math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 9.3s

Alternatives: 22

Speedup: 1.0×

• 49.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (exp((0.0d0 - im)) + exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (Math.exp((0.0 - im)) + Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.sin(re)) * (math.exp((0.0 - im)) + math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (exp((0.0d0 - im)) + exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (Math.exp((0.0 - im)) + Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.sin(re)) * (math.exp((0.0 - im)) + math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp(-im) + exp(im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (exp(-im) + exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (Math.exp(-im) + Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.sin(re)) * (math.exp(-im) + math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) + exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (exp(-im) + exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Step-by-step derivation

   1. sub0-neg 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 92.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.5:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(\left(2 + im \cdot im\right) + {im}^{4} \cdot 0.08333333333333333\right)\\ \mathbf{elif}\;im \leq 1.2 \cdot 10^{+77}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.5)
   (*
    (* 0.5 (sin re))
    (+ (+ 2.0 (* im im)) (* (pow im 4.0) 0.08333333333333333)))
   (if (<= im 1.2e+77)
     (* (* 0.5 re) (+ (exp (- im)) (exp im)))
     (* 0.041666666666666664 (* (sin re) (pow im 4.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.5) {
		tmp = (0.5 * sin(re)) * ((2.0 + (im * im)) + (pow(im, 4.0) * 0.08333333333333333));
	} else if (im <= 1.2e+77) {
		tmp = (0.5 * re) * (exp(-im) + exp(im));
	} else {
		tmp = 0.041666666666666664 * (sin(re) * pow(im, 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 2.5d0) then
        tmp = (0.5d0 * sin(re)) * ((2.0d0 + (im * im)) + ((im ** 4.0d0) * 0.08333333333333333d0))
    else if (im <= 1.2d+77) then
        tmp = (0.5d0 * re) * (exp(-im) + exp(im))
    else
        tmp = 0.041666666666666664d0 * (sin(re) * (im ** 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.5) {
		tmp = (0.5 * Math.sin(re)) * ((2.0 + (im * im)) + (Math.pow(im, 4.0) * 0.08333333333333333));
	} else if (im <= 1.2e+77) {
		tmp = (0.5 * re) * (Math.exp(-im) + Math.exp(im));
	} else {
		tmp = 0.041666666666666664 * (Math.sin(re) * Math.pow(im, 4.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.5:
		tmp = (0.5 * math.sin(re)) * ((2.0 + (im * im)) + (math.pow(im, 4.0) * 0.08333333333333333))
	elif im <= 1.2e+77:
		tmp = (0.5 * re) * (math.exp(-im) + math.exp(im))
	else:
		tmp = 0.041666666666666664 * (math.sin(re) * math.pow(im, 4.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.5)
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(Float64(2.0 + Float64(im * im)) + Float64((im ^ 4.0) * 0.08333333333333333)));
	elseif (im <= 1.2e+77)
		tmp = Float64(Float64(0.5 * re) * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64(0.041666666666666664 * Float64(sin(re) * (im ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.5)
		tmp = (0.5 * sin(re)) * ((2.0 + (im * im)) + ((im ^ 4.0) * 0.08333333333333333));
	elseif (im <= 1.2e+77)
		tmp = (0.5 * re) * (exp(-im) + exp(im));
	else
		tmp = 0.041666666666666664 * (sin(re) * (im ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.5], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision] + N[(N[Power[im, 4.0], $MachinePrecision] * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.2e+77], N[(N[(0.5 * re), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.041666666666666664 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 2.5

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 87.7%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]

   6. Simplified 87.7%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left(2 + im \cdot im\right) + {im}^{4} \cdot 0.08333333333333333\right)} \]

   if 2.5 < im < 1.1999999999999999e77

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 76.9%

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

   6. Simplified 76.9%

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

   if 1.1999999999999999e77 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]

   6. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left(2 + im \cdot im\right) + {im}^{4} \cdot 0.08333333333333333\right)} \]

   7. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto 0.041666666666666664 \cdot \color{blue}{\left({im}^{4} \cdot \sin re\right)} \]

   9. Simplified 100.0%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left({im}^{4} \cdot \sin re\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.5:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(\left(2 + im \cdot im\right) + {im}^{4} \cdot 0.08333333333333333\right)\\ \mathbf{elif}\;im \leq 1.2 \cdot 10^{+77}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\\ \end{array} \]

Alternative 3: 86.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.5:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.2 \cdot 10^{+77}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.5)
   (* (* 0.5 (sin re)) (+ 2.0 (* im im)))
   (if (<= im 1.2e+77)
     (* (* 0.5 re) (+ (exp (- im)) (exp im)))
     (* 0.041666666666666664 (* (sin re) (pow im 4.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.5) {
		tmp = (0.5 * sin(re)) * (2.0 + (im * im));
	} else if (im <= 1.2e+77) {
		tmp = (0.5 * re) * (exp(-im) + exp(im));
	} else {
		tmp = 0.041666666666666664 * (sin(re) * pow(im, 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 2.5d0) then
        tmp = (0.5d0 * sin(re)) * (2.0d0 + (im * im))
    else if (im <= 1.2d+77) then
        tmp = (0.5d0 * re) * (exp(-im) + exp(im))
    else
        tmp = 0.041666666666666664d0 * (sin(re) * (im ** 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.5) {
		tmp = (0.5 * Math.sin(re)) * (2.0 + (im * im));
	} else if (im <= 1.2e+77) {
		tmp = (0.5 * re) * (Math.exp(-im) + Math.exp(im));
	} else {
		tmp = 0.041666666666666664 * (Math.sin(re) * Math.pow(im, 4.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.5:
		tmp = (0.5 * math.sin(re)) * (2.0 + (im * im))
	elif im <= 1.2e+77:
		tmp = (0.5 * re) * (math.exp(-im) + math.exp(im))
	else:
		tmp = 0.041666666666666664 * (math.sin(re) * math.pow(im, 4.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.5)
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(2.0 + Float64(im * im)));
	elseif (im <= 1.2e+77)
		tmp = Float64(Float64(0.5 * re) * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64(0.041666666666666664 * Float64(sin(re) * (im ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.5)
		tmp = (0.5 * sin(re)) * (2.0 + (im * im));
	elseif (im <= 1.2e+77)
		tmp = (0.5 * re) * (exp(-im) + exp(im));
	else
		tmp = 0.041666666666666664 * (sin(re) * (im ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.5], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.2e+77], N[(N[(0.5 * re), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.041666666666666664 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 2.5

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 82.3%

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

   6. Simplified 82.3%

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

   if 2.5 < im < 1.1999999999999999e77

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 76.9%

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

   6. Simplified 76.9%

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

   if 1.1999999999999999e77 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]

   6. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left(2 + im \cdot im\right) + {im}^{4} \cdot 0.08333333333333333\right)} \]

   7. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto 0.041666666666666664 \cdot \color{blue}{\left({im}^{4} \cdot \sin re\right)} \]

   9. Simplified 100.0%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left({im}^{4} \cdot \sin re\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.5:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.2 \cdot 10^{+77}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\\ \end{array} \]

Alternative 4: 86.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.5:\\ \;\;\;\;\sin re + \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.2 \cdot 10^{+77}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.5)
   (+ (sin re) (* (* 0.5 (sin re)) (* im im)))
   (if (<= im 1.2e+77)
     (* (* 0.5 re) (+ (exp (- im)) (exp im)))
     (* 0.041666666666666664 (* (sin re) (pow im 4.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.5) {
		tmp = sin(re) + ((0.5 * sin(re)) * (im * im));
	} else if (im <= 1.2e+77) {
		tmp = (0.5 * re) * (exp(-im) + exp(im));
	} else {
		tmp = 0.041666666666666664 * (sin(re) * pow(im, 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 2.5d0) then
        tmp = sin(re) + ((0.5d0 * sin(re)) * (im * im))
    else if (im <= 1.2d+77) then
        tmp = (0.5d0 * re) * (exp(-im) + exp(im))
    else
        tmp = 0.041666666666666664d0 * (sin(re) * (im ** 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.5) {
		tmp = Math.sin(re) + ((0.5 * Math.sin(re)) * (im * im));
	} else if (im <= 1.2e+77) {
		tmp = (0.5 * re) * (Math.exp(-im) + Math.exp(im));
	} else {
		tmp = 0.041666666666666664 * (Math.sin(re) * Math.pow(im, 4.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.5:
		tmp = math.sin(re) + ((0.5 * math.sin(re)) * (im * im))
	elif im <= 1.2e+77:
		tmp = (0.5 * re) * (math.exp(-im) + math.exp(im))
	else:
		tmp = 0.041666666666666664 * (math.sin(re) * math.pow(im, 4.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.5)
		tmp = Float64(sin(re) + Float64(Float64(0.5 * sin(re)) * Float64(im * im)));
	elseif (im <= 1.2e+77)
		tmp = Float64(Float64(0.5 * re) * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64(0.041666666666666664 * Float64(sin(re) * (im ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.5)
		tmp = sin(re) + ((0.5 * sin(re)) * (im * im));
	elseif (im <= 1.2e+77)
		tmp = (0.5 * re) * (exp(-im) + exp(im));
	else
		tmp = 0.041666666666666664 * (sin(re) * (im ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.5], N[(N[Sin[re], $MachinePrecision] + N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.2e+77], N[(N[(0.5 * re), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.041666666666666664 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 2.5

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 82.3%

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

   6. Simplified 82.3%

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

   if 2.5 < im < 1.1999999999999999e77

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 76.9%

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

   6. Simplified 76.9%

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

   if 1.1999999999999999e77 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]

   6. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left(2 + im \cdot im\right) + {im}^{4} \cdot 0.08333333333333333\right)} \]

   7. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto 0.041666666666666664 \cdot \color{blue}{\left({im}^{4} \cdot \sin re\right)} \]

   9. Simplified 100.0%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left({im}^{4} \cdot \sin re\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.5:\\ \;\;\;\;\sin re + \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.2 \cdot 10^{+77}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\\ \end{array} \]

Alternative 5: 84.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 95000:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 2.15 \cdot 10^{+59}:\\ \;\;\;\;{\left(\sin re \cdot 1.9380669946781485 \cdot 10^{-10}\right)}^{-512}\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 95000.0)
   (* (* 0.5 (sin re)) (+ 2.0 (* im im)))
   (if (<= im 2.15e+59)
     (pow (* (sin re) 1.9380669946781485e-10) -512.0)
     (* 0.041666666666666664 (* (sin re) (pow im 4.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 95000.0) {
		tmp = (0.5 * sin(re)) * (2.0 + (im * im));
	} else if (im <= 2.15e+59) {
		tmp = pow((sin(re) * 1.9380669946781485e-10), -512.0);
	} else {
		tmp = 0.041666666666666664 * (sin(re) * pow(im, 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 95000.0d0) then
        tmp = (0.5d0 * sin(re)) * (2.0d0 + (im * im))
    else if (im <= 2.15d+59) then
        tmp = (sin(re) * 1.9380669946781485d-10) ** (-512.0d0)
    else
        tmp = 0.041666666666666664d0 * (sin(re) * (im ** 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 95000.0) {
		tmp = (0.5 * Math.sin(re)) * (2.0 + (im * im));
	} else if (im <= 2.15e+59) {
		tmp = Math.pow((Math.sin(re) * 1.9380669946781485e-10), -512.0);
	} else {
		tmp = 0.041666666666666664 * (Math.sin(re) * Math.pow(im, 4.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 95000.0:
		tmp = (0.5 * math.sin(re)) * (2.0 + (im * im))
	elif im <= 2.15e+59:
		tmp = math.pow((math.sin(re) * 1.9380669946781485e-10), -512.0)
	else:
		tmp = 0.041666666666666664 * (math.sin(re) * math.pow(im, 4.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 95000.0)
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(2.0 + Float64(im * im)));
	elseif (im <= 2.15e+59)
		tmp = Float64(sin(re) * 1.9380669946781485e-10) ^ -512.0;
	else
		tmp = Float64(0.041666666666666664 * Float64(sin(re) * (im ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 95000.0)
		tmp = (0.5 * sin(re)) * (2.0 + (im * im));
	elseif (im <= 2.15e+59)
		tmp = (sin(re) * 1.9380669946781485e-10) ^ -512.0;
	else
		tmp = 0.041666666666666664 * (sin(re) * (im ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 95000.0], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.15e+59], N[Power[N[(N[Sin[re], $MachinePrecision] * 1.9380669946781485e-10), $MachinePrecision], -512.0], $MachinePrecision], N[(0.041666666666666664 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 95000

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 81.1%

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

   6. Simplified 81.1%

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

   if 95000 < im < 2.15000000000000012e59

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 3.3%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]

   6. Simplified 3.3%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left(2 + im \cdot im\right) + {im}^{4} \cdot 0.08333333333333333\right)} \]

   8. Applied egg-rr 50.0%

      \[\leadsto \color{blue}{{\left(\sin re \cdot 1.9380669946781485 \cdot 10^{-10}\right)}^{-512}} \]

   if 2.15000000000000012e59 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 95.6%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]

   6. Simplified 95.6%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left(2 + im \cdot im\right) + {im}^{4} \cdot 0.08333333333333333\right)} \]

   7. Taylor expanded in im around inf 95.6%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 95.6%

         \[\leadsto 0.041666666666666664 \cdot \color{blue}{\left({im}^{4} \cdot \sin re\right)} \]

   9. Simplified 95.6%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left({im}^{4} \cdot \sin re\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 95000:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 2.15 \cdot 10^{+59}:\\ \;\;\;\;{\left(\sin re \cdot 1.9380669946781485 \cdot 10^{-10}\right)}^{-512}\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\\ \end{array} \]

Alternative 6: 81.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 95000 \lor \neg \left(im \leq 1.22 \cdot 10^{+150}\right):\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\sin re \cdot 1.9380669946781485 \cdot 10^{-10}\right)}^{-512}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 95000.0) (not (<= im 1.22e+150)))
   (* (* 0.5 (sin re)) (+ 2.0 (* im im)))
   (pow (* (sin re) 1.9380669946781485e-10) -512.0)))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 95000.0) || !(im <= 1.22e+150)) {
		tmp = (0.5 * sin(re)) * (2.0 + (im * im));
	} else {
		tmp = pow((sin(re) * 1.9380669946781485e-10), -512.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= 95000.0d0) .or. (.not. (im <= 1.22d+150))) then
        tmp = (0.5d0 * sin(re)) * (2.0d0 + (im * im))
    else
        tmp = (sin(re) * 1.9380669946781485d-10) ** (-512.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= 95000.0) || !(im <= 1.22e+150)) {
		tmp = (0.5 * Math.sin(re)) * (2.0 + (im * im));
	} else {
		tmp = Math.pow((Math.sin(re) * 1.9380669946781485e-10), -512.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= 95000.0) or not (im <= 1.22e+150):
		tmp = (0.5 * math.sin(re)) * (2.0 + (im * im))
	else:
		tmp = math.pow((math.sin(re) * 1.9380669946781485e-10), -512.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 95000.0) || !(im <= 1.22e+150))
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(2.0 + Float64(im * im)));
	else
		tmp = Float64(sin(re) * 1.9380669946781485e-10) ^ -512.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= 95000.0) || ~((im <= 1.22e+150)))
		tmp = (0.5 * sin(re)) * (2.0 + (im * im));
	else
		tmp = (sin(re) * 1.9380669946781485e-10) ^ -512.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 95000.0], N[Not[LessEqual[im, 1.22e+150]], $MachinePrecision]], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[Sin[re], $MachinePrecision] * 1.9380669946781485e-10), $MachinePrecision], -512.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 95000 or 1.22e150 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 82.1%

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

   6. Simplified 82.1%

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

   if 95000 < im < 1.22e150

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 58.3%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]

   6. Simplified 58.3%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left(2 + im \cdot im\right) + {im}^{4} \cdot 0.08333333333333333\right)} \]

   8. Applied egg-rr 60.9%

      \[\leadsto \color{blue}{{\left(\sin re \cdot 1.9380669946781485 \cdot 10^{-10}\right)}^{-512}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 95000 \lor \neg \left(im \leq 1.22 \cdot 10^{+150}\right):\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\sin re \cdot 1.9380669946781485 \cdot 10^{-10}\right)}^{-512}\\ \end{array} \]

Alternative 7: 78.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 95000 \lor \neg \left(im \leq 2.05 \cdot 10^{+136}\right):\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;{\sin re}^{-512}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 95000.0) (not (<= im 2.05e+136)))
   (* (* 0.5 (sin re)) (+ 2.0 (* im im)))
   (pow (sin re) -512.0)))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 95000.0) || !(im <= 2.05e+136)) {
		tmp = (0.5 * sin(re)) * (2.0 + (im * im));
	} else {
		tmp = pow(sin(re), -512.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= 95000.0d0) .or. (.not. (im <= 2.05d+136))) then
        tmp = (0.5d0 * sin(re)) * (2.0d0 + (im * im))
    else
        tmp = sin(re) ** (-512.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= 95000.0) || !(im <= 2.05e+136)) {
		tmp = (0.5 * Math.sin(re)) * (2.0 + (im * im));
	} else {
		tmp = Math.pow(Math.sin(re), -512.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= 95000.0) or not (im <= 2.05e+136):
		tmp = (0.5 * math.sin(re)) * (2.0 + (im * im))
	else:
		tmp = math.pow(math.sin(re), -512.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 95000.0) || !(im <= 2.05e+136))
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(2.0 + Float64(im * im)));
	else
		tmp = sin(re) ^ -512.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= 95000.0) || ~((im <= 2.05e+136)))
		tmp = (0.5 * sin(re)) * (2.0 + (im * im));
	else
		tmp = sin(re) ^ -512.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 95000.0], N[Not[LessEqual[im, 2.05e+136]], $MachinePrecision]], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[Sin[re], $MachinePrecision], -512.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 95000 or 2.0499999999999999e136 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 81.8%

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

   6. Simplified 81.8%

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

   if 95000 < im < 2.0499999999999999e136

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 4.3%

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

   6. Simplified 4.3%

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

   8. Applied egg-rr 28.9%

      \[\leadsto \color{blue}{{\sin re}^{-512}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 95000 \lor \neg \left(im \leq 2.05 \cdot 10^{+136}\right):\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;{\sin re}^{-512}\\ \end{array} \]

Alternative 8: 63.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 95000:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 3.7 \cdot 10^{+118}:\\ \;\;\;\;{re}^{-512}\\ \mathbf{elif}\;im \leq 3.9 \cdot 10^{+218}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(2 + im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(\sin re \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 95000.0)
   (sin re)
   (if (<= im 3.7e+118)
     (pow re -512.0)
     (if (<= im 3.9e+218)
       (* 0.5 (* re (+ 2.0 (* im im))))
       (* im (* 0.5 (* (sin re) im)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 95000.0) {
		tmp = sin(re);
	} else if (im <= 3.7e+118) {
		tmp = pow(re, -512.0);
	} else if (im <= 3.9e+218) {
		tmp = 0.5 * (re * (2.0 + (im * im)));
	} else {
		tmp = im * (0.5 * (sin(re) * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 95000.0d0) then
        tmp = sin(re)
    else if (im <= 3.7d+118) then
        tmp = re ** (-512.0d0)
    else if (im <= 3.9d+218) then
        tmp = 0.5d0 * (re * (2.0d0 + (im * im)))
    else
        tmp = im * (0.5d0 * (sin(re) * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 95000.0) {
		tmp = Math.sin(re);
	} else if (im <= 3.7e+118) {
		tmp = Math.pow(re, -512.0);
	} else if (im <= 3.9e+218) {
		tmp = 0.5 * (re * (2.0 + (im * im)));
	} else {
		tmp = im * (0.5 * (Math.sin(re) * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 95000.0:
		tmp = math.sin(re)
	elif im <= 3.7e+118:
		tmp = math.pow(re, -512.0)
	elif im <= 3.9e+218:
		tmp = 0.5 * (re * (2.0 + (im * im)))
	else:
		tmp = im * (0.5 * (math.sin(re) * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 95000.0)
		tmp = sin(re);
	elseif (im <= 3.7e+118)
		tmp = re ^ -512.0;
	elseif (im <= 3.9e+218)
		tmp = Float64(0.5 * Float64(re * Float64(2.0 + Float64(im * im))));
	else
		tmp = Float64(im * Float64(0.5 * Float64(sin(re) * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 95000.0)
		tmp = sin(re);
	elseif (im <= 3.7e+118)
		tmp = re ^ -512.0;
	elseif (im <= 3.9e+218)
		tmp = 0.5 * (re * (2.0 + (im * im)));
	else
		tmp = im * (0.5 * (sin(re) * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 95000.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 3.7e+118], N[Power[re, -512.0], $MachinePrecision], If[LessEqual[im, 3.9e+218], N[(0.5 * N[(re * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(0.5 * N[(N[Sin[re], $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < 95000

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 69.1%

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

   if 95000 < im < 3.69999999999999987e118

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 73.7%

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

   6. Simplified 73.7%

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

   8. Applied egg-rr 32.2%

      \[\leadsto \color{blue}{{re}^{-512}} \]

   if 3.69999999999999987e118 < im < 3.9000000000000002e218

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 68.8%

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

   6. Simplified 68.8%

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

   7. Taylor expanded in re around 0 67.2%

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

      1. *-commutative 67.2%

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

      2. unpow2 67.2%

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

   9. Simplified 67.2%

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

   if 3.9000000000000002e218 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in im around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. unpow2 100.0%

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

      3. associate-*r* 90.8%

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

      4. *-commutative 90.8%

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

      5. associate-*l* 90.8%

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

      6. *-commutative 90.8%

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

   9. Simplified 90.8%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 95000:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 3.7 \cdot 10^{+118}:\\ \;\;\;\;{re}^{-512}\\ \mathbf{elif}\;im \leq 3.9 \cdot 10^{+218}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(2 + im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(\sin re \cdot im\right)\right)\\ \end{array} \]

Alternative 9: 78.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 95000 \lor \neg \left(im \leq 1.22 \cdot 10^{+150}\right):\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 95000.0) (not (<= im 1.22e+150)))
   (* (* 0.5 (sin re)) (+ 2.0 (* im im)))
   (+
    0.08333333333333333
    (+ (/ 0.25 (* re re)) (* (* re re) 0.016666666666666666)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 95000.0) || !(im <= 1.22e+150)) {
		tmp = (0.5 * sin(re)) * (2.0 + (im * im));
	} else {
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= 95000.0d0) .or. (.not. (im <= 1.22d+150))) then
        tmp = (0.5d0 * sin(re)) * (2.0d0 + (im * im))
    else
        tmp = 0.08333333333333333d0 + ((0.25d0 / (re * re)) + ((re * re) * 0.016666666666666666d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= 95000.0) || !(im <= 1.22e+150)) {
		tmp = (0.5 * Math.sin(re)) * (2.0 + (im * im));
	} else {
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= 95000.0) or not (im <= 1.22e+150):
		tmp = (0.5 * math.sin(re)) * (2.0 + (im * im))
	else:
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 95000.0) || !(im <= 1.22e+150))
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(2.0 + Float64(im * im)));
	else
		tmp = Float64(0.08333333333333333 + Float64(Float64(0.25 / Float64(re * re)) + Float64(Float64(re * re) * 0.016666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= 95000.0) || ~((im <= 1.22e+150)))
		tmp = (0.5 * sin(re)) * (2.0 + (im * im));
	else
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 95000.0], N[Not[LessEqual[im, 1.22e+150]], $MachinePrecision]], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.08333333333333333 + N[(N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision] + N[(N[(re * re), $MachinePrecision] * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 95000 or 1.22e150 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 82.1%

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

   6. Simplified 82.1%

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

   if 95000 < im < 1.22e150

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 19.0%

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

   6. Taylor expanded in re around 0 32.0%

      \[\leadsto \color{blue}{0.08333333333333333 + \left(0.25 \cdot \frac{1}{{re}^{2}} + 0.016666666666666666 \cdot {re}^{2}\right)} \]
   7. Step-by-step derivation

      1. associate-*r/ 32.0%

         \[\leadsto 0.08333333333333333 + \left(\color{blue}{\frac{0.25 \cdot 1}{{re}^{2}}} + 0.016666666666666666 \cdot {re}^{2}\right) \]

      2. metadata-eval 32.0%

         \[\leadsto 0.08333333333333333 + \left(\frac{\color{blue}{0.25}}{{re}^{2}} + 0.016666666666666666 \cdot {re}^{2}\right) \]

      3. unpow2 32.0%

         \[\leadsto 0.08333333333333333 + \left(\frac{0.25}{\color{blue}{re \cdot re}} + 0.016666666666666666 \cdot {re}^{2}\right) \]

      4. *-commutative 32.0%

         \[\leadsto 0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \color{blue}{{re}^{2} \cdot 0.016666666666666666}\right) \]

      5. unpow2 32.0%

         \[\leadsto 0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \color{blue}{\left(re \cdot re\right)} \cdot 0.016666666666666666\right) \]

   8. Simplified 32.0%

      \[\leadsto \color{blue}{0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 95000 \lor \neg \left(im \leq 1.22 \cdot 10^{+150}\right):\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)\\ \end{array} \]

Alternative 10: 62.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 95000:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 6.2 \cdot 10^{+118}:\\ \;\;\;\;{re}^{-512}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(2 + im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 95000.0)
   (sin re)
   (if (<= im 6.2e+118) (pow re -512.0) (* 0.5 (* re (+ 2.0 (* im im)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 95000.0) {
		tmp = sin(re);
	} else if (im <= 6.2e+118) {
		tmp = pow(re, -512.0);
	} else {
		tmp = 0.5 * (re * (2.0 + (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 (im <= 95000.0d0) then
        tmp = sin(re)
    else if (im <= 6.2d+118) then
        tmp = re ** (-512.0d0)
    else
        tmp = 0.5d0 * (re * (2.0d0 + (im * im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 95000.0) {
		tmp = Math.sin(re);
	} else if (im <= 6.2e+118) {
		tmp = Math.pow(re, -512.0);
	} else {
		tmp = 0.5 * (re * (2.0 + (im * im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 95000.0:
		tmp = math.sin(re)
	elif im <= 6.2e+118:
		tmp = math.pow(re, -512.0)
	else:
		tmp = 0.5 * (re * (2.0 + (im * im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 95000.0)
		tmp = sin(re);
	elseif (im <= 6.2e+118)
		tmp = re ^ -512.0;
	else
		tmp = Float64(0.5 * Float64(re * Float64(2.0 + Float64(im * im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 95000.0)
		tmp = sin(re);
	elseif (im <= 6.2e+118)
		tmp = re ^ -512.0;
	else
		tmp = 0.5 * (re * (2.0 + (im * im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 95000.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 6.2e+118], N[Power[re, -512.0], $MachinePrecision], N[(0.5 * N[(re * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 95000

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 69.1%

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

   if 95000 < im < 6.19999999999999973e118

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 73.7%

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

   6. Simplified 73.7%

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

   8. Applied egg-rr 32.2%

      \[\leadsto \color{blue}{{re}^{-512}} \]

   if 6.19999999999999973e118 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 78.9%

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

   6. Simplified 78.9%

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

   7. Taylor expanded in re around 0 61.7%

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

      1. *-commutative 61.7%

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

      2. unpow2 61.7%

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

   9. Simplified 61.7%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 95000:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 6.2 \cdot 10^{+118}:\\ \;\;\;\;{re}^{-512}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(2 + im \cdot im\right)\right)\\ \end{array} \]

Alternative 11: 62.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1300000:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+120}:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(2 + im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1300000.0)
   (sin re)
   (if (<= im 4.5e+120)
     (+
      0.08333333333333333
      (+ (/ 0.25 (* re re)) (* (* re re) 0.016666666666666666)))
     (* 0.5 (* re (+ 2.0 (* im im)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1300000.0) {
		tmp = sin(re);
	} else if (im <= 4.5e+120) {
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	} else {
		tmp = 0.5 * (re * (2.0 + (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 (im <= 1300000.0d0) then
        tmp = sin(re)
    else if (im <= 4.5d+120) then
        tmp = 0.08333333333333333d0 + ((0.25d0 / (re * re)) + ((re * re) * 0.016666666666666666d0))
    else
        tmp = 0.5d0 * (re * (2.0d0 + (im * im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1300000.0) {
		tmp = Math.sin(re);
	} else if (im <= 4.5e+120) {
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	} else {
		tmp = 0.5 * (re * (2.0 + (im * im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1300000.0:
		tmp = math.sin(re)
	elif im <= 4.5e+120:
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666))
	else:
		tmp = 0.5 * (re * (2.0 + (im * im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1300000.0)
		tmp = sin(re);
	elseif (im <= 4.5e+120)
		tmp = Float64(0.08333333333333333 + Float64(Float64(0.25 / Float64(re * re)) + Float64(Float64(re * re) * 0.016666666666666666)));
	else
		tmp = Float64(0.5 * Float64(re * Float64(2.0 + Float64(im * im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1300000.0)
		tmp = sin(re);
	elseif (im <= 4.5e+120)
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	else
		tmp = 0.5 * (re * (2.0 + (im * im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1300000.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 4.5e+120], N[(0.08333333333333333 + N[(N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision] + N[(N[(re * re), $MachinePrecision] * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(re * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.3e6

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 69.1%

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

   if 1.3e6 < im < 4.49999999999999977e120

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 22.7%

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

   6. Taylor expanded in re around 0 33.2%

      \[\leadsto \color{blue}{0.08333333333333333 + \left(0.25 \cdot \frac{1}{{re}^{2}} + 0.016666666666666666 \cdot {re}^{2}\right)} \]
   7. Step-by-step derivation

      1. associate-*r/ 33.2%

         \[\leadsto 0.08333333333333333 + \left(\color{blue}{\frac{0.25 \cdot 1}{{re}^{2}}} + 0.016666666666666666 \cdot {re}^{2}\right) \]

      2. metadata-eval 33.2%

         \[\leadsto 0.08333333333333333 + \left(\frac{\color{blue}{0.25}}{{re}^{2}} + 0.016666666666666666 \cdot {re}^{2}\right) \]

      3. unpow2 33.2%

         \[\leadsto 0.08333333333333333 + \left(\frac{0.25}{\color{blue}{re \cdot re}} + 0.016666666666666666 \cdot {re}^{2}\right) \]

      4. *-commutative 33.2%

         \[\leadsto 0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \color{blue}{{re}^{2} \cdot 0.016666666666666666}\right) \]

      5. unpow2 33.2%

         \[\leadsto 0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \color{blue}{\left(re \cdot re\right)} \cdot 0.016666666666666666\right) \]

   8. Simplified 33.2%

      \[\leadsto \color{blue}{0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)} \]

   if 4.49999999999999977e120 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 78.9%

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

   6. Simplified 78.9%

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

   7. Taylor expanded in re around 0 61.7%

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

      1. *-commutative 61.7%

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

      2. unpow2 61.7%

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

   9. Simplified 61.7%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1300000:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+120}:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(2 + im \cdot im\right)\right)\\ \end{array} \]

Alternative 12: 48.9% accurate, 18.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 95000:\\ \;\;\;\;re + 0.5 \cdot \left(re \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 7.8 \cdot 10^{+121}:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(2 + im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 95000.0)
   (+ re (* 0.5 (* re (* im im))))
   (if (<= im 7.8e+121)
     (+
      0.08333333333333333
      (+ (/ 0.25 (* re re)) (* (* re re) 0.016666666666666666)))
     (* 0.5 (* re (+ 2.0 (* im im)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 95000.0) {
		tmp = re + (0.5 * (re * (im * im)));
	} else if (im <= 7.8e+121) {
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	} else {
		tmp = 0.5 * (re * (2.0 + (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 (im <= 95000.0d0) then
        tmp = re + (0.5d0 * (re * (im * im)))
    else if (im <= 7.8d+121) then
        tmp = 0.08333333333333333d0 + ((0.25d0 / (re * re)) + ((re * re) * 0.016666666666666666d0))
    else
        tmp = 0.5d0 * (re * (2.0d0 + (im * im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 95000.0) {
		tmp = re + (0.5 * (re * (im * im)));
	} else if (im <= 7.8e+121) {
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	} else {
		tmp = 0.5 * (re * (2.0 + (im * im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 95000.0:
		tmp = re + (0.5 * (re * (im * im)))
	elif im <= 7.8e+121:
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666))
	else:
		tmp = 0.5 * (re * (2.0 + (im * im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 95000.0)
		tmp = Float64(re + Float64(0.5 * Float64(re * Float64(im * im))));
	elseif (im <= 7.8e+121)
		tmp = Float64(0.08333333333333333 + Float64(Float64(0.25 / Float64(re * re)) + Float64(Float64(re * re) * 0.016666666666666666)));
	else
		tmp = Float64(0.5 * Float64(re * Float64(2.0 + Float64(im * im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 95000.0)
		tmp = re + (0.5 * (re * (im * im)));
	elseif (im <= 7.8e+121)
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	else
		tmp = 0.5 * (re * (2.0 + (im * im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 95000.0], N[(re + N[(0.5 * N[(re * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 7.8e+121], N[(0.08333333333333333 + N[(N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision] + N[(N[(re * re), $MachinePrecision] * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(re * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 95000

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 81.1%

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

   6. Simplified 81.1%

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

   7. Taylor expanded in re around 0 51.5%

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

      1. associate-*r* 51.5%

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

      2. *-commutative 51.5%

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

      3. +-commutative 51.5%

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

      4. distribute-rgt-in 51.5%

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

      5. metadata-eval 51.5%

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

      6. distribute-lft-in 51.5%

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

      7. *-rgt-identity 51.5%

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

      8. fma-def 51.5%

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, {im}^{2} \cdot 0.5, re\right)} \]

      9. unpow2 51.5%

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

      10. associate-*l* 51.5%

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

   9. Simplified 51.5%

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

       1. fma-udef 51.5%

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

       2. associate-*r* 51.5%

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

       3. associate-*r* 51.5%

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

   11. Applied egg-rr 51.5%

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

   if 95000 < im < 7.79999999999999967e121

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 22.7%

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

   6. Taylor expanded in re around 0 33.2%

      \[\leadsto \color{blue}{0.08333333333333333 + \left(0.25 \cdot \frac{1}{{re}^{2}} + 0.016666666666666666 \cdot {re}^{2}\right)} \]
   7. Step-by-step derivation

      1. associate-*r/ 33.2%

         \[\leadsto 0.08333333333333333 + \left(\color{blue}{\frac{0.25 \cdot 1}{{re}^{2}}} + 0.016666666666666666 \cdot {re}^{2}\right) \]

      2. metadata-eval 33.2%

         \[\leadsto 0.08333333333333333 + \left(\frac{\color{blue}{0.25}}{{re}^{2}} + 0.016666666666666666 \cdot {re}^{2}\right) \]

      3. unpow2 33.2%

         \[\leadsto 0.08333333333333333 + \left(\frac{0.25}{\color{blue}{re \cdot re}} + 0.016666666666666666 \cdot {re}^{2}\right) \]

      4. *-commutative 33.2%

         \[\leadsto 0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \color{blue}{{re}^{2} \cdot 0.016666666666666666}\right) \]

      5. unpow2 33.2%

         \[\leadsto 0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \color{blue}{\left(re \cdot re\right)} \cdot 0.016666666666666666\right) \]

   8. Simplified 33.2%

      \[\leadsto \color{blue}{0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)} \]

   if 7.79999999999999967e121 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 78.9%

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

   6. Simplified 78.9%

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

   7. Taylor expanded in re around 0 61.7%

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

      1. *-commutative 61.7%

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

      2. unpow2 61.7%

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

   9. Simplified 61.7%

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

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 95000:\\ \;\;\;\;re + 0.5 \cdot \left(re \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 7.8 \cdot 10^{+121}:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(2 + im \cdot im\right)\right)\\ \end{array} \]

Alternative 13: 48.0% accurate, 23.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.032 \lor \neg \left(im \leq 1.9 \cdot 10^{+100}\right):\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(2 + im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.08333333333333333 + \frac{\frac{0.25}{re}}{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 0.032) (not (<= im 1.9e+100)))
   (* 0.5 (* re (+ 2.0 (* im im))))
   (+ 0.08333333333333333 (/ (/ 0.25 re) re))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 0.032) || !(im <= 1.9e+100)) {
		tmp = 0.5 * (re * (2.0 + (im * im)));
	} else {
		tmp = 0.08333333333333333 + ((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 ((im <= 0.032d0) .or. (.not. (im <= 1.9d+100))) then
        tmp = 0.5d0 * (re * (2.0d0 + (im * im)))
    else
        tmp = 0.08333333333333333d0 + ((0.25d0 / re) / re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= 0.032) || !(im <= 1.9e+100)) {
		tmp = 0.5 * (re * (2.0 + (im * im)));
	} else {
		tmp = 0.08333333333333333 + ((0.25 / re) / re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= 0.032) or not (im <= 1.9e+100):
		tmp = 0.5 * (re * (2.0 + (im * im)))
	else:
		tmp = 0.08333333333333333 + ((0.25 / re) / re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 0.032) || !(im <= 1.9e+100))
		tmp = Float64(0.5 * Float64(re * Float64(2.0 + Float64(im * im))));
	else
		tmp = Float64(0.08333333333333333 + Float64(Float64(0.25 / re) / re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= 0.032) || ~((im <= 1.9e+100)))
		tmp = 0.5 * (re * (2.0 + (im * im)));
	else
		tmp = 0.08333333333333333 + ((0.25 / re) / re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 0.032], N[Not[LessEqual[im, 1.9e+100]], $MachinePrecision]], N[(0.5 * N[(re * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.08333333333333333 + N[(N[(0.25 / re), $MachinePrecision] / re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 0.032000000000000001 or 1.89999999999999982e100 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 80.9%

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

   6. Simplified 80.9%

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

   7. Taylor expanded in re around 0 53.4%

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

      1. *-commutative 53.4%

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

      2. unpow2 53.4%

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

   9. Simplified 53.4%

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

   if 0.032000000000000001 < im < 1.89999999999999982e100

   1. Initial program 99.9%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Applied egg-rr 23.2%

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

   6. Taylor expanded in re around 0 22.5%

      \[\leadsto \color{blue}{0.08333333333333333 + 0.25 \cdot \frac{1}{{re}^{2}}} \]
   7. Step-by-step derivation

      1. associate-*r/ 22.5%

         \[\leadsto 0.08333333333333333 + \color{blue}{\frac{0.25 \cdot 1}{{re}^{2}}} \]

      2. metadata-eval 22.5%

         \[\leadsto 0.08333333333333333 + \frac{\color{blue}{0.25}}{{re}^{2}} \]

      3. unpow2 22.5%

         \[\leadsto 0.08333333333333333 + \frac{0.25}{\color{blue}{re \cdot re}} \]

      4. associate-/r* 22.5%

         \[\leadsto 0.08333333333333333 + \color{blue}{\frac{\frac{0.25}{re}}{re}} \]

   8. Simplified 22.5%

      \[\leadsto \color{blue}{0.08333333333333333 + \frac{\frac{0.25}{re}}{re}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.032 \lor \neg \left(im \leq 1.9 \cdot 10^{+100}\right):\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(2 + im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.08333333333333333 + \frac{\frac{0.25}{re}}{re}\\ \end{array} \]

Alternative 14: 47.9% accurate, 23.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.032:\\ \;\;\;\;re + 0.5 \cdot \left(re \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+103}:\\ \;\;\;\;0.08333333333333333 + \frac{\frac{0.25}{re}}{re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(2 + im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.032)
   (+ re (* 0.5 (* re (* im im))))
   (if (<= im 1.4e+103)
     (+ 0.08333333333333333 (/ (/ 0.25 re) re))
     (* 0.5 (* re (+ 2.0 (* im im)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.032) {
		tmp = re + (0.5 * (re * (im * im)));
	} else if (im <= 1.4e+103) {
		tmp = 0.08333333333333333 + ((0.25 / re) / re);
	} else {
		tmp = 0.5 * (re * (2.0 + (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 (im <= 0.032d0) then
        tmp = re + (0.5d0 * (re * (im * im)))
    else if (im <= 1.4d+103) then
        tmp = 0.08333333333333333d0 + ((0.25d0 / re) / re)
    else
        tmp = 0.5d0 * (re * (2.0d0 + (im * im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.032) {
		tmp = re + (0.5 * (re * (im * im)));
	} else if (im <= 1.4e+103) {
		tmp = 0.08333333333333333 + ((0.25 / re) / re);
	} else {
		tmp = 0.5 * (re * (2.0 + (im * im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.032:
		tmp = re + (0.5 * (re * (im * im)))
	elif im <= 1.4e+103:
		tmp = 0.08333333333333333 + ((0.25 / re) / re)
	else:
		tmp = 0.5 * (re * (2.0 + (im * im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.032)
		tmp = Float64(re + Float64(0.5 * Float64(re * Float64(im * im))));
	elseif (im <= 1.4e+103)
		tmp = Float64(0.08333333333333333 + Float64(Float64(0.25 / re) / re));
	else
		tmp = Float64(0.5 * Float64(re * Float64(2.0 + Float64(im * im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.032)
		tmp = re + (0.5 * (re * (im * im)));
	elseif (im <= 1.4e+103)
		tmp = 0.08333333333333333 + ((0.25 / re) / re);
	else
		tmp = 0.5 * (re * (2.0 + (im * im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.032], N[(re + N[(0.5 * N[(re * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.4e+103], N[(0.08333333333333333 + N[(N[(0.25 / re), $MachinePrecision] / re), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(re * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.032000000000000001

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 82.7%

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

   6. Simplified 82.7%

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

   7. Taylor expanded in re around 0 52.7%

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

      1. associate-*r* 52.7%

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

      2. *-commutative 52.7%

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

      3. +-commutative 52.7%

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

      4. distribute-rgt-in 52.7%

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

      5. metadata-eval 52.7%

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

      6. distribute-lft-in 52.7%

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

      7. *-rgt-identity 52.7%

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

      8. fma-def 52.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, {im}^{2} \cdot 0.5, re\right)} \]

      9. unpow2 52.7%

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

      10. associate-*l* 52.7%

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

   9. Simplified 52.7%

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

       1. fma-udef 52.7%

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

       2. associate-*r* 52.7%

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

       3. associate-*r* 52.7%

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

   11. Applied egg-rr 52.7%

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

   if 0.032000000000000001 < im < 1.40000000000000004e103

   1. Initial program 99.9%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Applied egg-rr 23.2%

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

   6. Taylor expanded in re around 0 22.5%

      \[\leadsto \color{blue}{0.08333333333333333 + 0.25 \cdot \frac{1}{{re}^{2}}} \]
   7. Step-by-step derivation

      1. associate-*r/ 22.5%

         \[\leadsto 0.08333333333333333 + \color{blue}{\frac{0.25 \cdot 1}{{re}^{2}}} \]

      2. metadata-eval 22.5%

         \[\leadsto 0.08333333333333333 + \frac{\color{blue}{0.25}}{{re}^{2}} \]

      3. unpow2 22.5%

         \[\leadsto 0.08333333333333333 + \frac{0.25}{\color{blue}{re \cdot re}} \]

      4. associate-/r* 22.5%

         \[\leadsto 0.08333333333333333 + \color{blue}{\frac{\frac{0.25}{re}}{re}} \]

   8. Simplified 22.5%

      \[\leadsto \color{blue}{0.08333333333333333 + \frac{\frac{0.25}{re}}{re}} \]

   if 1.40000000000000004e103 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 70.5%

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

   6. Simplified 70.5%

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

   7. Taylor expanded in re around 0 57.6%

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

      1. *-commutative 57.6%

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

      2. unpow2 57.6%

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

   9. Simplified 57.6%

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

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.032:\\ \;\;\;\;re + 0.5 \cdot \left(re \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+103}:\\ \;\;\;\;0.08333333333333333 + \frac{\frac{0.25}{re}}{re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(2 + im \cdot im\right)\right)\\ \end{array} \]

Alternative 15: 34.1% accurate, 27.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 60000:\\ \;\;\;\;re\\ \mathbf{elif}\;im \leq 1.06 \cdot 10^{+102}:\\ \;\;\;\;\frac{0.25}{re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(re \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 60000.0)
   re
   (if (<= im 1.06e+102) (/ 0.25 (* re re)) (* 0.5 (* im (* re im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 60000.0) {
		tmp = re;
	} else if (im <= 1.06e+102) {
		tmp = 0.25 / (re * re);
	} else {
		tmp = 0.5 * (im * (re * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 60000.0d0) then
        tmp = re
    else if (im <= 1.06d+102) then
        tmp = 0.25d0 / (re * re)
    else
        tmp = 0.5d0 * (im * (re * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 60000.0) {
		tmp = re;
	} else if (im <= 1.06e+102) {
		tmp = 0.25 / (re * re);
	} else {
		tmp = 0.5 * (im * (re * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 60000.0:
		tmp = re
	elif im <= 1.06e+102:
		tmp = 0.25 / (re * re)
	else:
		tmp = 0.5 * (im * (re * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 60000.0)
		tmp = re;
	elseif (im <= 1.06e+102)
		tmp = Float64(0.25 / Float64(re * re));
	else
		tmp = Float64(0.5 * Float64(im * Float64(re * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 60000.0)
		tmp = re;
	elseif (im <= 1.06e+102)
		tmp = 0.25 / (re * re);
	else
		tmp = 0.5 * (im * (re * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 60000.0], re, If[LessEqual[im, 1.06e+102], N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[(re * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 6e4

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 65.8%

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

   6. Simplified 65.8%

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

   7. Taylor expanded in im around 0 41.3%

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

   if 6e4 < im < 1.06000000000000001e102

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 26.1%

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

   6. Taylor expanded in re around 0 26.0%

      \[\leadsto \color{blue}{\frac{0.25}{{re}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 26.0%

         \[\leadsto \frac{0.25}{\color{blue}{re \cdot re}} \]

   8. Simplified 26.0%

      \[\leadsto \color{blue}{\frac{0.25}{re \cdot re}} \]

   if 1.06000000000000001e102 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 70.5%

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

   6. Simplified 70.5%

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

   7. Taylor expanded in re around 0 57.6%

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

      1. associate-*r* 57.6%

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

      2. *-commutative 57.6%

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

      3. +-commutative 57.6%

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

      4. distribute-rgt-in 57.6%

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

      5. metadata-eval 57.6%

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

      6. distribute-lft-in 57.6%

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

      7. *-rgt-identity 57.6%

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

      8. fma-def 57.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, {im}^{2} \cdot 0.5, re\right)} \]

      9. unpow2 57.6%

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

      10. associate-*l* 57.6%

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

   9. Simplified 57.6%

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

   10. Taylor expanded in im around inf 57.6%

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

       1. *-commutative 57.6%

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

       2. unpow2 57.6%

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

       3. associate-*l* 39.1%

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

   12. Simplified 39.1%

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

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 60000:\\ \;\;\;\;re\\ \mathbf{elif}\;im \leq 1.06 \cdot 10^{+102}:\\ \;\;\;\;\frac{0.25}{re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(re \cdot im\right)\right)\\ \end{array} \]

Alternative 16: 37.0% accurate, 27.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 60000:\\ \;\;\;\;re\\ \mathbf{elif}\;im \leq 1.25 \cdot 10^{+104}:\\ \;\;\;\;\frac{0.25}{re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 60000.0)
   re
   (if (<= im 1.25e+104) (/ 0.25 (* re re)) (* (* im im) (* 0.5 re)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 60000.0) {
		tmp = re;
	} else if (im <= 1.25e+104) {
		tmp = 0.25 / (re * re);
	} else {
		tmp = (im * im) * (0.5 * re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 60000.0d0) then
        tmp = re
    else if (im <= 1.25d+104) then
        tmp = 0.25d0 / (re * re)
    else
        tmp = (im * im) * (0.5d0 * re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 60000.0) {
		tmp = re;
	} else if (im <= 1.25e+104) {
		tmp = 0.25 / (re * re);
	} else {
		tmp = (im * im) * (0.5 * re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 60000.0:
		tmp = re
	elif im <= 1.25e+104:
		tmp = 0.25 / (re * re)
	else:
		tmp = (im * im) * (0.5 * re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 60000.0)
		tmp = re;
	elseif (im <= 1.25e+104)
		tmp = Float64(0.25 / Float64(re * re));
	else
		tmp = Float64(Float64(im * im) * Float64(0.5 * re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 60000.0)
		tmp = re;
	elseif (im <= 1.25e+104)
		tmp = 0.25 / (re * re);
	else
		tmp = (im * im) * (0.5 * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 60000.0], re, If[LessEqual[im, 1.25e+104], N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision], N[(N[(im * im), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 6e4

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 65.8%

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

   6. Simplified 65.8%

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

   7. Taylor expanded in im around 0 41.3%

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

   if 6e4 < im < 1.2499999999999999e104

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 26.1%

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

   6. Taylor expanded in re around 0 26.0%

      \[\leadsto \color{blue}{\frac{0.25}{{re}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 26.0%

         \[\leadsto \frac{0.25}{\color{blue}{re \cdot re}} \]

   8. Simplified 26.0%

      \[\leadsto \color{blue}{\frac{0.25}{re \cdot re}} \]

   if 1.2499999999999999e104 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 70.5%

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

   6. Simplified 70.5%

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

   7. Taylor expanded in re around 0 57.6%

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

      1. associate-*r* 57.6%

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

      2. *-commutative 57.6%

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

      3. +-commutative 57.6%

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

      4. distribute-rgt-in 57.6%

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

      5. metadata-eval 57.6%

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

      6. distribute-lft-in 57.6%

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

      7. *-rgt-identity 57.6%

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

      8. fma-def 57.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, {im}^{2} \cdot 0.5, re\right)} \]

      9. unpow2 57.6%

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

      10. associate-*l* 57.6%

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

   9. Simplified 57.6%

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

   10. Taylor expanded in im around inf 57.6%

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

       1. associate-*r* 57.6%

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

       2. *-commutative 57.6%

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

       3. unpow2 57.6%

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

   12. Simplified 57.6%

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

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 60000:\\ \;\;\;\;re\\ \mathbf{elif}\;im \leq 1.25 \cdot 10^{+104}:\\ \;\;\;\;\frac{0.25}{re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \]

Alternative 17: 37.0% accurate, 27.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.032:\\ \;\;\;\;re\\ \mathbf{elif}\;im \leq 1.95 \cdot 10^{+100}:\\ \;\;\;\;0.08333333333333333 + \frac{\frac{0.25}{re}}{re}\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.032)
   re
   (if (<= im 1.95e+100)
     (+ 0.08333333333333333 (/ (/ 0.25 re) re))
     (* (* im im) (* 0.5 re)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.032) {
		tmp = re;
	} else if (im <= 1.95e+100) {
		tmp = 0.08333333333333333 + ((0.25 / re) / re);
	} else {
		tmp = (im * im) * (0.5 * re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.032d0) then
        tmp = re
    else if (im <= 1.95d+100) then
        tmp = 0.08333333333333333d0 + ((0.25d0 / re) / re)
    else
        tmp = (im * im) * (0.5d0 * re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.032) {
		tmp = re;
	} else if (im <= 1.95e+100) {
		tmp = 0.08333333333333333 + ((0.25 / re) / re);
	} else {
		tmp = (im * im) * (0.5 * re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.032:
		tmp = re
	elif im <= 1.95e+100:
		tmp = 0.08333333333333333 + ((0.25 / re) / re)
	else:
		tmp = (im * im) * (0.5 * re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.032)
		tmp = re;
	elseif (im <= 1.95e+100)
		tmp = Float64(0.08333333333333333 + Float64(Float64(0.25 / re) / re));
	else
		tmp = Float64(Float64(im * im) * Float64(0.5 * re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.032)
		tmp = re;
	elseif (im <= 1.95e+100)
		tmp = 0.08333333333333333 + ((0.25 / re) / re);
	else
		tmp = (im * im) * (0.5 * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.032], re, If[LessEqual[im, 1.95e+100], N[(0.08333333333333333 + N[(N[(0.25 / re), $MachinePrecision] / re), $MachinePrecision]), $MachinePrecision], N[(N[(im * im), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.032000000000000001

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 66.1%

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

   6. Simplified 66.1%

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

   7. Taylor expanded in im around 0 42.0%

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

   if 0.032000000000000001 < im < 1.95e100

   1. Initial program 99.9%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Applied egg-rr 23.2%

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

   6. Taylor expanded in re around 0 22.5%

      \[\leadsto \color{blue}{0.08333333333333333 + 0.25 \cdot \frac{1}{{re}^{2}}} \]
   7. Step-by-step derivation

      1. associate-*r/ 22.5%

         \[\leadsto 0.08333333333333333 + \color{blue}{\frac{0.25 \cdot 1}{{re}^{2}}} \]

      2. metadata-eval 22.5%

         \[\leadsto 0.08333333333333333 + \frac{\color{blue}{0.25}}{{re}^{2}} \]

      3. unpow2 22.5%

         \[\leadsto 0.08333333333333333 + \frac{0.25}{\color{blue}{re \cdot re}} \]

      4. associate-/r* 22.5%

         \[\leadsto 0.08333333333333333 + \color{blue}{\frac{\frac{0.25}{re}}{re}} \]

   8. Simplified 22.5%

      \[\leadsto \color{blue}{0.08333333333333333 + \frac{\frac{0.25}{re}}{re}} \]

   if 1.95e100 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 70.5%

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

   6. Simplified 70.5%

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

   7. Taylor expanded in re around 0 57.6%

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

      1. associate-*r* 57.6%

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

      2. *-commutative 57.6%

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

      3. +-commutative 57.6%

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

      4. distribute-rgt-in 57.6%

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

      5. metadata-eval 57.6%

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

      6. distribute-lft-in 57.6%

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

      7. *-rgt-identity 57.6%

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

      8. fma-def 57.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, {im}^{2} \cdot 0.5, re\right)} \]

      9. unpow2 57.6%

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

      10. associate-*l* 57.6%

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

   9. Simplified 57.6%

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

   10. Taylor expanded in im around inf 57.6%

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

       1. associate-*r* 57.6%

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

       2. *-commutative 57.6%

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

       3. unpow2 57.6%

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

   12. Simplified 57.6%

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

4. Final simplification 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.032:\\ \;\;\;\;re\\ \mathbf{elif}\;im \leq 1.95 \cdot 10^{+100}:\\ \;\;\;\;0.08333333333333333 + \frac{\frac{0.25}{re}}{re}\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \]

Alternative 18: 28.7% accurate, 43.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 75000:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{re \cdot re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 75000.0) re (/ 0.25 (* re re))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 75000.0) {
		tmp = re;
	} else {
		tmp = 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 (im <= 75000.0d0) then
        tmp = re
    else
        tmp = 0.25d0 / (re * re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 75000.0) {
		tmp = re;
	} else {
		tmp = 0.25 / (re * re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 75000.0:
		tmp = re
	else:
		tmp = 0.25 / (re * re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 75000.0)
		tmp = re;
	else
		tmp = Float64(0.25 / Float64(re * re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 75000.0)
		tmp = re;
	else
		tmp = 0.25 / (re * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 75000.0], re, N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 75000

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 65.8%

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

   6. Simplified 65.8%

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

   7. Taylor expanded in im around 0 41.3%

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

   if 75000 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 11.6%

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

   6. Taylor expanded in re around 0 11.5%

      \[\leadsto \color{blue}{\frac{0.25}{{re}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 11.5%

         \[\leadsto \frac{0.25}{\color{blue}{re \cdot re}} \]

   8. Simplified 11.5%

      \[\leadsto \color{blue}{\frac{0.25}{re \cdot re}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 75000:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{re \cdot re}\\ \end{array} \]

Alternative 19: 3.6% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ 1.9380669946781485 \cdot 10^{-10} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 1.9380669946781485e-10)

C

double code(double re, double im) {
	return 1.9380669946781485e-10;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 1.9380669946781485d-10
end function

Java

public static double code(double re, double im) {
	return 1.9380669946781485e-10;
}

Python

def code(re, im):
	return 1.9380669946781485e-10

Julia

function code(re, im)
	return 1.9380669946781485e-10
end

MATLAB

function tmp = code(re, im)
	tmp = 1.9380669946781485e-10;
end

Wolfram

code[re_, im_] := 1.9380669946781485e-10

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Step-by-step derivation

   1. sub0-neg 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in im around 0 85.4%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]

6. Simplified 85.4%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left(2 + im \cdot im\right) + {im}^{4} \cdot 0.08333333333333333\right)} \]

8. Applied egg-rr 3.5%

   \[\leadsto \color{blue}{\frac{\sin re \cdot 1.9380669946781485 \cdot 10^{-10}}{\sin re + \left(\sin re \cdot 1.9380669946781485 \cdot 10^{-10} - \sin re \cdot 1.9380669946781485 \cdot 10^{-10}\right)}} \]
9. Step-by-step derivation

   1. +-inverses 3.5%

      \[\leadsto \frac{\sin re \cdot 1.9380669946781485 \cdot 10^{-10}}{\sin re + \color{blue}{0}} \]

   2. +-rgt-identity 3.5%

      \[\leadsto \frac{\sin re \cdot 1.9380669946781485 \cdot 10^{-10}}{\color{blue}{\sin re}} \]

   3. associate-*l/ 3.5%

      \[\leadsto \color{blue}{\frac{\sin re}{\sin re} \cdot 1.9380669946781485 \cdot 10^{-10}} \]

   4. *-inverses 3.5%

      \[\leadsto \color{blue}{1} \cdot 1.9380669946781485 \cdot 10^{-10} \]

   5. metadata-eval 3.5%

      \[\leadsto \color{blue}{1.9380669946781485 \cdot 10^{-10}} \]

10. Simplified 3.5%

    \[\leadsto \color{blue}{1.9380669946781485 \cdot 10^{-10}} \]

11. Final simplification 3.5%

    \[\leadsto 1.9380669946781485 \cdot 10^{-10} \]

Alternative 20: 4.3% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ 0.08333333333333333 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.08333333333333333)

C

double code(double re, double im) {
	return 0.08333333333333333;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.08333333333333333d0
end function

Java

public static double code(double re, double im) {
	return 0.08333333333333333;
}

Python

def code(re, im):
	return 0.08333333333333333

Julia

function code(re, im)
	return 0.08333333333333333
end

MATLAB

function tmp = code(re, im)
	tmp = 0.08333333333333333;
end

Wolfram

code[re_, im_] := 0.08333333333333333

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Step-by-step derivation

   1. sub0-neg 100.0%

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

3. Simplified 100.0%

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

5. Applied egg-rr 9.2%

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

6. Taylor expanded in re around 0 8.9%

   \[\leadsto \color{blue}{0.08333333333333333 + 0.25 \cdot \frac{1}{{re}^{2}}} \]
7. Step-by-step derivation

   1. associate-*r/ 8.9%

      \[\leadsto 0.08333333333333333 + \color{blue}{\frac{0.25 \cdot 1}{{re}^{2}}} \]

   2. metadata-eval 8.9%

      \[\leadsto 0.08333333333333333 + \frac{\color{blue}{0.25}}{{re}^{2}} \]

   3. unpow2 8.9%

      \[\leadsto 0.08333333333333333 + \frac{0.25}{\color{blue}{re \cdot re}} \]

   4. associate-/r* 8.9%

      \[\leadsto 0.08333333333333333 + \color{blue}{\frac{\frac{0.25}{re}}{re}} \]

8. Simplified 8.9%

   \[\leadsto \color{blue}{0.08333333333333333 + \frac{\frac{0.25}{re}}{re}} \]

9. Taylor expanded in re around inf 4.0%

   \[\leadsto \color{blue}{0.08333333333333333} \]

10. Final simplification 4.0%

    \[\leadsto 0.08333333333333333 \]

Alternative 21: 4.8% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 1.0)

C

double code(double re, double im) {
	return 1.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 1.0d0
end function

Java

public static double code(double re, double im) {
	return 1.0;
}

Python

def code(re, im):
	return 1.0

Julia

function code(re, im)
	return 1.0
end

MATLAB

function tmp = code(re, im)
	tmp = 1.0;
end

Wolfram

code[re_, im_] := 1.0

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Step-by-step derivation

   1. sub0-neg 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in im around 0 85.4%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]

6. Simplified 85.4%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left(2 + im \cdot im\right) + {im}^{4} \cdot 0.08333333333333333\right)} \]

8. Applied egg-rr 4.5%

   \[\leadsto \color{blue}{\frac{\sin re - \sin re \cdot 1.9380669946781485 \cdot 10^{-10}}{\sin re - \sin re \cdot 1.9380669946781485 \cdot 10^{-10}}} \]
9. Step-by-step derivation

   1. *-inverses 4.5%

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

10. Simplified 4.5%

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

11. Final simplification 4.5%

    \[\leadsto 1 \]

Alternative 22: 26.2% accurate, 309.0× 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%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Step-by-step derivation

   1. sub0-neg 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in re around 0 66.8%

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

6. Simplified 66.8%

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

7. Taylor expanded in im around 0 33.5%

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

8. Final simplification 33.5%

   \[\leadsto re \]

Reproduce

herbie shell --seed 2023199 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))