math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 6.5s

Alternatives: 8

Speedup: 1.0×

• 25.3% 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: 91.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.00012:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.00023:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 7 \cdot 10^{+269}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.00012)
   (exp re)
   (if (<= re 0.00023)
     (* (cos im) (+ re 1.0))
     (if (<= re 7e+269) (exp re) (* (+ 1.0 (* -0.5 (* im im))) (+ re 1.0))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.00012) {
		tmp = exp(re);
	} else if (re <= 0.00023) {
		tmp = cos(im) * (re + 1.0);
	} else if (re <= 7e+269) {
		tmp = exp(re);
	} else {
		tmp = (1.0 + (-0.5 * (im * im))) * (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 <= (-0.00012d0)) then
        tmp = exp(re)
    else if (re <= 0.00023d0) then
        tmp = cos(im) * (re + 1.0d0)
    else if (re <= 7d+269) then
        tmp = exp(re)
    else
        tmp = (1.0d0 + ((-0.5d0) * (im * im))) * (re + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.00012) {
		tmp = Math.exp(re);
	} else if (re <= 0.00023) {
		tmp = Math.cos(im) * (re + 1.0);
	} else if (re <= 7e+269) {
		tmp = Math.exp(re);
	} else {
		tmp = (1.0 + (-0.5 * (im * im))) * (re + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.00012:
		tmp = math.exp(re)
	elif re <= 0.00023:
		tmp = math.cos(im) * (re + 1.0)
	elif re <= 7e+269:
		tmp = math.exp(re)
	else:
		tmp = (1.0 + (-0.5 * (im * im))) * (re + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.00012)
		tmp = exp(re);
	elseif (re <= 0.00023)
		tmp = Float64(cos(im) * Float64(re + 1.0));
	elseif (re <= 7e+269)
		tmp = exp(re);
	else
		tmp = Float64(Float64(1.0 + Float64(-0.5 * Float64(im * im))) * Float64(re + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.00012)
		tmp = exp(re);
	elseif (re <= 0.00023)
		tmp = cos(im) * (re + 1.0);
	elseif (re <= 7e+269)
		tmp = exp(re);
	else
		tmp = (1.0 + (-0.5 * (im * im))) * (re + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.00012], N[Exp[re], $MachinePrecision], If[LessEqual[re, 0.00023], N[(N[Cos[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 7e+269], N[Exp[re], $MachinePrecision], N[(N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.20000000000000003e-4 or 2.3000000000000001e-4 < re < 7.0000000000000003e269

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 91.0%

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

   if -1.20000000000000003e-4 < re < 2.3000000000000001e-4

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.8%

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

      1. distribute-rgt1-in 99.8%

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

   4. Simplified 99.8%

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

   if 7.0000000000000003e269 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 11.1%

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

      1. distribute-rgt1-in 11.1%

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

   4. Simplified 11.1%

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

   5. Taylor expanded in im around 0 87.9%

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

      1. associate-+r+ 87.9%

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

      2. +-commutative 87.9%

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

      3. associate-*r* 87.9%

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

      4. +-commutative 87.9%

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

      5. distribute-rgt1-in 87.9%

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

   7. Simplified 87.9%

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

      1. unpow2 87.9%

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

   9. Applied egg-rr 87.9%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.00012:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.00023:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 7 \cdot 10^{+269}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(re + 1\right)\\ \end{array} \]

Alternative 3: 91.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -7.2 \cdot 10^{-10}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.000172:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;re \leq 7 \cdot 10^{+269}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -7.2e-10)
   (exp re)
   (if (<= re 0.000172)
     (cos im)
     (if (<= re 7e+269) (exp re) (* (+ 1.0 (* -0.5 (* im im))) (+ re 1.0))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -7.2e-10) {
		tmp = exp(re);
	} else if (re <= 0.000172) {
		tmp = cos(im);
	} else if (re <= 7e+269) {
		tmp = exp(re);
	} else {
		tmp = (1.0 + (-0.5 * (im * im))) * (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 <= (-7.2d-10)) then
        tmp = exp(re)
    else if (re <= 0.000172d0) then
        tmp = cos(im)
    else if (re <= 7d+269) then
        tmp = exp(re)
    else
        tmp = (1.0d0 + ((-0.5d0) * (im * im))) * (re + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -7.2e-10) {
		tmp = Math.exp(re);
	} else if (re <= 0.000172) {
		tmp = Math.cos(im);
	} else if (re <= 7e+269) {
		tmp = Math.exp(re);
	} else {
		tmp = (1.0 + (-0.5 * (im * im))) * (re + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -7.2e-10:
		tmp = math.exp(re)
	elif re <= 0.000172:
		tmp = math.cos(im)
	elif re <= 7e+269:
		tmp = math.exp(re)
	else:
		tmp = (1.0 + (-0.5 * (im * im))) * (re + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -7.2e-10)
		tmp = exp(re);
	elseif (re <= 0.000172)
		tmp = cos(im);
	elseif (re <= 7e+269)
		tmp = exp(re);
	else
		tmp = Float64(Float64(1.0 + Float64(-0.5 * Float64(im * im))) * Float64(re + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -7.2e-10)
		tmp = exp(re);
	elseif (re <= 0.000172)
		tmp = cos(im);
	elseif (re <= 7e+269)
		tmp = exp(re);
	else
		tmp = (1.0 + (-0.5 * (im * im))) * (re + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -7.2e-10], N[Exp[re], $MachinePrecision], If[LessEqual[re, 0.000172], N[Cos[im], $MachinePrecision], If[LessEqual[re, 7e+269], N[Exp[re], $MachinePrecision], N[(N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -7.2e-10 or 1.7200000000000001e-4 < re < 7.0000000000000003e269

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 91.0%

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

   if -7.2e-10 < re < 1.7200000000000001e-4

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.2%

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

   if 7.0000000000000003e269 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 11.1%

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

      1. distribute-rgt1-in 11.1%

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

   4. Simplified 11.1%

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

   5. Taylor expanded in im around 0 87.9%

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

      1. associate-+r+ 87.9%

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

      2. +-commutative 87.9%

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

      3. associate-*r* 87.9%

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

      4. +-commutative 87.9%

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

      5. distribute-rgt1-in 87.9%

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

   7. Simplified 87.9%

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

      1. unpow2 87.9%

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

   9. Applied egg-rr 87.9%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -7.2 \cdot 10^{-10}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.000172:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;re \leq 7 \cdot 10^{+269}:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(re + 1\right)\\ \end{array} \]

Alternative 4: 60.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -480:\\ \;\;\;\;-0.5 \cdot \left(\left(im \cdot im\right) \cdot \left(re + 1\right)\right)\\ \mathbf{elif}\;re \leq 0.00072:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -480.0)
   (* -0.5 (* (* im im) (+ re 1.0)))
   (if (<= re 0.00072) (cos im) (* (+ 1.0 (* -0.5 (* im im))) (+ re 1.0)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -480.0) {
		tmp = -0.5 * ((im * im) * (re + 1.0));
	} else if (re <= 0.00072) {
		tmp = cos(im);
	} else {
		tmp = (1.0 + (-0.5 * (im * im))) * (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 <= (-480.0d0)) then
        tmp = (-0.5d0) * ((im * im) * (re + 1.0d0))
    else if (re <= 0.00072d0) then
        tmp = cos(im)
    else
        tmp = (1.0d0 + ((-0.5d0) * (im * im))) * (re + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -480.0) {
		tmp = -0.5 * ((im * im) * (re + 1.0));
	} else if (re <= 0.00072) {
		tmp = Math.cos(im);
	} else {
		tmp = (1.0 + (-0.5 * (im * im))) * (re + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -480.0:
		tmp = -0.5 * ((im * im) * (re + 1.0))
	elif re <= 0.00072:
		tmp = math.cos(im)
	else:
		tmp = (1.0 + (-0.5 * (im * im))) * (re + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -480.0)
		tmp = Float64(-0.5 * Float64(Float64(im * im) * Float64(re + 1.0)));
	elseif (re <= 0.00072)
		tmp = cos(im);
	else
		tmp = Float64(Float64(1.0 + Float64(-0.5 * Float64(im * im))) * Float64(re + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -480.0)
		tmp = -0.5 * ((im * im) * (re + 1.0));
	elseif (re <= 0.00072)
		tmp = cos(im);
	else
		tmp = (1.0 + (-0.5 * (im * im))) * (re + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -480.0], N[(-0.5 * N[(N[(im * im), $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 0.00072], N[Cos[im], $MachinePrecision], N[(N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -480

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 2.2%

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

      1. distribute-rgt1-in 2.2%

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

   4. Simplified 2.2%

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

   5. Taylor expanded in im around 0 1.9%

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

      1. associate-+r+ 1.9%

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

      2. +-commutative 1.9%

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

      3. associate-*r* 1.9%

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

      4. +-commutative 1.9%

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

      5. distribute-rgt1-in 1.9%

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

   7. Simplified 1.9%

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

   8. Taylor expanded in im around inf 28.0%

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

      1. +-commutative 28.0%

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

   10. Simplified 28.0%

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

       1. unpow2 1.9%

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

   12. Applied egg-rr 28.0%

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

   if -480 < re < 7.20000000000000045e-4

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 99.2%

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

   if 7.20000000000000045e-4 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 6.1%

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

      1. distribute-rgt1-in 6.1%

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

   4. Simplified 6.1%

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

   5. Taylor expanded in im around 0 23.2%

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

      1. associate-+r+ 23.2%

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

      2. +-commutative 23.2%

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

      3. associate-*r* 23.2%

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

      4. +-commutative 23.2%

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

      5. distribute-rgt1-in 23.2%

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

   7. Simplified 23.2%

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

      1. unpow2 23.2%

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

   9. Applied egg-rr 23.2%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -480:\\ \;\;\;\;-0.5 \cdot \left(\left(im \cdot im\right) \cdot \left(re + 1\right)\right)\\ \mathbf{elif}\;re \leq 0.00072:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(re + 1\right)\\ \end{array} \]

Alternative 5: 39.5% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -12.5:\\ \;\;\;\;-0.5 \cdot \left(\left(im \cdot im\right) \cdot \left(re + 1\right)\right)\\ \mathbf{elif}\;re \leq 9 \cdot 10^{+22}:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -12.5)
   (* -0.5 (* (* im im) (+ re 1.0)))
   (if (<= re 9e+22) (+ re 1.0) (* (+ 1.0 (* -0.5 (* im im))) (+ re 1.0)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -12.5) {
		tmp = -0.5 * ((im * im) * (re + 1.0));
	} else if (re <= 9e+22) {
		tmp = re + 1.0;
	} else {
		tmp = (1.0 + (-0.5 * (im * im))) * (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 <= (-12.5d0)) then
        tmp = (-0.5d0) * ((im * im) * (re + 1.0d0))
    else if (re <= 9d+22) then
        tmp = re + 1.0d0
    else
        tmp = (1.0d0 + ((-0.5d0) * (im * im))) * (re + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -12.5) {
		tmp = -0.5 * ((im * im) * (re + 1.0));
	} else if (re <= 9e+22) {
		tmp = re + 1.0;
	} else {
		tmp = (1.0 + (-0.5 * (im * im))) * (re + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -12.5:
		tmp = -0.5 * ((im * im) * (re + 1.0))
	elif re <= 9e+22:
		tmp = re + 1.0
	else:
		tmp = (1.0 + (-0.5 * (im * im))) * (re + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -12.5)
		tmp = Float64(-0.5 * Float64(Float64(im * im) * Float64(re + 1.0)));
	elseif (re <= 9e+22)
		tmp = Float64(re + 1.0);
	else
		tmp = Float64(Float64(1.0 + Float64(-0.5 * Float64(im * im))) * Float64(re + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -12.5)
		tmp = -0.5 * ((im * im) * (re + 1.0));
	elseif (re <= 9e+22)
		tmp = re + 1.0;
	else
		tmp = (1.0 + (-0.5 * (im * im))) * (re + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -12.5], N[(-0.5 * N[(N[(im * im), $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 9e+22], N[(re + 1.0), $MachinePrecision], N[(N[(1.0 + N[(-0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -12.5

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 2.2%

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

      1. distribute-rgt1-in 2.2%

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

   4. Simplified 2.2%

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

   5. Taylor expanded in im around 0 1.9%

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

      1. associate-+r+ 1.9%

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

      2. +-commutative 1.9%

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

      3. associate-*r* 1.9%

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

      4. +-commutative 1.9%

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

      5. distribute-rgt1-in 1.9%

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

   7. Simplified 1.9%

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

   8. Taylor expanded in im around inf 28.0%

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

      1. +-commutative 28.0%

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

   10. Simplified 28.0%

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

       1. unpow2 1.9%

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

   12. Applied egg-rr 28.0%

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

   if -12.5 < re < 8.9999999999999996e22

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 92.9%

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

      1. distribute-rgt1-in 92.9%

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

   4. Simplified 92.9%

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

   5. Taylor expanded in im around 0 43.1%

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

      1. associate-+r+ 43.1%

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

      2. +-commutative 43.1%

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

      3. associate-*r* 43.1%

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

      4. +-commutative 43.1%

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

      5. distribute-rgt1-in 43.1%

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

   7. Simplified 43.1%

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

      1. unpow2 43.1%

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

   9. Applied egg-rr 43.1%

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

   10. Taylor expanded in im around 0 48.9%

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

   if 8.9999999999999996e22 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 5.3%

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

      1. distribute-rgt1-in 5.3%

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

   4. Simplified 5.3%

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

   5. Taylor expanded in im around 0 25.9%

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

      1. associate-+r+ 25.9%

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

      2. +-commutative 25.9%

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

      3. associate-*r* 25.9%

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

      4. +-commutative 25.9%

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

      5. distribute-rgt1-in 25.9%

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

   7. Simplified 25.9%

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

      1. unpow2 25.9%

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

   9. Applied egg-rr 25.9%

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

4. Final simplification 39.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -12.5:\\ \;\;\;\;-0.5 \cdot \left(\left(im \cdot im\right) \cdot \left(re + 1\right)\right)\\ \mathbf{elif}\;re \leq 9 \cdot 10^{+22}:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(re + 1\right)\\ \end{array} \]

Alternative 6: 39.0% accurate, 15.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -6.2 \lor \neg \left(re \leq 1.85 \cdot 10^{+28}\right):\\ \;\;\;\;-0.5 \cdot \left(\left(im \cdot im\right) \cdot \left(re + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -6.2) (not (<= re 1.85e+28)))
   (* -0.5 (* (* im im) (+ re 1.0)))
   (+ re 1.0)))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -6.2) || !(re <= 1.85e+28)) {
		tmp = -0.5 * ((im * im) * (re + 1.0));
	} 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 <= (-6.2d0)) .or. (.not. (re <= 1.85d+28))) then
        tmp = (-0.5d0) * ((im * im) * (re + 1.0d0))
    else
        tmp = re + 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -6.2) || !(re <= 1.85e+28)) {
		tmp = -0.5 * ((im * im) * (re + 1.0));
	} else {
		tmp = re + 1.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -6.2) or not (re <= 1.85e+28):
		tmp = -0.5 * ((im * im) * (re + 1.0))
	else:
		tmp = re + 1.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -6.2) || !(re <= 1.85e+28))
		tmp = Float64(-0.5 * Float64(Float64(im * im) * Float64(re + 1.0)));
	else
		tmp = Float64(re + 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -6.2) || ~((re <= 1.85e+28)))
		tmp = -0.5 * ((im * im) * (re + 1.0));
	else
		tmp = re + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -6.2], N[Not[LessEqual[re, 1.85e+28]], $MachinePrecision]], N[(-0.5 * N[(N[(im * im), $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -6.20000000000000018 or 1.85e28 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 3.8%

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

      1. distribute-rgt1-in 3.8%

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

   4. Simplified 3.8%

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

   5. Taylor expanded in im around 0 14.0%

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

      1. associate-+r+ 14.0%

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

      2. +-commutative 14.0%

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

      3. associate-*r* 14.0%

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

      4. +-commutative 14.0%

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

      5. distribute-rgt1-in 14.0%

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

   7. Simplified 14.0%

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

   8. Taylor expanded in im around inf 26.5%

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

      1. +-commutative 26.5%

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

   10. Simplified 26.5%

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

       1. unpow2 14.0%

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

   12. Applied egg-rr 26.5%

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

   if -6.20000000000000018 < re < 1.85e28

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 91.7%

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

      1. distribute-rgt1-in 91.7%

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

   4. Simplified 91.7%

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

   5. Taylor expanded in im around 0 42.6%

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

      1. associate-+r+ 42.6%

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

      2. +-commutative 42.6%

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

      3. associate-*r* 42.6%

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

      4. +-commutative 42.6%

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

      5. distribute-rgt1-in 42.6%

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

   7. Simplified 42.6%

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

      1. unpow2 42.6%

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

   9. Applied egg-rr 42.6%

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

   10. Taylor expanded in im around 0 48.2%

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

4. Final simplification 38.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -6.2 \lor \neg \left(re \leq 1.85 \cdot 10^{+28}\right):\\ \;\;\;\;-0.5 \cdot \left(\left(im \cdot im\right) \cdot \left(re + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re + 1\\ \end{array} \]

Alternative 7: 29.1% 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 53.6%

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

   1. distribute-rgt1-in 53.6%

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

4. Simplified 53.6%

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

5. Taylor expanded in im around 0 30.2%

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

   1. associate-+r+ 30.2%

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

   2. +-commutative 30.2%

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

   3. associate-*r* 30.2%

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

   4. +-commutative 30.2%

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

   5. distribute-rgt1-in 30.2%

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

7. Simplified 30.2%

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

   1. unpow2 30.2%

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

9. Applied egg-rr 30.2%

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

10. Taylor expanded in im around 0 28.6%

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

11. Final simplification 28.6%

    \[\leadsto re + 1 \]

Alternative 8: 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 re around 0 53.6%

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

   1. distribute-rgt1-in 53.6%

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

4. Simplified 53.6%

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

5. Taylor expanded in re around inf 3.9%

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

6. Taylor expanded in im around 0 3.5%

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

7. Final simplification 3.5%

   \[\leadsto re \]

Reproduce

herbie shell --seed 2023321 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))