      2. associate-*r/ 75.0%

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

      3. metadata-eval 75.0%

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

   9. Applied egg-rr 75.0%

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

   10. Taylor expanded in im around 0 62.5%

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

       1. unpow2 62.5%

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

   12. Simplified 62.5%

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

4. Final simplification 40.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.056:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 1400000:\\ \;\;\;\;re + 1\\ \mathbf{elif}\;re \leq 3.3 \cdot 10^{+165}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(\left(-0.5 + re \cdot \left(re \cdot -0.25\right)\right) + re \cdot \left(-0.5 + \left(re \cdot re\right) \cdot -0.08333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - re \cdot re}{1 - re}\\ \end{array} \]

Alternative 5: 45.3% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.056:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 80000000:\\ \;\;\;\;re + 1\\ \mathbf{elif}\;re \leq 3.3 \cdot 10^{+165}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - re \cdot re}{1 - re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.056)
   (* -0.5 (* im im))
   (if (<= re 80000000.0)
     (+ re 1.0)
     (if (<= re 3.3e+165)
       (* (* im im) (* -0.25 (* re re)))
       (/ (- 1.0 (* re re)) (- 1.0 re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.056) {
		tmp = -0.5 * (im * im);
	} else if (re <= 80000000.0) {
		tmp = re + 1.0;
	} else if (re <= 3.3e+165) {
		tmp = (im * im) * (-0.25 * (re * re));
	} 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 <= (-0.056d0)) then
        tmp = (-0.5d0) * (im * im)
    else if (re <= 80000000.0d0) then
        tmp = re + 1.0d0
    else if (re <= 3.3d+165) then
        tmp = (im * im) * ((-0.25d0) * (re * re))
    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 <= -0.056) {
		tmp = -0.5 * (im * im);
	} else if (re <= 80000000.0) {
		tmp = re + 1.0;
	} else if (re <= 3.3e+165) {
		tmp = (im * im) * (-0.25 * (re * re));
	} else {
		tmp = (1.0 - (re * re)) / (1.0 - re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.056:
		tmp = -0.5 * (im * im)
	elif re <= 80000000.0:
		tmp = re + 1.0
	elif re <= 3.3e+165:
		tmp = (im * im) * (-0.25 * (re * re))
	else:
		tmp = (1.0 - (re * re)) / (1.0 - re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.056)
		tmp = Float64(-0.5 * Float64(im * im));
	elseif (re <= 80000000.0)
		tmp = Float64(re + 1.0);
	elseif (re <= 3.3e+165)
		tmp = Float64(Float64(im * im) * Float64(-0.25 * Float64(re * re)));
	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 <= -0.056)
		tmp = -0.5 * (im * im);
	elseif (re <= 80000000.0)
		tmp = re + 1.0;
	elseif (re <= 3.3e+165)
		tmp = (im * im) * (-0.25 * (re * re));
	else
		tmp = (1.0 - (re * re)) / (1.0 - re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.056], N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 80000000.0], N[(re + 1.0), $MachinePrecision], If[LessEqual[re, 3.3e+165], N[(N[(im * im), $MachinePrecision] * N[(-0.25 * N[(re * re), $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 < -0.0560000000000000012

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 80.9%

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

      1. unpow2 80.9%

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

   4. Simplified 80.9%

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

   5. Taylor expanded in im around inf 78.2%

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

      1. unpow2 78.2%

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

      2. associate-*r* 78.2%

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

      3. *-commutative 78.2%

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

      4. associate-*r* 78.2%

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

   7. Simplified 78.2%

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

   8. Taylor expanded in re around 0 29.1%

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

      1. unpow2 29.1%

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

   10. Simplified 29.1%

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

   if -0.0560000000000000012 < re < 8e7

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 41.9%

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

      1. unpow2 41.9%

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

   4. Simplified 41.9%

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

   5. Taylor expanded in re around 0 40.2%

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

      1. associate-+r+ 40.2%

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

      2. *-commutative 40.2%

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

      3. distribute-rgt1-in 40.4%

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

      4. *-commutative 40.4%

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

      5. +-commutative 40.4%

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

      6. unpow2 40.4%

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

      7. fma-udef 40.4%

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

      8. +-commutative 40.4%

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

   7. Simplified 40.4%

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

   8. Taylor expanded in im around 0 45.4%

      \[\leadsto \color{blue}{1 + re} \]
   9. Step-by-step derivation

      1. +-commutative 45.4%

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

   10. Simplified 45.4%

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

   if 8e7 < re < 3.2999999999999999e165

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 69.7%

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

      1. unpow2 69.7%

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

   4. Simplified 69.7%

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

   5. Taylor expanded in im around inf 33.3%

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

      1. unpow2 33.3%

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

      2. associate-*r* 33.3%

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

      3. *-commutative 33.3%

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

      4. associate-*r* 33.3%

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

   7. Simplified 33.3%

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

   8. Taylor expanded in re around 0 31.0%

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

      1. +-commutative 31.0%

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

      2. +-commutative 31.0%

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

      3. associate-+l+ 31.0%

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

      4. associate-*r* 31.0%

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

      5. *-commutative 31.0%

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

      6. associate-*r* 31.0%

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

      7. distribute-rgt-out 31.0%

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

      8. metadata-eval 31.0%

         \[\leadsto {im}^{2} \cdot \left(-0.25 \cdot {re}^{2}\right) + {im}^{2} \cdot \left(\color{blue}{-0.5 \cdot 1} + -0.5 \cdot re\right) \]

      9. distribute-lft-in 31.0%

         \[\leadsto {im}^{2} \cdot \left(-0.25 \cdot {re}^{2}\right) + {im}^{2} \cdot \color{blue}{\left(-0.5 \cdot \left(1 + re\right)\right)} \]

      10. distribute-lft-out 31.0%

          \[\leadsto \color{blue}{{im}^{2} \cdot \left(-0.25 \cdot {re}^{2} + -0.5 \cdot \left(1 + re\right)\right)} \]

      11. unpow2 31.0%

          \[\leadsto \color{blue}{\left(im \cdot im\right)} \cdot \left(-0.25 \cdot {re}^{2} + -0.5 \cdot \left(1 + re\right)\right) \]

      12. unpow2 31.0%

          \[\leadsto \left(im \cdot im\right) \cdot \left(-0.25 \cdot \color{blue}{\left(re \cdot re\right)} + -0.5 \cdot \left(1 + re\right)\right) \]

      13. associate-*r* 31.0%

          \[\leadsto \left(im \cdot im\right) \cdot \left(\color{blue}{\left(-0.25 \cdot re\right) \cdot re} + -0.5 \cdot \left(1 + re\right)\right) \]

      14. *-commutative 31.0%

          \[\leadsto \left(im \cdot im\right) \cdot \left(\color{blue}{re \cdot \left(-0.25 \cdot re\right)} + -0.5 \cdot \left(1 + re\right)\right) \]

      15. *-commutative 31.0%

          \[\leadsto \left(im \cdot im\right) \cdot \left(re \cdot \color{blue}{\left(re \cdot -0.25\right)} + -0.5 \cdot \left(1 + re\right)\right) \]

      16. distribute-lft-in 31.0%

          \[\leadsto \left(im \cdot im\right) \cdot \left(re \cdot \left(re \cdot -0.25\right) + \color{blue}{\left(-0.5 \cdot 1 + -0.5 \cdot re\right)}\right) \]

      17. metadata-eval 31.0%

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

   10. Simplified 31.0%

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

   11. Taylor expanded in re around inf 31.0%

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

       1. *-commutative 31.0%

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

       2. unpow2 31.0%

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

   13. Simplified 31.0%

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

   if 3.2999999999999999e165 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 75.0%

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

      1. unpow2 75.0%

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

   4. Simplified 75.0%

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

   5. Taylor expanded in re around 0 36.0%

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

      1. associate-+r+ 36.0%

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

      2. *-commutative 36.0%

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

      3. distribute-rgt1-in 36.0%

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

      4. *-commutative 36.0%

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

      5. +-commutative 36.0%

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

      6. unpow2 36.0%

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

      7. fma-udef 36.0%

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

      8. +-commutative 36.0%

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

   7. Simplified 36.0%

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

      1. flip-+ 75.0%

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

      2. associate-*r/ 75.0%

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

      3. metadata-eval 75.0%

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

   9. Applied egg-rr 75.0%

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

   10. Taylor expanded in im around 0 62.5%

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

       1. unpow2 62.5%

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

   12. Simplified 62.5%

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

4. Final simplification 40.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.056:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 80000000:\\ \;\;\;\;re + 1\\ \mathbf{elif}\;re \leq 3.3 \cdot 10^{+165}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - re \cdot re}{1 - re}\\ \end{array} \]

Alternative 6: 39.4% accurate, 15.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.056:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 1500000:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.056)
   (* -0.5 (* im im))
   (if (<= re 1500000.0) (+ re 1.0) (* (* im im) (* -0.25 (* re re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.056) {
		tmp = -0.5 * (im * im);
	} else if (re <= 1500000.0) {
		tmp = re + 1.0;
	} else {
		tmp = (im * im) * (-0.25 * (re * 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 <= (-0.056d0)) then
        tmp = (-0.5d0) * (im * im)
    else if (re <= 1500000.0d0) then
        tmp = re + 1.0d0
    else
        tmp = (im * im) * ((-0.25d0) * (re * re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.056) {
		tmp = -0.5 * (im * im);
	} else if (re <= 1500000.0) {
		tmp = re + 1.0;
	} else {
		tmp = (im * im) * (-0.25 * (re * re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.056:
		tmp = -0.5 * (im * im)
	elif re <= 1500000.0:
		tmp = re + 1.0
	else:
		tmp = (im * im) * (-0.25 * (re * re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.056)
		tmp = Float64(-0.5 * Float64(im * im));
	elseif (re <= 1500000.0)
		tmp = Float64(re + 1.0);
	else
		tmp = Float64(Float64(im * im) * Float64(-0.25 * Float64(re * re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.056)
		tmp = -0.5 * (im * im);
	elseif (re <= 1500000.0)
		tmp = re + 1.0;
	else
		tmp = (im * im) * (-0.25 * (re * re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.056], N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1500000.0], N[(re + 1.0), $MachinePrecision], N[(N[(im * im), $MachinePrecision] * N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.0560000000000000012

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 80.9%

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

      1. unpow2 80.9%

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

   4. Simplified 80.9%

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

   5. Taylor expanded in im around inf 78.2%

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

      1. unpow2 78.2%

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

      2. associate-*r* 78.2%

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

      3. *-commutative 78.2%

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

      4. associate-*r* 78.2%

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

   7. Simplified 78.2%

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

   8. Taylor expanded in re around 0 29.1%

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

      1. unpow2 29.1%

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

   10. Simplified 29.1%

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

   if -0.0560000000000000012 < re < 1.5e6

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 41.9%

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

      1. unpow2 41.9%

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

   4. Simplified 41.9%

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

   5. Taylor expanded in re around 0 40.2%

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

      1. associate-+r+ 40.2%

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

      2. *-commutative 40.2%

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

      3. distribute-rgt1-in 40.4%

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

      4. *-commutative 40.4%

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

      5. +-commutative 40.4%

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

      6. unpow2 40.4%

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

      7. fma-udef 40.4%

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

      8. +-commutative 40.4%

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

   7. Simplified 40.4%

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

   8. Taylor expanded in im around 0 45.4%

      \[\leadsto \color{blue}{1 + re} \]
   9. Step-by-step derivation

      1. +-commutative 45.4%

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

   10. Simplified 45.4%

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

   if 1.5e6 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 71.9%

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

      1. unpow2 71.9%

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

   4. Simplified 71.9%

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

   5. Taylor expanded in im around inf 35.1%

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

      1. unpow2 35.1%

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

      2. associate-*r* 35.1%

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

      3. *-commutative 35.1%

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

      4. associate-*r* 35.1%

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

   7. Simplified 35.1%

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

   8. Taylor expanded in re around 0 33.8%

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

      1. +-commutative 33.8%

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

      2. +-commutative 33.8%

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

      3. associate-+l+ 33.8%

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

      4. associate-*r* 33.8%

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

      5. *-commutative 33.8%

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

      6. associate-*r* 33.8%

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

      7. distribute-rgt-out 33.8%

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

      8. metadata-eval 33.8%

         \[\leadsto {im}^{2} \cdot \left(-0.25 \cdot {re}^{2}\right) + {im}^{2} \cdot \left(\color{blue}{-0.5 \cdot 1} + -0.5 \cdot re\right) \]

      9. distribute-lft-in 33.8%

         \[\leadsto {im}^{2} \cdot \left(-0.25 \cdot {re}^{2}\right) + {im}^{2} \cdot \color{blue}{\left(-0.5 \cdot \left(1 + re\right)\right)} \]

      10. distribute-lft-out 33.8%

          \[\leadsto \color{blue}{{im}^{2} \cdot \left(-0.25 \cdot {re}^{2} + -0.5 \cdot \left(1 + re\right)\right)} \]

      11. unpow2 33.8%

          \[\leadsto \color{blue}{\left(im \cdot im\right)} \cdot \left(-0.25 \cdot {re}^{2} + -0.5 \cdot \left(1 + re\right)\right) \]

      12. unpow2 33.8%

          \[\leadsto \left(im \cdot im\right) \cdot \left(-0.25 \cdot \color{blue}{\left(re \cdot re\right)} + -0.5 \cdot \left(1 + re\right)\right) \]

      13. associate-*r* 33.8%

          \[\leadsto \left(im \cdot im\right) \cdot \left(\color{blue}{\left(-0.25 \cdot re\right) \cdot re} + -0.5 \cdot \left(1 + re\right)\right) \]

      14. *-commutative 33.8%

          \[\leadsto \left(im \cdot im\right) \cdot \left(\color{blue}{re \cdot \left(-0.25 \cdot re\right)} + -0.5 \cdot \left(1 + re\right)\right) \]

      15. *-commutative 33.8%

          \[\leadsto \left(im \cdot im\right) \cdot \left(re \cdot \color{blue}{\left(re \cdot -0.25\right)} + -0.5 \cdot \left(1 + re\right)\right) \]

      16. distribute-lft-in 33.8%

          \[\leadsto \left(im \cdot im\right) \cdot \left(re \cdot \left(re \cdot -0.25\right) + \color{blue}{\left(-0.5 \cdot 1 + -0.5 \cdot re\right)}\right) \]

      17. metadata-eval 33.8%

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

   10. Simplified 33.8%

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

   11. Taylor expanded in re around inf 33.8%

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

       1. *-commutative 33.8%

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

       2. unpow2 33.8%

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

   13. Simplified 33.8%

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

4. Final simplification 38.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.056:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 1500000:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]

Alternative 7: 38.8% accurate, 18.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.056:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 6700000:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(re \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.056)
   (* -0.5 (* im im))
   (if (<= re 6700000.0) (+ re 1.0) (* (* im im) (* re -0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.056) {
		tmp = -0.5 * (im * im);
	} else if (re <= 6700000.0) {
		tmp = re + 1.0;
	} else {
		tmp = (im * im) * (re * -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-0.056d0)) then
        tmp = (-0.5d0) * (im * im)
    else if (re <= 6700000.0d0) then
        tmp = re + 1.0d0
    else
        tmp = (im * im) * (re * (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.056) {
		tmp = -0.5 * (im * im);
	} else if (re <= 6700000.0) {
		tmp = re + 1.0;
	} else {
		tmp = (im * im) * (re * -0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.056:
		tmp = -0.5 * (im * im)
	elif re <= 6700000.0:
		tmp = re + 1.0
	else:
		tmp = (im * im) * (re * -0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.056)
		tmp = Float64(-0.5 * Float64(im * im));
	elseif (re <= 6700000.0)
		tmp = Float64(re + 1.0);
	else
		tmp = Float64(Float64(im * im) * Float64(re * -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.056)
		tmp = -0.5 * (im * im);
	elseif (re <= 6700000.0)
		tmp = re + 1.0;
	else
		tmp = (im * im) * (re * -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.056], N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 6700000.0], N[(re + 1.0), $MachinePrecision], N[(N[(im * im), $MachinePrecision] * N[(re * -0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.0560000000000000012

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 80.9%

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

      1. unpow2 80.9%

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

   4. Simplified 80.9%

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

   5. Taylor expanded in im around inf 78.2%

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

      1. unpow2 78.2%

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

      2. associate-*r* 78.2%

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

      3. *-commutative 78.2%

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

      4. associate-*r* 78.2%

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

   7. Simplified 78.2%

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

   8. Taylor expanded in re around 0 29.1%

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

      1. unpow2 29.1%

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

   10. Simplified 29.1%

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

   if -0.0560000000000000012 < re < 6.7e6

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 41.9%

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

      1. unpow2 41.9%

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

   4. Simplified 41.9%

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

   5. Taylor expanded in re around 0 40.2%

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

      1. associate-+r+ 40.2%

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

      2. *-commutative 40.2%

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

      3. distribute-rgt1-in 40.4%

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

      4. *-commutative 40.4%

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

      5. +-commutative 40.4%

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

      6. unpow2 40.4%

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

      7. fma-udef 40.4%

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

      8. +-commutative 40.4%

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

   7. Simplified 40.4%

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

   8. Taylor expanded in im around 0 45.4%

      \[\leadsto \color{blue}{1 + re} \]
   9. Step-by-step derivation

      1. +-commutative 45.4%

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

   10. Simplified 45.4%

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

   if 6.7e6 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 71.9%

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

      1. unpow2 71.9%

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

   4. Simplified 71.9%

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

   5. Taylor expanded in im around inf 35.1%

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

      1. unpow2 35.1%

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

      2. associate-*r* 35.1%

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

      3. *-commutative 35.1%

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

      4. associate-*r* 35.1%

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

   7. Simplified 35.1%

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

   8. Taylor expanded in re around 0 30.6%

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

      1. +-commutative 30.6%

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

      2. associate-*r* 30.6%

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

      3. distribute-rgt-out 30.6%

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

      4. unpow2 30.6%

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

   10. Simplified 30.6%

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

   11. Taylor expanded in re around inf 30.6%

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

       1. unpow2 30.6%

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

       2. associate-*r* 30.6%

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

       3. *-commutative 30.6%

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

   13. Simplified 30.6%

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

4. Final simplification 37.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.056:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 6700000:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(re \cdot -0.5\right)\\ \end{array} \]

Alternative 8: 37.4% accurate, 22.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.056 \lor \neg \left(re \leq 1400000\right):\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;re + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.056) (not (<= re 1400000.0)))
   (* -0.5 (* im im))
   (+ re 1.0)))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.056) || !(re <= 1400000.0)) {
		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 <= (-0.056d0)) .or. (.not. (re <= 1400000.0d0))) 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 <= -0.056) || !(re <= 1400000.0)) {
		tmp = -0.5 * (im * im);
	} else {
		tmp = re + 1.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.056) or not (re <= 1400000.0):
		tmp = -0.5 * (im * im)
	else:
		tmp = re + 1.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.056) || !(re <= 1400000.0))
		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 <= -0.056) || ~((re <= 1400000.0)))
		tmp = -0.5 * (im * im);
	else
		tmp = re + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.056], N[Not[LessEqual[re, 1400000.0]], $MachinePrecision]], N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision], N[(re + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -0.0560000000000000012 or 1.4e6 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 77.0%

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

      1. unpow2 77.0%

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

   4. Simplified 77.0%

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

   5. Taylor expanded in im around inf 59.3%

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

      1. unpow2 59.3%

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

      2. associate-*r* 59.3%

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

      3. *-commutative 59.3%

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

      4. associate-*r* 59.3%

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

   7. Simplified 59.3%

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

   8. Taylor expanded in re around 0 23.9%

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

      1. unpow2 23.9%

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

   10. Simplified 23.9%

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

   if -0.0560000000000000012 < re < 1.4e6

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 41.9%

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

      1. unpow2 41.9%

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

   4. Simplified 41.9%

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

   5. Taylor expanded in re around 0 40.2%

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

      1. associate-+r+ 40.2%

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

      2. *-commutative 40.2%

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

      3. distribute-rgt1-in 40.4%

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

      4. *-commutative 40.4%

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

      5. +-commutative 40.4%

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

      6. unpow2 40.4%

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

      7. fma-udef 40.4%

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

      8. +-commutative 40.4%

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

   7. Simplified 40.4%

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

   8. Taylor expanded in im around 0 45.4%

      \[\leadsto \color{blue}{1 + re} \]
   9. Step-by-step derivation

      1. +-commutative 45.4%

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

   10. Simplified 45.4%

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

4. Final simplification 34.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.056 \lor \neg \left(re \leq 1400000\right):\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;re + 1\\ \end{array} \]

Alternative 9: 29.2% 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 im around 0 59.7%

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

   1. unpow2 59.7%

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

4. Simplified 59.7%

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

5. Taylor expanded in re around 0 27.4%

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

   1. associate-+r+ 27.4%

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

   2. *-commutative 27.4%

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

   3. distribute-rgt1-in 27.6%

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

   4. *-commutative 27.6%

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

   5. +-commutative 27.6%

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

   6. unpow2 27.6%

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

   7. fma-udef 27.6%

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

   8. +-commutative 27.6%

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

7. Simplified 27.6%

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

8. Taylor expanded in im around 0 23.7%

   \[\leadsto \color{blue}{1 + re} \]
9. Step-by-step derivation

   1. +-commutative 23.7%

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

10. Simplified 23.7%

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

11. Final simplification 23.7%

    \[\leadsto re + 1 \]

Alternative 10: 3.5% accurate, 203.0× speedup

Math

\[\begin{array}{l} \\ re \end{array} \]

FPCore

(FPCore (re im) :precision binary64 re)

C

double code(double re, double im) {
	return re;
}

Fortran

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

Java

public static double code(double re, double im) {
	return re;
}

Python

def code(re, im):
	return re

Julia

function code(re, im)
	return re
end

MATLAB

function tmp = code(re, im)
	tmp = re;
end

Wolfram

code[re_, im_] := re

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in im around 0 59.7%

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

   1. unpow2 59.7%

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

4. Simplified 59.7%

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

5. Taylor expanded in re around 0 27.4%

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

   1. associate-+r+ 27.4%

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

   2. *-commutative 27.4%

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

   3. distribute-rgt1-in 27.6%

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

   4. *-commutative 27.6%

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

   5. +-commutative 27.6%

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

   6. unpow2 27.6%

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

   7. fma-udef 27.6%

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

   8. +-commutative 27.6%

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

7. Simplified 27.6%

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

8. Taylor expanded in re around inf 9.5%

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

   1. *-commutative 9.5%

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

   2. +-commutative 9.5%

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

   3. unpow2 9.5%

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

   4. fma-udef 9.5%

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

10. Simplified 9.5%

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

11. Taylor expanded in im around 0 3.3%

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

12. Final simplification 3.3%

    \[\leadsto re \]

Reproduce

herbie shell --seed 2023195 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))