math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 8.2s

Alternatives: 17

Speedup: 1.0×

• 49.2% 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: 86.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.058:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im + 2\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \left(\frac{re}{e^{im}} + re \cdot e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{4} \cdot \left(\sin re \cdot 0.041666666666666664\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.058)
   (* (* 0.5 (sin re)) (+ (* im im) 2.0))
   (if (<= im 1.15e+77)
     (* 0.5 (+ (/ re (exp im)) (* re (exp im))))
     (* (pow im 4.0) (* (sin re) 0.041666666666666664)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.058) {
		tmp = (0.5 * sin(re)) * ((im * im) + 2.0);
	} else if (im <= 1.15e+77) {
		tmp = 0.5 * ((re / exp(im)) + (re * exp(im)));
	} else {
		tmp = pow(im, 4.0) * (sin(re) * 0.041666666666666664);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.058d0) then
        tmp = (0.5d0 * sin(re)) * ((im * im) + 2.0d0)
    else if (im <= 1.15d+77) then
        tmp = 0.5d0 * ((re / exp(im)) + (re * exp(im)))
    else
        tmp = (im ** 4.0d0) * (sin(re) * 0.041666666666666664d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.058) {
		tmp = (0.5 * Math.sin(re)) * ((im * im) + 2.0);
	} else if (im <= 1.15e+77) {
		tmp = 0.5 * ((re / Math.exp(im)) + (re * Math.exp(im)));
	} else {
		tmp = Math.pow(im, 4.0) * (Math.sin(re) * 0.041666666666666664);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.058:
		tmp = (0.5 * math.sin(re)) * ((im * im) + 2.0)
	elif im <= 1.15e+77:
		tmp = 0.5 * ((re / math.exp(im)) + (re * math.exp(im)))
	else:
		tmp = math.pow(im, 4.0) * (math.sin(re) * 0.041666666666666664)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.058)
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(Float64(im * im) + 2.0));
	elseif (im <= 1.15e+77)
		tmp = Float64(0.5 * Float64(Float64(re / exp(im)) + Float64(re * exp(im))));
	else
		tmp = Float64((im ^ 4.0) * Float64(sin(re) * 0.041666666666666664));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.058)
		tmp = (0.5 * sin(re)) * ((im * im) + 2.0);
	elseif (im <= 1.15e+77)
		tmp = 0.5 * ((re / exp(im)) + (re * exp(im)));
	else
		tmp = (im ^ 4.0) * (sin(re) * 0.041666666666666664);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.058], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.15e+77], N[(0.5 * N[(N[(re / N[Exp[im], $MachinePrecision]), $MachinePrecision] + N[(re * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[im, 4.0], $MachinePrecision] * N[(N[Sin[re], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.0580000000000000029

   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 86.3%

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

      1. unpow2 86.3%

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

   6. Simplified 86.3%

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

   if 0.0580000000000000029 < im < 1.14999999999999997e77

   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 92.6%

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

      1. distribute-rgt-in 92.6%

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

      2. fma-def 92.6%

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

      3. exp-neg 92.6%

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

      4. associate-*l/ 92.6%

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

      5. *-lft-identity 92.6%

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

   6. Simplified 92.6%

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

   7. Taylor expanded in im around inf 92.6%

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

   if 1.14999999999999997e77 < 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%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]
   5. Step-by-step derivation

      1. associate-+r+ 100.0%

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

      2. unpow2 100.0%

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

   6. Simplified 100.0%

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

   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%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

   9. Simplified 100.0%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.058:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im + 2\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \left(\frac{re}{e^{im}} + re \cdot e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{4} \cdot \left(\sin re \cdot 0.041666666666666664\right)\\ \end{array} \]

Alternative 3: 86.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.105:\\ \;\;\;\;\sin re + 0.5 \cdot \left(\sin re \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \left(\frac{re}{e^{im}} + re \cdot e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{4} \cdot \left(\sin re \cdot 0.041666666666666664\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.105)
   (+ (sin re) (* 0.5 (* (sin re) (* im im))))
   (if (<= im 1.15e+77)
     (* 0.5 (+ (/ re (exp im)) (* re (exp im))))
     (* (pow im 4.0) (* (sin re) 0.041666666666666664)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.105) {
		tmp = sin(re) + (0.5 * (sin(re) * (im * im)));
	} else if (im <= 1.15e+77) {
		tmp = 0.5 * ((re / exp(im)) + (re * exp(im)));
	} else {
		tmp = pow(im, 4.0) * (sin(re) * 0.041666666666666664);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.105d0) then
        tmp = sin(re) + (0.5d0 * (sin(re) * (im * im)))
    else if (im <= 1.15d+77) then
        tmp = 0.5d0 * ((re / exp(im)) + (re * exp(im)))
    else
        tmp = (im ** 4.0d0) * (sin(re) * 0.041666666666666664d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.105) {
		tmp = Math.sin(re) + (0.5 * (Math.sin(re) * (im * im)));
	} else if (im <= 1.15e+77) {
		tmp = 0.5 * ((re / Math.exp(im)) + (re * Math.exp(im)));
	} else {
		tmp = Math.pow(im, 4.0) * (Math.sin(re) * 0.041666666666666664);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.105:
		tmp = math.sin(re) + (0.5 * (math.sin(re) * (im * im)))
	elif im <= 1.15e+77:
		tmp = 0.5 * ((re / math.exp(im)) + (re * math.exp(im)))
	else:
		tmp = math.pow(im, 4.0) * (math.sin(re) * 0.041666666666666664)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.105)
		tmp = Float64(sin(re) + Float64(0.5 * Float64(sin(re) * Float64(im * im))));
	elseif (im <= 1.15e+77)
		tmp = Float64(0.5 * Float64(Float64(re / exp(im)) + Float64(re * exp(im))));
	else
		tmp = Float64((im ^ 4.0) * Float64(sin(re) * 0.041666666666666664));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.105)
		tmp = sin(re) + (0.5 * (sin(re) * (im * im)));
	elseif (im <= 1.15e+77)
		tmp = 0.5 * ((re / exp(im)) + (re * exp(im)));
	else
		tmp = (im ^ 4.0) * (sin(re) * 0.041666666666666664);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.105], N[(N[Sin[re], $MachinePrecision] + N[(0.5 * N[(N[Sin[re], $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.15e+77], N[(0.5 * N[(N[(re / N[Exp[im], $MachinePrecision]), $MachinePrecision] + N[(re * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[im, 4.0], $MachinePrecision] * N[(N[Sin[re], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.104999999999999996

   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 86.3%

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

      1. *-commutative 86.3%

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

      2. unpow2 86.3%

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

   6. Simplified 86.3%

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

   if 0.104999999999999996 < im < 1.14999999999999997e77

   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 92.6%

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

      1. distribute-rgt-in 92.6%

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

      2. fma-def 92.6%

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

      3. exp-neg 92.6%

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

      4. associate-*l/ 92.6%

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

      5. *-lft-identity 92.6%

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

   6. Simplified 92.6%

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

   7. Taylor expanded in im around inf 92.6%

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

   if 1.14999999999999997e77 < 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%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]
   5. Step-by-step derivation

      1. associate-+r+ 100.0%

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

      2. unpow2 100.0%

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

   6. Simplified 100.0%

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

   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%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

   9. Simplified 100.0%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.105:\\ \;\;\;\;\sin re + 0.5 \cdot \left(\sin re \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \left(\frac{re}{e^{im}} + re \cdot e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{4} \cdot \left(\sin re \cdot 0.041666666666666664\right)\\ \end{array} \]

Alternative 4: 86.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.06:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im + 2\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(2 \cdot \cosh im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{4} \cdot \left(\sin re \cdot 0.041666666666666664\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.06)
   (* (* 0.5 (sin re)) (+ (* im im) 2.0))
   (if (<= im 1.15e+77)
     (* 0.5 (* re (* 2.0 (cosh im))))
     (* (pow im 4.0) (* (sin re) 0.041666666666666664)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.06) {
		tmp = (0.5 * sin(re)) * ((im * im) + 2.0);
	} else if (im <= 1.15e+77) {
		tmp = 0.5 * (re * (2.0 * cosh(im)));
	} else {
		tmp = pow(im, 4.0) * (sin(re) * 0.041666666666666664);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.06d0) then
        tmp = (0.5d0 * sin(re)) * ((im * im) + 2.0d0)
    else if (im <= 1.15d+77) then
        tmp = 0.5d0 * (re * (2.0d0 * cosh(im)))
    else
        tmp = (im ** 4.0d0) * (sin(re) * 0.041666666666666664d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.06) {
		tmp = (0.5 * Math.sin(re)) * ((im * im) + 2.0);
	} else if (im <= 1.15e+77) {
		tmp = 0.5 * (re * (2.0 * Math.cosh(im)));
	} else {
		tmp = Math.pow(im, 4.0) * (Math.sin(re) * 0.041666666666666664);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.06:
		tmp = (0.5 * math.sin(re)) * ((im * im) + 2.0)
	elif im <= 1.15e+77:
		tmp = 0.5 * (re * (2.0 * math.cosh(im)))
	else:
		tmp = math.pow(im, 4.0) * (math.sin(re) * 0.041666666666666664)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.06)
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(Float64(im * im) + 2.0));
	elseif (im <= 1.15e+77)
		tmp = Float64(0.5 * Float64(re * Float64(2.0 * cosh(im))));
	else
		tmp = Float64((im ^ 4.0) * Float64(sin(re) * 0.041666666666666664));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.06)
		tmp = (0.5 * sin(re)) * ((im * im) + 2.0);
	elseif (im <= 1.15e+77)
		tmp = 0.5 * (re * (2.0 * cosh(im)));
	else
		tmp = (im ^ 4.0) * (sin(re) * 0.041666666666666664);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.06], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.15e+77], N[(0.5 * N[(re * N[(2.0 * N[Cosh[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[im, 4.0], $MachinePrecision] * N[(N[Sin[re], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.059999999999999998

   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 86.3%

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

      1. unpow2 86.3%

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

   6. Simplified 86.3%

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

   if 0.059999999999999998 < im < 1.14999999999999997e77

   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 92.6%

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

      1. distribute-rgt-in 92.6%

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

      2. fma-def 92.6%

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

      3. exp-neg 92.6%

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

      4. associate-*l/ 92.6%

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

      5. *-lft-identity 92.6%

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

   6. Simplified 92.6%

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

   7. Taylor expanded in im around inf 92.6%

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

      1. expm1-log1p-u 46.2%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{re}{e^{im}} + e^{im} \cdot re\right)\right)} \]

      2. expm1-udef 46.2%

         \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{re}{e^{im}} + e^{im} \cdot re\right)} - 1\right)} \]

      3. +-commutative 46.2%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{e^{im} \cdot re + \frac{re}{e^{im}}}\right)} - 1\right) \]

      4. *-commutative 46.2%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{re \cdot e^{im}} + \frac{re}{e^{im}}\right)} - 1\right) \]

      5. div-inv 46.2%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(re \cdot e^{im} + \color{blue}{re \cdot \frac{1}{e^{im}}}\right)} - 1\right) \]

      6. distribute-lft-out 46.2%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{re \cdot \left(e^{im} + \frac{1}{e^{im}}\right)}\right)} - 1\right) \]

      7. rec-exp 46.2%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(re \cdot \left(e^{im} + \color{blue}{e^{-im}}\right)\right)} - 1\right) \]

      8. cosh-undef 46.2%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(re \cdot \color{blue}{\left(2 \cdot \cosh im\right)}\right)} - 1\right) \]

   9. Applied egg-rr 46.2%

      \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(re \cdot \left(2 \cdot \cosh im\right)\right)} - 1\right)} \]
   10. Step-by-step derivation

       1. expm1-def 46.2%

          \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(re \cdot \left(2 \cdot \cosh im\right)\right)\right)} \]

       2. expm1-log1p 92.6%

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

   11. Simplified 92.6%

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

   if 1.14999999999999997e77 < 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%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]
   5. Step-by-step derivation

      1. associate-+r+ 100.0%

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

      2. unpow2 100.0%

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

   6. Simplified 100.0%

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

   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%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

   9. Simplified 100.0%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.06:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im + 2\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(2 \cdot \cosh im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{4} \cdot \left(\sin re \cdot 0.041666666666666664\right)\\ \end{array} \]

Alternative 5: 84.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.044 \lor \neg \left(im \leq 2.6 \cdot 10^{+152}\right):\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im + 2\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(2 \cdot \cosh im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 0.044) (not (<= im 2.6e+152)))
   (* (* 0.5 (sin re)) (+ (* im im) 2.0))
   (* 0.5 (* re (* 2.0 (cosh im))))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 0.044) || !(im <= 2.6e+152)) {
		tmp = (0.5 * sin(re)) * ((im * im) + 2.0);
	} else {
		tmp = 0.5 * (re * (2.0 * cosh(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 <= 0.044d0) .or. (.not. (im <= 2.6d+152))) then
        tmp = (0.5d0 * sin(re)) * ((im * im) + 2.0d0)
    else
        tmp = 0.5d0 * (re * (2.0d0 * cosh(im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= 0.044) || !(im <= 2.6e+152)) {
		tmp = (0.5 * Math.sin(re)) * ((im * im) + 2.0);
	} else {
		tmp = 0.5 * (re * (2.0 * Math.cosh(im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= 0.044) or not (im <= 2.6e+152):
		tmp = (0.5 * math.sin(re)) * ((im * im) + 2.0)
	else:
		tmp = 0.5 * (re * (2.0 * math.cosh(im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 0.044) || !(im <= 2.6e+152))
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(Float64(im * im) + 2.0));
	else
		tmp = Float64(0.5 * Float64(re * Float64(2.0 * cosh(im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= 0.044) || ~((im <= 2.6e+152)))
		tmp = (0.5 * sin(re)) * ((im * im) + 2.0);
	else
		tmp = 0.5 * (re * (2.0 * cosh(im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 0.044], N[Not[LessEqual[im, 2.6e+152]], $MachinePrecision]], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(re * N[(2.0 * N[Cosh[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 0.043999999999999997 or 2.6000000000000001e152 < 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 88.1%

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

      1. unpow2 88.1%

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

   6. Simplified 88.1%

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

   if 0.043999999999999997 < im < 2.6000000000000001e152

   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 81.4%

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

      1. distribute-rgt-in 81.4%

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

      2. fma-def 81.4%

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

      3. exp-neg 81.4%

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

      4. associate-*l/ 81.4%

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

      5. *-lft-identity 81.4%

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

   6. Simplified 81.4%

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

   7. Taylor expanded in im around inf 81.4%

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

      1. expm1-log1p-u 40.6%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{re}{e^{im}} + e^{im} \cdot re\right)\right)} \]

      2. expm1-udef 40.6%

         \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{re}{e^{im}} + e^{im} \cdot re\right)} - 1\right)} \]

      3. +-commutative 40.6%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{e^{im} \cdot re + \frac{re}{e^{im}}}\right)} - 1\right) \]

      4. *-commutative 40.6%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{re \cdot e^{im}} + \frac{re}{e^{im}}\right)} - 1\right) \]

      5. div-inv 40.6%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(re \cdot e^{im} + \color{blue}{re \cdot \frac{1}{e^{im}}}\right)} - 1\right) \]

      6. distribute-lft-out 40.6%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{re \cdot \left(e^{im} + \frac{1}{e^{im}}\right)}\right)} - 1\right) \]

      7. rec-exp 40.6%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(re \cdot \left(e^{im} + \color{blue}{e^{-im}}\right)\right)} - 1\right) \]

      8. cosh-undef 40.6%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(re \cdot \color{blue}{\left(2 \cdot \cosh im\right)}\right)} - 1\right) \]

   9. Applied egg-rr 40.6%

      \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(re \cdot \left(2 \cdot \cosh im\right)\right)} - 1\right)} \]
   10. Step-by-step derivation

       1. expm1-def 40.6%

          \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(re \cdot \left(2 \cdot \cosh im\right)\right)\right)} \]

       2. expm1-log1p 81.4%

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

   11. Simplified 81.4%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.044 \lor \neg \left(im \leq 2.6 \cdot 10^{+152}\right):\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im + 2\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(2 \cdot \cosh im\right)\right)\\ \end{array} \]

Alternative 6: 69.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.01:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 7 \cdot 10^{+215}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(2 \cdot \cosh im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(0.5 \cdot \sin re\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.01)
   (sin re)
   (if (<= im 7e+215)
     (* 0.5 (* re (* 2.0 (cosh im))))
     (* im (* (* 0.5 (sin re)) im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.01) {
		tmp = sin(re);
	} else if (im <= 7e+215) {
		tmp = 0.5 * (re * (2.0 * cosh(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 <= 0.01d0) then
        tmp = sin(re)
    else if (im <= 7d+215) then
        tmp = 0.5d0 * (re * (2.0d0 * cosh(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 <= 0.01) {
		tmp = Math.sin(re);
	} else if (im <= 7e+215) {
		tmp = 0.5 * (re * (2.0 * Math.cosh(im)));
	} else {
		tmp = im * ((0.5 * Math.sin(re)) * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.01:
		tmp = math.sin(re)
	elif im <= 7e+215:
		tmp = 0.5 * (re * (2.0 * math.cosh(im)))
	else:
		tmp = im * ((0.5 * math.sin(re)) * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.01)
		tmp = sin(re);
	elseif (im <= 7e+215)
		tmp = Float64(0.5 * Float64(re * Float64(2.0 * cosh(im))));
	else
		tmp = Float64(im * Float64(Float64(0.5 * sin(re)) * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.01)
		tmp = sin(re);
	elseif (im <= 7e+215)
		tmp = 0.5 * (re * (2.0 * cosh(im)));
	else
		tmp = im * ((0.5 * sin(re)) * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.01], N[Sin[re], $MachinePrecision], If[LessEqual[im, 7e+215], N[(0.5 * N[(re * N[(2.0 * N[Cosh[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.0100000000000000002

   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 67.9%

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

   if 0.0100000000000000002 < im < 6.99999999999999954e215

   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 81.5%

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

      1. distribute-rgt-in 81.5%

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

      2. fma-def 81.5%

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

      3. exp-neg 81.5%

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

      4. associate-*l/ 81.5%

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

      5. *-lft-identity 81.5%

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

   6. Simplified 81.5%

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

   7. Taylor expanded in im around inf 81.5%

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

      1. expm1-log1p-u 37.2%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{re}{e^{im}} + e^{im} \cdot re\right)\right)} \]

      2. expm1-udef 37.2%

         \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{re}{e^{im}} + e^{im} \cdot re\right)} - 1\right)} \]

      3. +-commutative 37.2%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{e^{im} \cdot re + \frac{re}{e^{im}}}\right)} - 1\right) \]

      4. *-commutative 37.2%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{re \cdot e^{im}} + \frac{re}{e^{im}}\right)} - 1\right) \]

      5. div-inv 37.2%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(re \cdot e^{im} + \color{blue}{re \cdot \frac{1}{e^{im}}}\right)} - 1\right) \]

      6. distribute-lft-out 37.2%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{re \cdot \left(e^{im} + \frac{1}{e^{im}}\right)}\right)} - 1\right) \]

      7. rec-exp 37.2%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(re \cdot \left(e^{im} + \color{blue}{e^{-im}}\right)\right)} - 1\right) \]

      8. cosh-undef 37.2%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(re \cdot \color{blue}{\left(2 \cdot \cosh im\right)}\right)} - 1\right) \]

   9. Applied egg-rr 37.2%

      \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(re \cdot \left(2 \cdot \cosh im\right)\right)} - 1\right)} \]
   10. Step-by-step derivation

       1. expm1-def 37.2%

          \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(re \cdot \left(2 \cdot \cosh im\right)\right)\right)} \]

       2. expm1-log1p 81.5%

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

   11. Simplified 81.5%

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

   if 6.99999999999999954e215 < 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%

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

      1. unpow2 100.0%

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

   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. *-commutative 100.0%

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

      3. unpow2 100.0%

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

      4. associate-*l* 85.8%

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

   9. Simplified 85.8%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.01:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 7 \cdot 10^{+215}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(2 \cdot \cosh im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(0.5 \cdot \sin re\right) \cdot im\right)\\ \end{array} \]

Alternative 7: 69.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.018:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(2 \cdot \cosh im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.018) (sin re) (* 0.5 (* re (* 2.0 (cosh im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.018) {
		tmp = sin(re);
	} else {
		tmp = 0.5 * (re * (2.0 * cosh(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 <= 0.018d0) then
        tmp = sin(re)
    else
        tmp = 0.5d0 * (re * (2.0d0 * cosh(im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.018) {
		tmp = Math.sin(re);
	} else {
		tmp = 0.5 * (re * (2.0 * Math.cosh(im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.018:
		tmp = math.sin(re)
	else:
		tmp = 0.5 * (re * (2.0 * math.cosh(im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.018)
		tmp = sin(re);
	else
		tmp = Float64(0.5 * Float64(re * Float64(2.0 * cosh(im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.018)
		tmp = sin(re);
	else
		tmp = 0.5 * (re * (2.0 * cosh(im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.018], N[Sin[re], $MachinePrecision], N[(0.5 * N[(re * N[(2.0 * N[Cosh[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 0.0179999999999999986

   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 67.9%

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

   if 0.0179999999999999986 < 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 re around 0 80.7%

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

      1. distribute-rgt-in 80.7%

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

      2. fma-def 80.7%

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

      3. exp-neg 80.7%

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

      4. associate-*l/ 80.7%

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

      5. *-lft-identity 80.7%

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

   6. Simplified 80.7%

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

   7. Taylor expanded in im around inf 80.7%

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

      1. expm1-log1p-u 41.9%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{re}{e^{im}} + e^{im} \cdot re\right)\right)} \]

      2. expm1-udef 41.9%

         \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{re}{e^{im}} + e^{im} \cdot re\right)} - 1\right)} \]

      3. +-commutative 41.9%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{e^{im} \cdot re + \frac{re}{e^{im}}}\right)} - 1\right) \]

      4. *-commutative 41.9%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{re \cdot e^{im}} + \frac{re}{e^{im}}\right)} - 1\right) \]

      5. div-inv 41.9%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(re \cdot e^{im} + \color{blue}{re \cdot \frac{1}{e^{im}}}\right)} - 1\right) \]

      6. distribute-lft-out 41.9%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{re \cdot \left(e^{im} + \frac{1}{e^{im}}\right)}\right)} - 1\right) \]

      7. rec-exp 41.9%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(re \cdot \left(e^{im} + \color{blue}{e^{-im}}\right)\right)} - 1\right) \]

      8. cosh-undef 41.9%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(re \cdot \color{blue}{\left(2 \cdot \cosh im\right)}\right)} - 1\right) \]

   9. Applied egg-rr 41.9%

      \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(re \cdot \left(2 \cdot \cosh im\right)\right)} - 1\right)} \]
   10. Step-by-step derivation

       1. expm1-def 41.9%

          \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(re \cdot \left(2 \cdot \cosh im\right)\right)\right)} \]

       2. expm1-log1p 80.7%

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

   11. Simplified 80.7%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.018:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(2 \cdot \cosh im\right)\right)\\ \end{array} \]

Alternative 8: 65.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 8.2 \cdot 10^{+15}:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(re \cdot {im}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 8.2e+15) (sin re) (* 0.041666666666666664 (* re (pow im 4.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 8.2e+15) {
		tmp = sin(re);
	} else {
		tmp = 0.041666666666666664 * (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 <= 8.2d+15) then
        tmp = sin(re)
    else
        tmp = 0.041666666666666664d0 * (re * (im ** 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 8.2e+15) {
		tmp = Math.sin(re);
	} else {
		tmp = 0.041666666666666664 * (re * Math.pow(im, 4.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 8.2e+15:
		tmp = math.sin(re)
	else:
		tmp = 0.041666666666666664 * (re * math.pow(im, 4.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 8.2e+15)
		tmp = sin(re);
	else
		tmp = Float64(0.041666666666666664 * Float64(re * (im ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 8.2e+15)
		tmp = sin(re);
	else
		tmp = 0.041666666666666664 * (re * (im ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 8.2e+15], N[Sin[re], $MachinePrecision], N[(0.041666666666666664 * N[(re * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 8.2e15

   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 67.3%

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

   if 8.2e15 < 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 82.6%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]
   5. Step-by-step derivation

      1. associate-+r+ 82.6%

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

      2. unpow2 82.6%

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

   6. Simplified 82.6%

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

   7. Taylor expanded in im around inf 82.6%

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

      1. *-commutative 82.6%

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

      2. *-commutative 82.6%

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

      3. associate-*l* 82.6%

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

   9. Simplified 82.6%

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

   10. Taylor expanded in re around 0 68.9%

       \[\leadsto \color{blue}{0.041666666666666664 \cdot \left(re \cdot {im}^{4}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 8.2 \cdot 10^{+15}:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(re \cdot {im}^{4}\right)\\ \end{array} \]

Alternative 9: 61.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.2 \cdot 10^{+25}:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im \cdot im + 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.2e+25) (sin re) (* 0.5 (* re (+ (* im im) 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.2e+25) {
		tmp = sin(re);
	} else {
		tmp = 0.5 * (re * ((im * im) + 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.2d+25) then
        tmp = sin(re)
    else
        tmp = 0.5d0 * (re * ((im * im) + 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.2e+25) {
		tmp = Math.sin(re);
	} else {
		tmp = 0.5 * (re * ((im * im) + 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.2e+25:
		tmp = math.sin(re)
	else:
		tmp = 0.5 * (re * ((im * im) + 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.2e+25)
		tmp = sin(re);
	else
		tmp = Float64(0.5 * Float64(re * Float64(Float64(im * im) + 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.2e+25)
		tmp = sin(re);
	else
		tmp = 0.5 * (re * ((im * im) + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.2e+25], N[Sin[re], $MachinePrecision], N[(0.5 * N[(re * N[(N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.19999999999999998e25

   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 66.3%

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

   if 1.19999999999999998e25 < 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 54.8%

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

      1. unpow2 54.8%

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

   6. Simplified 54.8%

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

   7. Taylor expanded in re around 0 55.4%

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

      1. *-commutative 55.4%

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

      2. unpow2 55.4%

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

   9. Simplified 55.4%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.2 \cdot 10^{+25}:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im \cdot im + 2\right)\right)\\ \end{array} \]

Alternative 10: 29.2% accurate, 34.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1900000000000:\\ \;\;\;\;0.5 \cdot \left(re + re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(4 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1900000000000.0) (* 0.5 (+ re re)) (* 0.5 (* 4.0 (* re re)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1900000000000.0) {
		tmp = 0.5 * (re + re);
	} else {
		tmp = 0.5 * (4.0 * (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 <= 1900000000000.0d0) then
        tmp = 0.5d0 * (re + re)
    else
        tmp = 0.5d0 * (4.0d0 * (re * re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1900000000000.0) {
		tmp = 0.5 * (re + re);
	} else {
		tmp = 0.5 * (4.0 * (re * re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1900000000000.0:
		tmp = 0.5 * (re + re)
	else:
		tmp = 0.5 * (4.0 * (re * re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1900000000000.0)
		tmp = Float64(0.5 * Float64(re + re));
	else
		tmp = Float64(0.5 * Float64(4.0 * Float64(re * re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1900000000000.0)
		tmp = 0.5 * (re + re);
	else
		tmp = 0.5 * (4.0 * (re * re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1900000000000.0], N[(0.5 * N[(re + re), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(4.0 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.9e12

   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 58.1%

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

      1. distribute-rgt-in 58.1%

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

      2. fma-def 58.1%

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

      3. exp-neg 58.1%

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

      4. associate-*l/ 58.1%

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

      5. *-lft-identity 58.1%

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

   6. Simplified 58.1%

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

   7. Taylor expanded in im around 0 34.4%

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

      1. count-2 34.4%

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

   9. Simplified 34.4%

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

   if 1.9e12 < 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 re around 0 81.7%

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

      1. distribute-rgt-in 81.7%

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

      2. fma-def 81.7%

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

      3. exp-neg 81.7%

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

      4. associate-*l/ 81.7%

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

      5. *-lft-identity 81.7%

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

   6. Simplified 81.7%

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

   8. Applied egg-rr 18.4%

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

      1. distribute-lft-in 18.4%

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

      2. count-2 18.4%

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

      3. count-2 18.4%

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

      4. swap-sqr 18.4%

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

      5. metadata-eval 18.4%

         \[\leadsto 0.5 \cdot \left(\color{blue}{4} \cdot \left(re \cdot re\right)\right) \]

   10. Simplified 18.4%

       \[\leadsto 0.5 \cdot \color{blue}{\left(4 \cdot \left(re \cdot re\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1900000000000:\\ \;\;\;\;0.5 \cdot \left(re + re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(4 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]

Alternative 11: 32.1% accurate, 34.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -7 \cdot 10^{+133}:\\ \;\;\;\;0.5 \cdot \left(4 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im + 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -7e+133) (* 0.5 (* 4.0 (* re re))) (* 0.5 (* re (+ im 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -7e+133) {
		tmp = 0.5 * (4.0 * (re * re));
	} else {
		tmp = 0.5 * (re * (im + 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-7d+133)) then
        tmp = 0.5d0 * (4.0d0 * (re * re))
    else
        tmp = 0.5d0 * (re * (im + 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -7e+133) {
		tmp = 0.5 * (4.0 * (re * re));
	} else {
		tmp = 0.5 * (re * (im + 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -7e+133:
		tmp = 0.5 * (4.0 * (re * re))
	else:
		tmp = 0.5 * (re * (im + 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -7e+133)
		tmp = Float64(0.5 * Float64(4.0 * Float64(re * re)));
	else
		tmp = Float64(0.5 * Float64(re * Float64(im + 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -7e+133)
		tmp = 0.5 * (4.0 * (re * re));
	else
		tmp = 0.5 * (re * (im + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -7e+133], N[(0.5 * N[(4.0 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(re * N[(im + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -6.9999999999999997e133

   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 19.1%

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

      1. distribute-rgt-in 19.1%

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

      2. fma-def 19.1%

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

      3. exp-neg 19.1%

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

      4. associate-*l/ 19.1%

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

      5. *-lft-identity 19.1%

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

   6. Simplified 19.1%

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

   8. Applied egg-rr 26.2%

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

      1. distribute-lft-in 26.2%

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

      2. count-2 26.2%

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

      3. count-2 26.2%

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

      4. swap-sqr 26.2%

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

      5. metadata-eval 26.2%

         \[\leadsto 0.5 \cdot \left(\color{blue}{4} \cdot \left(re \cdot re\right)\right) \]

   10. Simplified 26.2%

       \[\leadsto 0.5 \cdot \color{blue}{\left(4 \cdot \left(re \cdot re\right)\right)} \]

   if -6.9999999999999997e133 < re

   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 71.8%

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

      1. distribute-rgt-in 71.8%

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

      2. fma-def 71.8%

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

      3. exp-neg 71.8%

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

      4. associate-*l/ 71.8%

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

      5. *-lft-identity 71.8%

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

   6. Simplified 71.8%

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

   7. Taylor expanded in im around 0 51.1%

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

   8. Taylor expanded in im around 0 37.4%

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

      1. +-commutative 37.4%

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

      2. *-commutative 37.4%

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

      3. distribute-lft-out 37.4%

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

   10. Simplified 37.4%

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

4. Final simplification 35.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -7 \cdot 10^{+133}:\\ \;\;\;\;0.5 \cdot \left(4 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im + 2\right)\right)\\ \end{array} \]

Alternative 12: 48.6% accurate, 34.3× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(re \cdot \left(im \cdot im + 2\right)\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* 0.5 (* re (+ (* im im) 2.0))))

C

double code(double re, double im) {
	return 0.5 * (re * ((im * im) + 2.0));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * (re * ((im * im) + 2.0d0))
end function

Java

public static double code(double re, double im) {
	return 0.5 * (re * ((im * im) + 2.0));
}

Python

def code(re, im):
	return 0.5 * (re * ((im * im) + 2.0))

Julia

function code(re, im)
	return Float64(0.5 * Float64(re * Float64(Float64(im * im) + 2.0)))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * (re * ((im * im) + 2.0));
end

Wolfram

code[re_, im_] := N[(0.5 * N[(re * N[(N[(im * im), $MachinePrecision] + 2.0), $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. Taylor expanded in im around 0 77.7%

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

   1. unpow2 77.7%

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

6. Simplified 77.7%

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

7. Taylor expanded in re around 0 49.0%

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

   1. *-commutative 49.0%

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

   2. unpow2 49.0%

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

9. Simplified 49.0%

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

10. Final simplification 49.0%

    \[\leadsto 0.5 \cdot \left(re \cdot \left(im \cdot im + 2\right)\right) \]

Alternative 13: 29.4% accurate, 43.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 890:\\ \;\;\;\;0.5 \cdot \left(re + re\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.125}{re \cdot re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 890.0) (* 0.5 (+ re re)) (/ 0.125 (* re re))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 890.0) {
		tmp = 0.5 * (re + re);
	} else {
		tmp = 0.125 / (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 <= 890.0d0) then
        tmp = 0.5d0 * (re + re)
    else
        tmp = 0.125d0 / (re * re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 890.0) {
		tmp = 0.5 * (re + re);
	} else {
		tmp = 0.125 / (re * re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 890.0:
		tmp = 0.5 * (re + re)
	else:
		tmp = 0.125 / (re * re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 890.0)
		tmp = Float64(0.5 * Float64(re + re));
	else
		tmp = Float64(0.125 / Float64(re * re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 890.0)
		tmp = 0.5 * (re + re);
	else
		tmp = 0.125 / (re * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 890.0], N[(0.5 * N[(re + re), $MachinePrecision]), $MachinePrecision], N[(0.125 / N[(re * re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 890

   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 57.8%

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

      1. distribute-rgt-in 57.8%

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

      2. fma-def 57.8%

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

      3. exp-neg 57.8%

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

      4. associate-*l/ 57.8%

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

      5. *-lft-identity 57.8%

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

   6. Simplified 57.8%

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

   7. Taylor expanded in im around 0 34.6%

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

      1. count-2 34.6%

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

   9. Simplified 34.6%

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

   if 890 < 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 re around 0 82.0%

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

      1. distribute-rgt-in 82.0%

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

      2. fma-def 82.0%

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

      3. exp-neg 82.0%

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

      4. associate-*l/ 82.0%

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

      5. *-lft-identity 82.0%

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

   6. Simplified 82.0%

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

   8. Applied egg-rr 11.7%

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

   9. Taylor expanded in re around 0 11.7%

      \[\leadsto \color{blue}{\frac{0.125}{{re}^{2}}} \]
   10. Step-by-step derivation

       1. unpow2 11.7%

          \[\leadsto \frac{0.125}{\color{blue}{re \cdot re}} \]

   11. Simplified 11.7%

       \[\leadsto \color{blue}{\frac{0.125}{re \cdot re}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 29.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 890:\\ \;\;\;\;0.5 \cdot \left(re + re\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.125}{re \cdot re}\\ \end{array} \]

Alternative 14: 5.5% accurate, 61.8× speedup

Math

\[\begin{array}{l} \\ 0.001953125 \cdot \left(0.5 \cdot re\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* 0.001953125 (* 0.5 re)))

C

double code(double re, double im) {
	return 0.001953125 * (0.5 * re);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.001953125d0 * (0.5d0 * re)
end function

Java

public static double code(double re, double im) {
	return 0.001953125 * (0.5 * re);
}

Python

def code(re, im):
	return 0.001953125 * (0.5 * re)

Julia

function code(re, im)
	return Float64(0.001953125 * Float64(0.5 * re))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.001953125 * (0.5 * re);
end

Wolfram

code[re_, im_] := N[(0.001953125 * N[(0.5 * re), $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. Taylor expanded in re around 0 63.6%

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

   1. associate-*r* 63.6%

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

   2. *-commutative 63.6%

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

6. Simplified 63.6%

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

8. Applied egg-rr 5.6%

   \[\leadsto \color{blue}{0.001953125} \cdot \left(0.5 \cdot re\right) \]

9. Final simplification 5.6%

   \[\leadsto 0.001953125 \cdot \left(0.5 \cdot re\right) \]

Alternative 15: 26.7% accurate, 61.8× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(re + re\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* 0.5 (+ re re)))

C

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

Fortran

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

Java

public static double code(double re, double im) {
	return 0.5 * (re + re);
}

Python

def code(re, im):
	return 0.5 * (re + re)

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(0.5 * N[(re + re), $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. Taylor expanded in re around 0 63.6%

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

   1. distribute-rgt-in 63.6%

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

   2. fma-def 63.6%

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

   3. exp-neg 63.6%

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

   4. associate-*l/ 63.6%

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

   5. *-lft-identity 63.6%

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

6. Simplified 63.6%

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

7. Taylor expanded in im around 0 27.0%

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

   1. count-2 27.0%

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

9. Simplified 27.0%

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

10. Final simplification 27.0%

    \[\leadsto 0.5 \cdot \left(re + re\right) \]

Alternative 16: 3.5% 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 90.3%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]
5. Step-by-step derivation

   1. associate-+r+ 90.3%

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

   2. unpow2 90.3%

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

6. Simplified 90.3%

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

8. Applied egg-rr 3.3%

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

   1. *-commutative 3.3%

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

   2. +-inverses 3.3%

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

   3. +-rgt-identity 3.3%

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

   4. associate-/l* 3.3%

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

   5. *-inverses 3.3%

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

   6. metadata-eval 3.3%

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

10. Simplified 3.3%

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

11. Final simplification 3.3%

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

Alternative 17: 4.6% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.5)

C

double code(double re, double im) {
	return 0.5;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0
end function

Java

public static double code(double re, double im) {
	return 0.5;
}

Python

def code(re, im):
	return 0.5

Julia

function code(re, im)
	return 0.5
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5;
end

Wolfram

code[re_, im_] := 0.5

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 63.6%

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

   1. distribute-rgt-in 63.6%

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

   2. fma-def 63.6%

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

   3. exp-neg 63.6%

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

   4. associate-*l/ 63.6%

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

   5. *-lft-identity 63.6%

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

6. Simplified 63.6%

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

8. Applied egg-rr 8.1%

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

   1. sqr-pow 8.1%

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

   2. pow-prod-down 8.1%

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

   3. flip-+ 0.0%

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

   4. +-inverses 0.0%

      \[\leadsto 0.5 \cdot {\left(\left(re + re\right) \cdot \frac{\color{blue}{0}}{re - re}\right)}^{\left(\frac{-2}{2}\right)} \]

   5. +-inverses 0.0%

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

   6. associate-*r/ 0.0%

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

   7. difference-of-squares 0.0%

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

   8. flip-+ 3.6%

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

   9. metadata-eval 3.6%

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

   10. flip-+ 0.0%

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

   11. clear-num 0.0%

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

   12. +-inverses 0.0%

       \[\leadsto 0.5 \cdot {\left(\frac{1}{\frac{\color{blue}{0}}{re \cdot re - re \cdot re}}\right)}^{-1} \]

   13. +-inverses 0.0%

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

   14. +-inverses 0.0%

       \[\leadsto 0.5 \cdot {\left(\frac{1}{\frac{re \cdot re - re \cdot re}{\color{blue}{0}}}\right)}^{-1} \]

   15. +-inverses 0.0%

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

   16. flip-+ 27.0%

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

   17. unpow-1 27.0%

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

   18. metadata-eval 27.0%

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

   19. sqrt-pow1 14.0%

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

   20. metadata-eval 14.0%

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

   21. pow-prod-up 7.9%

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

10. Applied egg-rr 4.1%

    \[\leadsto \color{blue}{0.5} \]

11. Final simplification 4.1%

    \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023224 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))