math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 9.9s

Alternatives: 21

Speedup: 1.0×

• 49.9% 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 99.6%

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

   1. sub0-neg 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 2: 92.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.19:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(\left(2 + im \cdot im\right) + {im}^{4} \cdot 0.08333333333333333\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.19)
   (*
    (* 0.5 (sin re))
    (+ (+ 2.0 (* im im)) (* (pow im 4.0) 0.08333333333333333)))
   (if (<= im 1.15e+77)
     (* (* 0.5 re) (+ (exp (- im)) (exp im)))
     (* 0.041666666666666664 (* (sin re) (pow im 4.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.19) {
		tmp = (0.5 * sin(re)) * ((2.0 + (im * im)) + (pow(im, 4.0) * 0.08333333333333333));
	} else if (im <= 1.15e+77) {
		tmp = (0.5 * re) * (exp(-im) + exp(im));
	} else {
		tmp = 0.041666666666666664 * (sin(re) * pow(im, 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.19d0) then
        tmp = (0.5d0 * sin(re)) * ((2.0d0 + (im * im)) + ((im ** 4.0d0) * 0.08333333333333333d0))
    else if (im <= 1.15d+77) then
        tmp = (0.5d0 * re) * (exp(-im) + exp(im))
    else
        tmp = 0.041666666666666664d0 * (sin(re) * (im ** 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.19) {
		tmp = (0.5 * Math.sin(re)) * ((2.0 + (im * im)) + (Math.pow(im, 4.0) * 0.08333333333333333));
	} else if (im <= 1.15e+77) {
		tmp = (0.5 * re) * (Math.exp(-im) + Math.exp(im));
	} else {
		tmp = 0.041666666666666664 * (Math.sin(re) * Math.pow(im, 4.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.19:
		tmp = (0.5 * math.sin(re)) * ((2.0 + (im * im)) + (math.pow(im, 4.0) * 0.08333333333333333))
	elif im <= 1.15e+77:
		tmp = (0.5 * re) * (math.exp(-im) + math.exp(im))
	else:
		tmp = 0.041666666666666664 * (math.sin(re) * math.pow(im, 4.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.19)
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(Float64(2.0 + Float64(im * im)) + Float64((im ^ 4.0) * 0.08333333333333333)));
	elseif (im <= 1.15e+77)
		tmp = Float64(Float64(0.5 * re) * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64(0.041666666666666664 * Float64(sin(re) * (im ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.19)
		tmp = (0.5 * sin(re)) * ((2.0 + (im * im)) + ((im ^ 4.0) * 0.08333333333333333));
	elseif (im <= 1.15e+77)
		tmp = (0.5 * re) * (exp(-im) + exp(im));
	else
		tmp = 0.041666666666666664 * (sin(re) * (im ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.19], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision] + N[(N[Power[im, 4.0], $MachinePrecision] * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.15e+77], N[(N[(0.5 * re), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.041666666666666664 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.19

   1. Initial program 99.5%

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

      1. sub0-neg 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in im around 0 91.1%

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

   6. Simplified 91.1%

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

   if 0.19 < 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 70.6%

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

   6. Simplified 70.6%

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

   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%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in im around inf 100.0%

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

      1. *-commutative 100.0%

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

   9. Simplified 100.0%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.19:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(\left(2 + im \cdot im\right) + {im}^{4} \cdot 0.08333333333333333\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\\ \end{array} \]

Alternative 3: 86.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.05:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 5 \cdot 10^{+76}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.05)
   (* (* 0.5 (sin re)) (+ 2.0 (* im im)))
   (if (<= im 5e+76)
     (* (* 0.5 re) (+ (exp (- im)) (exp im)))
     (* 0.041666666666666664 (* (sin re) (pow im 4.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.05) {
		tmp = (0.5 * sin(re)) * (2.0 + (im * im));
	} else if (im <= 5e+76) {
		tmp = (0.5 * re) * (exp(-im) + exp(im));
	} else {
		tmp = 0.041666666666666664 * (sin(re) * pow(im, 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.05d0) then
        tmp = (0.5d0 * sin(re)) * (2.0d0 + (im * im))
    else if (im <= 5d+76) then
        tmp = (0.5d0 * re) * (exp(-im) + exp(im))
    else
        tmp = 0.041666666666666664d0 * (sin(re) * (im ** 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.05) {
		tmp = (0.5 * Math.sin(re)) * (2.0 + (im * im));
	} else if (im <= 5e+76) {
		tmp = (0.5 * re) * (Math.exp(-im) + Math.exp(im));
	} else {
		tmp = 0.041666666666666664 * (Math.sin(re) * Math.pow(im, 4.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.05:
		tmp = (0.5 * math.sin(re)) * (2.0 + (im * im))
	elif im <= 5e+76:
		tmp = (0.5 * re) * (math.exp(-im) + math.exp(im))
	else:
		tmp = 0.041666666666666664 * (math.sin(re) * math.pow(im, 4.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.05)
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(2.0 + Float64(im * im)));
	elseif (im <= 5e+76)
		tmp = Float64(Float64(0.5 * re) * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64(0.041666666666666664 * Float64(sin(re) * (im ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.05)
		tmp = (0.5 * sin(re)) * (2.0 + (im * im));
	elseif (im <= 5e+76)
		tmp = (0.5 * re) * (exp(-im) + exp(im));
	else
		tmp = 0.041666666666666664 * (sin(re) * (im ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.05], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 5e+76], N[(N[(0.5 * re), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.041666666666666664 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.050000000000000003

   1. Initial program 99.5%

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

      1. sub0-neg 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in im around 0 80.8%

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

   6. Simplified 80.8%

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

   if 0.050000000000000003 < im < 4.99999999999999991e76

   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 70.6%

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

   6. Simplified 70.6%

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

   if 4.99999999999999991e76 < im

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in im around inf 100.0%

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

      1. *-commutative 100.0%

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

   9. Simplified 100.0%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.05:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 5 \cdot 10^{+76}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\\ \end{array} \]

Alternative 4: 86.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.045:\\ \;\;\;\;\sin re + \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.045)
   (+ (sin re) (* (* 0.5 (sin re)) (* im im)))
   (if (<= im 1.15e+77)
     (* (* 0.5 re) (+ (exp (- im)) (exp im)))
     (* 0.041666666666666664 (* (sin re) (pow im 4.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.045) {
		tmp = sin(re) + ((0.5 * sin(re)) * (im * im));
	} else if (im <= 1.15e+77) {
		tmp = (0.5 * re) * (exp(-im) + exp(im));
	} else {
		tmp = 0.041666666666666664 * (sin(re) * pow(im, 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.045d0) then
        tmp = sin(re) + ((0.5d0 * sin(re)) * (im * im))
    else if (im <= 1.15d+77) then
        tmp = (0.5d0 * re) * (exp(-im) + exp(im))
    else
        tmp = 0.041666666666666664d0 * (sin(re) * (im ** 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.045) {
		tmp = Math.sin(re) + ((0.5 * Math.sin(re)) * (im * im));
	} else if (im <= 1.15e+77) {
		tmp = (0.5 * re) * (Math.exp(-im) + Math.exp(im));
	} else {
		tmp = 0.041666666666666664 * (Math.sin(re) * Math.pow(im, 4.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.045:
		tmp = math.sin(re) + ((0.5 * math.sin(re)) * (im * im))
	elif im <= 1.15e+77:
		tmp = (0.5 * re) * (math.exp(-im) + math.exp(im))
	else:
		tmp = 0.041666666666666664 * (math.sin(re) * math.pow(im, 4.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.045)
		tmp = Float64(sin(re) + Float64(Float64(0.5 * sin(re)) * Float64(im * im)));
	elseif (im <= 1.15e+77)
		tmp = Float64(Float64(0.5 * re) * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64(0.041666666666666664 * Float64(sin(re) * (im ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.045)
		tmp = sin(re) + ((0.5 * sin(re)) * (im * im));
	elseif (im <= 1.15e+77)
		tmp = (0.5 * re) * (exp(-im) + exp(im));
	else
		tmp = 0.041666666666666664 * (sin(re) * (im ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.045], N[(N[Sin[re], $MachinePrecision] + N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.15e+77], N[(N[(0.5 * re), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.041666666666666664 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.044999999999999998

   1. Initial program 99.5%

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

      1. sub0-neg 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in im around 0 80.8%

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

   6. Simplified 80.8%

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

   if 0.044999999999999998 < 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 70.6%

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

   6. Simplified 70.6%

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

   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%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in im around inf 100.0%

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

      1. *-commutative 100.0%

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

   9. Simplified 100.0%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.045:\\ \;\;\;\;\sin re + \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\\ \end{array} \]

Alternative 5: 84.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 115000:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 3.3 \cdot 10^{+75}:\\ \;\;\;\;{\left(\sin re \cdot 1.9380669946781485 \cdot 10^{-10}\right)}^{-512}\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 115000.0)
   (* (* 0.5 (sin re)) (+ 2.0 (* im im)))
   (if (<= im 3.3e+75)
     (pow (* (sin re) 1.9380669946781485e-10) -512.0)
     (* 0.041666666666666664 (* (sin re) (pow im 4.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 115000.0) {
		tmp = (0.5 * sin(re)) * (2.0 + (im * im));
	} else if (im <= 3.3e+75) {
		tmp = pow((sin(re) * 1.9380669946781485e-10), -512.0);
	} else {
		tmp = 0.041666666666666664 * (sin(re) * pow(im, 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 115000.0d0) then
        tmp = (0.5d0 * sin(re)) * (2.0d0 + (im * im))
    else if (im <= 3.3d+75) then
        tmp = (sin(re) * 1.9380669946781485d-10) ** (-512.0d0)
    else
        tmp = 0.041666666666666664d0 * (sin(re) * (im ** 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 115000.0) {
		tmp = (0.5 * Math.sin(re)) * (2.0 + (im * im));
	} else if (im <= 3.3e+75) {
		tmp = Math.pow((Math.sin(re) * 1.9380669946781485e-10), -512.0);
	} else {
		tmp = 0.041666666666666664 * (Math.sin(re) * Math.pow(im, 4.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 115000.0:
		tmp = (0.5 * math.sin(re)) * (2.0 + (im * im))
	elif im <= 3.3e+75:
		tmp = math.pow((math.sin(re) * 1.9380669946781485e-10), -512.0)
	else:
		tmp = 0.041666666666666664 * (math.sin(re) * math.pow(im, 4.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 115000.0)
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(2.0 + Float64(im * im)));
	elseif (im <= 3.3e+75)
		tmp = Float64(sin(re) * 1.9380669946781485e-10) ^ -512.0;
	else
		tmp = Float64(0.041666666666666664 * Float64(sin(re) * (im ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 115000.0)
		tmp = (0.5 * sin(re)) * (2.0 + (im * im));
	elseif (im <= 3.3e+75)
		tmp = (sin(re) * 1.9380669946781485e-10) ^ -512.0;
	else
		tmp = 0.041666666666666664 * (sin(re) * (im ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 115000.0], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 3.3e+75], N[Power[N[(N[Sin[re], $MachinePrecision] * 1.9380669946781485e-10), $MachinePrecision], -512.0], $MachinePrecision], N[(0.041666666666666664 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 115000

   1. Initial program 99.5%

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

      1. sub0-neg 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in im around 0 80.4%

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

   6. Simplified 80.4%

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

   if 115000 < im < 3.29999999999999998e75

   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 5.1%

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

   6. Simplified 5.1%

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

   8. Applied egg-rr 43.8%

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

   if 3.29999999999999998e75 < im

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in im around inf 100.0%

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

      1. *-commutative 100.0%

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

   9. Simplified 100.0%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 115000:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 3.3 \cdot 10^{+75}:\\ \;\;\;\;{\left(\sin re \cdot 1.9380669946781485 \cdot 10^{-10}\right)}^{-512}\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\\ \end{array} \]

Alternative 6: 82.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{if}\;im \leq 115000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 1.12 \cdot 10^{+70}:\\ \;\;\;\;{\left(\sin re \cdot 1.9380669946781485 \cdot 10^{-10}\right)}^{-512}\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \frac{re \cdot \left(4 - {im}^{4}\right)}{2 - im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (+ 2.0 (* im im)))))
   (if (<= im 115000.0)
     t_0
     (if (<= im 1.12e+70)
       (pow (* (sin re) 1.9380669946781485e-10) -512.0)
       (if (<= im 1.35e+154)
         (* 0.5 (/ (* re (- 4.0 (pow im 4.0))) (- 2.0 (* im im))))
         t_0)))))

C

double code(double re, double im) {
	double t_0 = (0.5 * sin(re)) * (2.0 + (im * im));
	double tmp;
	if (im <= 115000.0) {
		tmp = t_0;
	} else if (im <= 1.12e+70) {
		tmp = pow((sin(re) * 1.9380669946781485e-10), -512.0);
	} else if (im <= 1.35e+154) {
		tmp = 0.5 * ((re * (4.0 - pow(im, 4.0))) / (2.0 - (im * im)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (0.5d0 * sin(re)) * (2.0d0 + (im * im))
    if (im <= 115000.0d0) then
        tmp = t_0
    else if (im <= 1.12d+70) then
        tmp = (sin(re) * 1.9380669946781485d-10) ** (-512.0d0)
    else if (im <= 1.35d+154) then
        tmp = 0.5d0 * ((re * (4.0d0 - (im ** 4.0d0))) / (2.0d0 - (im * im)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (0.5 * Math.sin(re)) * (2.0 + (im * im));
	double tmp;
	if (im <= 115000.0) {
		tmp = t_0;
	} else if (im <= 1.12e+70) {
		tmp = Math.pow((Math.sin(re) * 1.9380669946781485e-10), -512.0);
	} else if (im <= 1.35e+154) {
		tmp = 0.5 * ((re * (4.0 - Math.pow(im, 4.0))) / (2.0 - (im * im)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (0.5 * math.sin(re)) * (2.0 + (im * im))
	tmp = 0
	if im <= 115000.0:
		tmp = t_0
	elif im <= 1.12e+70:
		tmp = math.pow((math.sin(re) * 1.9380669946781485e-10), -512.0)
	elif im <= 1.35e+154:
		tmp = 0.5 * ((re * (4.0 - math.pow(im, 4.0))) / (2.0 - (im * im)))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(2.0 + Float64(im * im)))
	tmp = 0.0
	if (im <= 115000.0)
		tmp = t_0;
	elseif (im <= 1.12e+70)
		tmp = Float64(sin(re) * 1.9380669946781485e-10) ^ -512.0;
	elseif (im <= 1.35e+154)
		tmp = Float64(0.5 * Float64(Float64(re * Float64(4.0 - (im ^ 4.0))) / Float64(2.0 - Float64(im * im))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (0.5 * sin(re)) * (2.0 + (im * im));
	tmp = 0.0;
	if (im <= 115000.0)
		tmp = t_0;
	elseif (im <= 1.12e+70)
		tmp = (sin(re) * 1.9380669946781485e-10) ^ -512.0;
	elseif (im <= 1.35e+154)
		tmp = 0.5 * ((re * (4.0 - (im ^ 4.0))) / (2.0 - (im * im)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 115000.0], t$95$0, If[LessEqual[im, 1.12e+70], N[Power[N[(N[Sin[re], $MachinePrecision] * 1.9380669946781485e-10), $MachinePrecision], -512.0], $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(0.5 * N[(N[(re * N[(4.0 - N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 - N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < 115000 or 1.35000000000000003e154 < im

   1. Initial program 99.6%

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

      1. sub0-neg 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in im around 0 83.0%

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

   6. Simplified 83.0%

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

   if 115000 < im < 1.11999999999999993e70

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 4.4%

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

   6. Simplified 4.4%

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

   8. Applied egg-rr 42.9%

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

   if 1.11999999999999993e70 < im < 1.35000000000000003e154

   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 5.2%

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

   6. Simplified 5.2%

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

   7. Taylor expanded in re around 0 28.9%

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

      1. *-commutative 28.9%

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

      2. unpow2 28.9%

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

   9. Simplified 28.9%

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

       1. *-commutative 28.9%

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

       2. flip-+ 89.5%

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

       3. associate-*l/ 89.5%

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

       4. metadata-eval 89.5%

          \[\leadsto 0.5 \cdot \frac{\left(\color{blue}{4} - \left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot re}{2 - im \cdot im} \]

       5. pow2 89.5%

          \[\leadsto 0.5 \cdot \frac{\left(4 - \color{blue}{{im}^{2}} \cdot \left(im \cdot im\right)\right) \cdot re}{2 - im \cdot im} \]

       6. pow2 89.5%

          \[\leadsto 0.5 \cdot \frac{\left(4 - {im}^{2} \cdot \color{blue}{{im}^{2}}\right) \cdot re}{2 - im \cdot im} \]

       7. pow-prod-up 89.5%

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

       8. metadata-eval 89.5%

          \[\leadsto 0.5 \cdot \frac{\left(4 - {im}^{\color{blue}{4}}\right) \cdot re}{2 - im \cdot im} \]

   11. Applied egg-rr 89.5%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 115000:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.12 \cdot 10^{+70}:\\ \;\;\;\;{\left(\sin re \cdot 1.9380669946781485 \cdot 10^{-10}\right)}^{-512}\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \frac{re \cdot \left(4 - {im}^{4}\right)}{2 - im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \end{array} \]

Alternative 7: 81.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{if}\;im \leq 115000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 8 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(re\right)\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \frac{re \cdot \left(4 - {im}^{4}\right)}{2 - im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (+ 2.0 (* im im)))))
   (if (<= im 115000.0)
     t_0
     (if (<= im 8e+41)
       (log1p (expm1 re))
       (if (<= im 1.35e+154)
         (* 0.5 (/ (* re (- 4.0 (pow im 4.0))) (- 2.0 (* im im))))
         t_0)))))

C

double code(double re, double im) {
	double t_0 = (0.5 * sin(re)) * (2.0 + (im * im));
	double tmp;
	if (im <= 115000.0) {
		tmp = t_0;
	} else if (im <= 8e+41) {
		tmp = log1p(expm1(re));
	} else if (im <= 1.35e+154) {
		tmp = 0.5 * ((re * (4.0 - pow(im, 4.0))) / (2.0 - (im * im)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = (0.5 * Math.sin(re)) * (2.0 + (im * im));
	double tmp;
	if (im <= 115000.0) {
		tmp = t_0;
	} else if (im <= 8e+41) {
		tmp = Math.log1p(Math.expm1(re));
	} else if (im <= 1.35e+154) {
		tmp = 0.5 * ((re * (4.0 - Math.pow(im, 4.0))) / (2.0 - (im * im)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (0.5 * math.sin(re)) * (2.0 + (im * im))
	tmp = 0
	if im <= 115000.0:
		tmp = t_0
	elif im <= 8e+41:
		tmp = math.log1p(math.expm1(re))
	elif im <= 1.35e+154:
		tmp = 0.5 * ((re * (4.0 - math.pow(im, 4.0))) / (2.0 - (im * im)))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(2.0 + Float64(im * im)))
	tmp = 0.0
	if (im <= 115000.0)
		tmp = t_0;
	elseif (im <= 8e+41)
		tmp = log1p(expm1(re));
	elseif (im <= 1.35e+154)
		tmp = Float64(0.5 * Float64(Float64(re * Float64(4.0 - (im ^ 4.0))) / Float64(2.0 - Float64(im * im))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 115000.0], t$95$0, If[LessEqual[im, 8e+41], N[Log[1 + N[(Exp[re] - 1), $MachinePrecision]], $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(0.5 * N[(N[(re * N[(4.0 - N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 - N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < 115000 or 1.35000000000000003e154 < im

   1. Initial program 99.6%

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

      1. sub0-neg 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in im around 0 83.0%

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

   6. Simplified 83.0%

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

   if 115000 < im < 8.00000000000000005e41

   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 60.0%

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

   6. Simplified 60.0%

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

   8. Applied egg-rr 60.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(re\right)\right)} \]

   if 8.00000000000000005e41 < im < 1.35000000000000003e154

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 4.5%

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

   6. Simplified 4.5%

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

   7. Taylor expanded in re around 0 27.4%

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

      1. *-commutative 27.4%

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

      2. unpow2 27.4%

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

   9. Simplified 27.4%

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

       1. *-commutative 27.4%

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

       2. flip-+ 68.5%

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

       3. associate-*l/ 71.9%

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

       4. metadata-eval 71.9%

          \[\leadsto 0.5 \cdot \frac{\left(\color{blue}{4} - \left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot re}{2 - im \cdot im} \]

       5. pow2 71.9%

          \[\leadsto 0.5 \cdot \frac{\left(4 - \color{blue}{{im}^{2}} \cdot \left(im \cdot im\right)\right) \cdot re}{2 - im \cdot im} \]

       6. pow2 71.9%

          \[\leadsto 0.5 \cdot \frac{\left(4 - {im}^{2} \cdot \color{blue}{{im}^{2}}\right) \cdot re}{2 - im \cdot im} \]

       7. pow-prod-up 71.9%

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

       8. metadata-eval 71.9%

          \[\leadsto 0.5 \cdot \frac{\left(4 - {im}^{\color{blue}{4}}\right) \cdot re}{2 - im \cdot im} \]

   11. Applied egg-rr 71.9%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 115000:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 8 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(re\right)\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \frac{re \cdot \left(4 - {im}^{4}\right)}{2 - im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \end{array} \]

Alternative 8: 80.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 - {im}^{4}\\ t_1 := \left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ t_2 := 2 - im \cdot im\\ \mathbf{if}\;im \leq 115000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 6 \cdot 10^{+39}:\\ \;\;\;\;0.5 \cdot \frac{t_0 \cdot \left(re \cdot re\right)}{re \cdot t_2}\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \frac{re \cdot t_0}{t_2}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- 4.0 (pow im 4.0)))
        (t_1 (* (* 0.5 (sin re)) (+ 2.0 (* im im))))
        (t_2 (- 2.0 (* im im))))
   (if (<= im 115000.0)
     t_1
     (if (<= im 6e+39)
       (* 0.5 (/ (* t_0 (* re re)) (* re t_2)))
       (if (<= im 1.35e+154) (* 0.5 (/ (* re t_0) t_2)) t_1)))))

C

double code(double re, double im) {
	double t_0 = 4.0 - pow(im, 4.0);
	double t_1 = (0.5 * sin(re)) * (2.0 + (im * im));
	double t_2 = 2.0 - (im * im);
	double tmp;
	if (im <= 115000.0) {
		tmp = t_1;
	} else if (im <= 6e+39) {
		tmp = 0.5 * ((t_0 * (re * re)) / (re * t_2));
	} else if (im <= 1.35e+154) {
		tmp = 0.5 * ((re * t_0) / t_2);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 4.0d0 - (im ** 4.0d0)
    t_1 = (0.5d0 * sin(re)) * (2.0d0 + (im * im))
    t_2 = 2.0d0 - (im * im)
    if (im <= 115000.0d0) then
        tmp = t_1
    else if (im <= 6d+39) then
        tmp = 0.5d0 * ((t_0 * (re * re)) / (re * t_2))
    else if (im <= 1.35d+154) then
        tmp = 0.5d0 * ((re * t_0) / t_2)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 4.0 - Math.pow(im, 4.0);
	double t_1 = (0.5 * Math.sin(re)) * (2.0 + (im * im));
	double t_2 = 2.0 - (im * im);
	double tmp;
	if (im <= 115000.0) {
		tmp = t_1;
	} else if (im <= 6e+39) {
		tmp = 0.5 * ((t_0 * (re * re)) / (re * t_2));
	} else if (im <= 1.35e+154) {
		tmp = 0.5 * ((re * t_0) / t_2);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 4.0 - math.pow(im, 4.0)
	t_1 = (0.5 * math.sin(re)) * (2.0 + (im * im))
	t_2 = 2.0 - (im * im)
	tmp = 0
	if im <= 115000.0:
		tmp = t_1
	elif im <= 6e+39:
		tmp = 0.5 * ((t_0 * (re * re)) / (re * t_2))
	elif im <= 1.35e+154:
		tmp = 0.5 * ((re * t_0) / t_2)
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(4.0 - (im ^ 4.0))
	t_1 = Float64(Float64(0.5 * sin(re)) * Float64(2.0 + Float64(im * im)))
	t_2 = Float64(2.0 - Float64(im * im))
	tmp = 0.0
	if (im <= 115000.0)
		tmp = t_1;
	elseif (im <= 6e+39)
		tmp = Float64(0.5 * Float64(Float64(t_0 * Float64(re * re)) / Float64(re * t_2)));
	elseif (im <= 1.35e+154)
		tmp = Float64(0.5 * Float64(Float64(re * t_0) / t_2));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 4.0 - (im ^ 4.0);
	t_1 = (0.5 * sin(re)) * (2.0 + (im * im));
	t_2 = 2.0 - (im * im);
	tmp = 0.0;
	if (im <= 115000.0)
		tmp = t_1;
	elseif (im <= 6e+39)
		tmp = 0.5 * ((t_0 * (re * re)) / (re * t_2));
	elseif (im <= 1.35e+154)
		tmp = 0.5 * ((re * t_0) / t_2);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(4.0 - N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 - N[(im * im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 115000.0], t$95$1, If[LessEqual[im, 6e+39], N[(0.5 * N[(N[(t$95$0 * N[(re * re), $MachinePrecision]), $MachinePrecision] / N[(re * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(0.5 * N[(N[(re * t$95$0), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < 115000 or 1.35000000000000003e154 < im

   1. Initial program 99.6%

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

      1. sub0-neg 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in im around 0 83.0%

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

   6. Simplified 83.0%

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

   if 115000 < im < 5.9999999999999999e39

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 3.5%

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

   6. Simplified 3.5%

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

   7. Taylor expanded in re around 0 3.0%

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

      1. *-commutative 3.0%

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

      2. unpow2 3.0%

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

   9. Simplified 3.0%

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

       1. distribute-lft-in 3.0%

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

       2. flip-+ 40.8%

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

   11. Applied egg-rr 40.8%

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

       1. swap-sqr 40.8%

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

       2. metadata-eval 40.8%

          \[\leadsto 0.5 \cdot \frac{\left(re \cdot re\right) \cdot \color{blue}{4} - \left(re \cdot \left(im \cdot im\right)\right) \cdot \left(re \cdot \left(im \cdot im\right)\right)}{re \cdot 2 - re \cdot \left(im \cdot im\right)} \]

       3. swap-sqr 40.8%

          \[\leadsto 0.5 \cdot \frac{\left(re \cdot re\right) \cdot 4 - \color{blue}{\left(re \cdot re\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)}}{re \cdot 2 - re \cdot \left(im \cdot im\right)} \]

       4. associate-*r* 40.8%

          \[\leadsto 0.5 \cdot \frac{\left(re \cdot re\right) \cdot 4 - \left(re \cdot re\right) \cdot \color{blue}{\left(\left(\left(im \cdot im\right) \cdot im\right) \cdot im\right)}}{re \cdot 2 - re \cdot \left(im \cdot im\right)} \]

       5. unpow3 40.8%

          \[\leadsto 0.5 \cdot \frac{\left(re \cdot re\right) \cdot 4 - \left(re \cdot re\right) \cdot \left(\color{blue}{{im}^{3}} \cdot im\right)}{re \cdot 2 - re \cdot \left(im \cdot im\right)} \]

       6. pow-plus 40.8%

          \[\leadsto 0.5 \cdot \frac{\left(re \cdot re\right) \cdot 4 - \left(re \cdot re\right) \cdot \color{blue}{{im}^{\left(3 + 1\right)}}}{re \cdot 2 - re \cdot \left(im \cdot im\right)} \]

       7. metadata-eval 40.8%

          \[\leadsto 0.5 \cdot \frac{\left(re \cdot re\right) \cdot 4 - \left(re \cdot re\right) \cdot {im}^{\color{blue}{4}}}{re \cdot 2 - re \cdot \left(im \cdot im\right)} \]

       8. distribute-lft-out-- 40.8%

          \[\leadsto 0.5 \cdot \frac{\color{blue}{\left(re \cdot re\right) \cdot \left(4 - {im}^{4}\right)}}{re \cdot 2 - re \cdot \left(im \cdot im\right)} \]

       9. distribute-lft-out-- 40.8%

          \[\leadsto 0.5 \cdot \frac{\left(re \cdot re\right) \cdot \left(4 - {im}^{4}\right)}{\color{blue}{re \cdot \left(2 - im \cdot im\right)}} \]

   13. Simplified 40.8%

       \[\leadsto 0.5 \cdot \color{blue}{\frac{\left(re \cdot re\right) \cdot \left(4 - {im}^{4}\right)}{re \cdot \left(2 - im \cdot im\right)}} \]

   if 5.9999999999999999e39 < im < 1.35000000000000003e154

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 4.5%

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

   6. Simplified 4.5%

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

   7. Taylor expanded in re around 0 27.4%

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

      1. *-commutative 27.4%

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

      2. unpow2 27.4%

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

   9. Simplified 27.4%

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

       1. *-commutative 27.4%

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

       2. flip-+ 68.5%

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

       3. associate-*l/ 71.9%

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

       4. metadata-eval 71.9%

          \[\leadsto 0.5 \cdot \frac{\left(\color{blue}{4} - \left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot re}{2 - im \cdot im} \]

       5. pow2 71.9%

          \[\leadsto 0.5 \cdot \frac{\left(4 - \color{blue}{{im}^{2}} \cdot \left(im \cdot im\right)\right) \cdot re}{2 - im \cdot im} \]

       6. pow2 71.9%

          \[\leadsto 0.5 \cdot \frac{\left(4 - {im}^{2} \cdot \color{blue}{{im}^{2}}\right) \cdot re}{2 - im \cdot im} \]

       7. pow-prod-up 71.9%

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

       8. metadata-eval 71.9%

          \[\leadsto 0.5 \cdot \frac{\left(4 - {im}^{\color{blue}{4}}\right) \cdot re}{2 - im \cdot im} \]

   11. Applied egg-rr 71.9%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 115000:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 6 \cdot 10^{+39}:\\ \;\;\;\;0.5 \cdot \frac{\left(4 - {im}^{4}\right) \cdot \left(re \cdot re\right)}{re \cdot \left(2 - im \cdot im\right)}\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \frac{re \cdot \left(4 - {im}^{4}\right)}{2 - im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \end{array} \]

Alternative 9: 81.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{if}\;im \leq 115000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 1.5 \cdot 10^{+80}:\\ \;\;\;\;\frac{0.25}{re \cdot re} + \mathsf{fma}\left(re, re \cdot 0.016666666666666666, 0.08333333333333333\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \frac{re}{\frac{2 - im \cdot im}{4 - {im}^{4}}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (+ 2.0 (* im im)))))
   (if (<= im 115000.0)
     t_0
     (if (<= im 1.5e+80)
       (+
        (/ 0.25 (* re re))
        (fma re (* re 0.016666666666666666) 0.08333333333333333))
       (if (<= im 1.35e+154)
         (* 0.5 (/ re (/ (- 2.0 (* im im)) (- 4.0 (pow im 4.0)))))
         t_0)))))

C

double code(double re, double im) {
	double t_0 = (0.5 * sin(re)) * (2.0 + (im * im));
	double tmp;
	if (im <= 115000.0) {
		tmp = t_0;
	} else if (im <= 1.5e+80) {
		tmp = (0.25 / (re * re)) + fma(re, (re * 0.016666666666666666), 0.08333333333333333);
	} else if (im <= 1.35e+154) {
		tmp = 0.5 * (re / ((2.0 - (im * im)) / (4.0 - pow(im, 4.0))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(2.0 + Float64(im * im)))
	tmp = 0.0
	if (im <= 115000.0)
		tmp = t_0;
	elseif (im <= 1.5e+80)
		tmp = Float64(Float64(0.25 / Float64(re * re)) + fma(re, Float64(re * 0.016666666666666666), 0.08333333333333333));
	elseif (im <= 1.35e+154)
		tmp = Float64(0.5 * Float64(re / Float64(Float64(2.0 - Float64(im * im)) / Float64(4.0 - (im ^ 4.0)))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 115000.0], t$95$0, If[LessEqual[im, 1.5e+80], N[(N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision] + N[(re * N[(re * 0.016666666666666666), $MachinePrecision] + 0.08333333333333333), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(0.5 * N[(re / N[(N[(2.0 - N[(im * im), $MachinePrecision]), $MachinePrecision] / N[(4.0 - N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < 115000 or 1.35000000000000003e154 < im

   1. Initial program 99.6%

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

      1. sub0-neg 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in im around 0 83.0%

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

   6. Simplified 83.0%

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

   if 115000 < im < 1.49999999999999993e80

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 7.6%

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

   6. Taylor expanded in re around 0 20.2%

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

      1. +-commutative 20.2%

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

      2. associate-+l+ 20.2%

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

      3. associate-*r/ 20.2%

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

      4. metadata-eval 20.2%

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

      5. unpow2 20.2%

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

      6. *-commutative 20.2%

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

      7. unpow2 20.2%

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

      8. associate-*l* 20.2%

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

      9. fma-def 20.2%

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

   8. Simplified 20.2%

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

   if 1.49999999999999993e80 < im < 1.35000000000000003e154

   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 5.2%

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

   6. Simplified 5.2%

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

   7. Taylor expanded in re around 0 26.5%

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

      1. *-commutative 26.5%

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

      2. unpow2 26.5%

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

   9. Simplified 26.5%

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

       1. flip-+ 94.1%

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

       2. associate-*r/ 94.1%

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

       3. metadata-eval 94.1%

          \[\leadsto 0.5 \cdot \frac{re \cdot \left(\color{blue}{4} - \left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)}{2 - im \cdot im} \]

       4. pow2 94.1%

          \[\leadsto 0.5 \cdot \frac{re \cdot \left(4 - \color{blue}{{im}^{2}} \cdot \left(im \cdot im\right)\right)}{2 - im \cdot im} \]

       5. pow2 94.1%

          \[\leadsto 0.5 \cdot \frac{re \cdot \left(4 - {im}^{2} \cdot \color{blue}{{im}^{2}}\right)}{2 - im \cdot im} \]

       6. pow-prod-up 94.1%

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

       7. metadata-eval 94.1%

          \[\leadsto 0.5 \cdot \frac{re \cdot \left(4 - {im}^{\color{blue}{4}}\right)}{2 - im \cdot im} \]

   11. Applied egg-rr 94.1%

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

       1. associate-/l* 94.1%

          \[\leadsto 0.5 \cdot \color{blue}{\frac{re}{\frac{2 - im \cdot im}{4 - {im}^{4}}}} \]

   13. Simplified 94.1%

       \[\leadsto 0.5 \cdot \color{blue}{\frac{re}{\frac{2 - im \cdot im}{4 - {im}^{4}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 115000:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.5 \cdot 10^{+80}:\\ \;\;\;\;\frac{0.25}{re \cdot re} + \mathsf{fma}\left(re, re \cdot 0.016666666666666666, 0.08333333333333333\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \frac{re}{\frac{2 - im \cdot im}{4 - {im}^{4}}}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \end{array} \]

Alternative 10: 80.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3250000 \lor \neg \left(im \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{re \cdot \left(4 - {im}^{4}\right)}{2 - im \cdot im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 3250000.0) (not (<= im 1.35e+154)))
   (* (* 0.5 (sin re)) (+ 2.0 (* im im)))
   (* 0.5 (/ (* re (- 4.0 (pow im 4.0))) (- 2.0 (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 3250000.0) || !(im <= 1.35e+154)) {
		tmp = (0.5 * sin(re)) * (2.0 + (im * im));
	} else {
		tmp = 0.5 * ((re * (4.0 - pow(im, 4.0))) / (2.0 - (im * im)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= 3250000.0d0) .or. (.not. (im <= 1.35d+154))) then
        tmp = (0.5d0 * sin(re)) * (2.0d0 + (im * im))
    else
        tmp = 0.5d0 * ((re * (4.0d0 - (im ** 4.0d0))) / (2.0d0 - (im * im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= 3250000.0) || !(im <= 1.35e+154)) {
		tmp = (0.5 * Math.sin(re)) * (2.0 + (im * im));
	} else {
		tmp = 0.5 * ((re * (4.0 - Math.pow(im, 4.0))) / (2.0 - (im * im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= 3250000.0) or not (im <= 1.35e+154):
		tmp = (0.5 * math.sin(re)) * (2.0 + (im * im))
	else:
		tmp = 0.5 * ((re * (4.0 - math.pow(im, 4.0))) / (2.0 - (im * im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 3250000.0) || !(im <= 1.35e+154))
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(2.0 + Float64(im * im)));
	else
		tmp = Float64(0.5 * Float64(Float64(re * Float64(4.0 - (im ^ 4.0))) / Float64(2.0 - Float64(im * im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= 3250000.0) || ~((im <= 1.35e+154)))
		tmp = (0.5 * sin(re)) * (2.0 + (im * im));
	else
		tmp = 0.5 * ((re * (4.0 - (im ^ 4.0))) / (2.0 - (im * im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 3250000.0], N[Not[LessEqual[im, 1.35e+154]], $MachinePrecision]], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(re * N[(4.0 - N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 - N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 3.25e6 or 1.35000000000000003e154 < im

   1. Initial program 99.6%

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

      1. sub0-neg 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in im around 0 83.0%

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

   6. Simplified 83.0%

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

   if 3.25e6 < im < 1.35000000000000003e154

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 4.4%

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

   6. Simplified 4.4%

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

   7. Taylor expanded in re around 0 23.7%

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

      1. *-commutative 23.7%

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

      2. unpow2 23.7%

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

   9. Simplified 23.7%

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

       1. *-commutative 23.7%

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

       2. flip-+ 58.6%

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

       3. associate-*l/ 61.4%

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

       4. metadata-eval 61.4%

          \[\leadsto 0.5 \cdot \frac{\left(\color{blue}{4} - \left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot re}{2 - im \cdot im} \]

       5. pow2 61.4%

          \[\leadsto 0.5 \cdot \frac{\left(4 - \color{blue}{{im}^{2}} \cdot \left(im \cdot im\right)\right) \cdot re}{2 - im \cdot im} \]

       6. pow2 61.4%

          \[\leadsto 0.5 \cdot \frac{\left(4 - {im}^{2} \cdot \color{blue}{{im}^{2}}\right) \cdot re}{2 - im \cdot im} \]

       7. pow-prod-up 61.4%

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

       8. metadata-eval 61.4%

          \[\leadsto 0.5 \cdot \frac{\left(4 - {im}^{\color{blue}{4}}\right) \cdot re}{2 - im \cdot im} \]

   11. Applied egg-rr 61.4%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3250000 \lor \neg \left(im \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{re \cdot \left(4 - {im}^{4}\right)}{2 - im \cdot im}\\ \end{array} \]

Alternative 11: 78.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + im \cdot im\\ t_1 := \left(0.5 \cdot \sin re\right) \cdot t_0\\ \mathbf{if}\;im \leq 115000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 2.9 \cdot 10^{+80}:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(re \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ 2.0 (* im im))) (t_1 (* (* 0.5 (sin re)) t_0)))
   (if (<= im 115000.0)
     t_1
     (if (<= im 2.9e+80)
       (+
        0.08333333333333333
        (+ (/ 0.25 (* re re)) (* (* re re) 0.016666666666666666)))
       (if (<= im 1.35e+154) (* 0.5 (* re t_0)) t_1)))))

C

double code(double re, double im) {
	double t_0 = 2.0 + (im * im);
	double t_1 = (0.5 * sin(re)) * t_0;
	double tmp;
	if (im <= 115000.0) {
		tmp = t_1;
	} else if (im <= 2.9e+80) {
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	} else if (im <= 1.35e+154) {
		tmp = 0.5 * (re * t_0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 + (im * im)
    t_1 = (0.5d0 * sin(re)) * t_0
    if (im <= 115000.0d0) then
        tmp = t_1
    else if (im <= 2.9d+80) then
        tmp = 0.08333333333333333d0 + ((0.25d0 / (re * re)) + ((re * re) * 0.016666666666666666d0))
    else if (im <= 1.35d+154) then
        tmp = 0.5d0 * (re * t_0)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 2.0 + (im * im);
	double t_1 = (0.5 * Math.sin(re)) * t_0;
	double tmp;
	if (im <= 115000.0) {
		tmp = t_1;
	} else if (im <= 2.9e+80) {
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	} else if (im <= 1.35e+154) {
		tmp = 0.5 * (re * t_0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 2.0 + (im * im)
	t_1 = (0.5 * math.sin(re)) * t_0
	tmp = 0
	if im <= 115000.0:
		tmp = t_1
	elif im <= 2.9e+80:
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666))
	elif im <= 1.35e+154:
		tmp = 0.5 * (re * t_0)
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(2.0 + Float64(im * im))
	t_1 = Float64(Float64(0.5 * sin(re)) * t_0)
	tmp = 0.0
	if (im <= 115000.0)
		tmp = t_1;
	elseif (im <= 2.9e+80)
		tmp = Float64(0.08333333333333333 + Float64(Float64(0.25 / Float64(re * re)) + Float64(Float64(re * re) * 0.016666666666666666)));
	elseif (im <= 1.35e+154)
		tmp = Float64(0.5 * Float64(re * t_0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 2.0 + (im * im);
	t_1 = (0.5 * sin(re)) * t_0;
	tmp = 0.0;
	if (im <= 115000.0)
		tmp = t_1;
	elseif (im <= 2.9e+80)
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	elseif (im <= 1.35e+154)
		tmp = 0.5 * (re * t_0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[im, 115000.0], t$95$1, If[LessEqual[im, 2.9e+80], N[(0.08333333333333333 + N[(N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision] + N[(N[(re * re), $MachinePrecision] * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(0.5 * N[(re * t$95$0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if im < 115000 or 1.35000000000000003e154 < im

   1. Initial program 99.6%

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

      1. sub0-neg 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in im around 0 83.0%

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

   6. Simplified 83.0%

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

   if 115000 < im < 2.89999999999999986e80

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 7.6%

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

   6. Taylor expanded in re around 0 20.2%

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

      1. associate-*r/ 20.2%

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

      2. metadata-eval 20.2%

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

      3. unpow2 20.2%

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

      4. *-commutative 20.2%

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

      5. unpow2 20.2%

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

   8. Simplified 20.2%

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

   if 2.89999999999999986e80 < im < 1.35000000000000003e154

   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 5.2%

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

   6. Simplified 5.2%

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

   7. Taylor expanded in re around 0 26.5%

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

      1. *-commutative 26.5%

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

      2. unpow2 26.5%

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

   9. Simplified 26.5%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 115000:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 2.9 \cdot 10^{+80}:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(2 + im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \end{array} \]

Alternative 12: 78.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + im \cdot im\\ t_1 := \left(0.5 \cdot \sin re\right) \cdot t_0\\ \mathbf{if}\;im \leq 115000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+80}:\\ \;\;\;\;\frac{0.25}{re \cdot re} + \mathsf{fma}\left(re, re \cdot 0.016666666666666666, 0.08333333333333333\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(re \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ 2.0 (* im im))) (t_1 (* (* 0.5 (sin re)) t_0)))
   (if (<= im 115000.0)
     t_1
     (if (<= im 1.9e+80)
       (+
        (/ 0.25 (* re re))
        (fma re (* re 0.016666666666666666) 0.08333333333333333))
       (if (<= im 1.35e+154) (* 0.5 (* re t_0)) t_1)))))

C

double code(double re, double im) {
	double t_0 = 2.0 + (im * im);
	double t_1 = (0.5 * sin(re)) * t_0;
	double tmp;
	if (im <= 115000.0) {
		tmp = t_1;
	} else if (im <= 1.9e+80) {
		tmp = (0.25 / (re * re)) + fma(re, (re * 0.016666666666666666), 0.08333333333333333);
	} else if (im <= 1.35e+154) {
		tmp = 0.5 * (re * t_0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(2.0 + Float64(im * im))
	t_1 = Float64(Float64(0.5 * sin(re)) * t_0)
	tmp = 0.0
	if (im <= 115000.0)
		tmp = t_1;
	elseif (im <= 1.9e+80)
		tmp = Float64(Float64(0.25 / Float64(re * re)) + fma(re, Float64(re * 0.016666666666666666), 0.08333333333333333));
	elseif (im <= 1.35e+154)
		tmp = Float64(0.5 * Float64(re * t_0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[im, 115000.0], t$95$1, If[LessEqual[im, 1.9e+80], N[(N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision] + N[(re * N[(re * 0.016666666666666666), $MachinePrecision] + 0.08333333333333333), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(0.5 * N[(re * t$95$0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if im < 115000 or 1.35000000000000003e154 < im

   1. Initial program 99.6%

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

      1. sub0-neg 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in im around 0 83.0%

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

   6. Simplified 83.0%

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

   if 115000 < im < 1.89999999999999999e80

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 7.6%

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

   6. Taylor expanded in re around 0 20.2%

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

      1. +-commutative 20.2%

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

      2. associate-+l+ 20.2%

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

      3. associate-*r/ 20.2%

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

      4. metadata-eval 20.2%

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

      5. unpow2 20.2%

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

      6. *-commutative 20.2%

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

      7. unpow2 20.2%

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

      8. associate-*l* 20.2%

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

      9. fma-def 20.2%

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

   8. Simplified 20.2%

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

   if 1.89999999999999999e80 < im < 1.35000000000000003e154

   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 5.2%

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

   6. Simplified 5.2%

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

   7. Taylor expanded in re around 0 26.5%

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

      1. *-commutative 26.5%

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

      2. unpow2 26.5%

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

   9. Simplified 26.5%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 115000:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+80}:\\ \;\;\;\;\frac{0.25}{re \cdot re} + \mathsf{fma}\left(re, re \cdot 0.016666666666666666, 0.08333333333333333\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(2 + im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \end{array} \]

Alternative 13: 63.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 120000:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2.9 \cdot 10^{+80}:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)\\ \mathbf{elif}\;im \leq 3.8 \cdot 10^{+215}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(2 + im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(\sin re \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 120000.0)
   (sin re)
   (if (<= im 2.9e+80)
     (+
      0.08333333333333333
      (+ (/ 0.25 (* re re)) (* (* re re) 0.016666666666666666)))
     (if (<= im 3.8e+215)
       (* 0.5 (* re (+ 2.0 (* im im))))
       (* im (* 0.5 (* (sin re) im)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 120000.0) {
		tmp = sin(re);
	} else if (im <= 2.9e+80) {
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	} else if (im <= 3.8e+215) {
		tmp = 0.5 * (re * (2.0 + (im * im)));
	} else {
		tmp = im * (0.5 * (sin(re) * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 120000.0d0) then
        tmp = sin(re)
    else if (im <= 2.9d+80) then
        tmp = 0.08333333333333333d0 + ((0.25d0 / (re * re)) + ((re * re) * 0.016666666666666666d0))
    else if (im <= 3.8d+215) then
        tmp = 0.5d0 * (re * (2.0d0 + (im * im)))
    else
        tmp = im * (0.5d0 * (sin(re) * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 120000.0) {
		tmp = Math.sin(re);
	} else if (im <= 2.9e+80) {
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	} else if (im <= 3.8e+215) {
		tmp = 0.5 * (re * (2.0 + (im * im)));
	} else {
		tmp = im * (0.5 * (Math.sin(re) * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 120000.0:
		tmp = math.sin(re)
	elif im <= 2.9e+80:
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666))
	elif im <= 3.8e+215:
		tmp = 0.5 * (re * (2.0 + (im * im)))
	else:
		tmp = im * (0.5 * (math.sin(re) * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 120000.0)
		tmp = sin(re);
	elseif (im <= 2.9e+80)
		tmp = Float64(0.08333333333333333 + Float64(Float64(0.25 / Float64(re * re)) + Float64(Float64(re * re) * 0.016666666666666666)));
	elseif (im <= 3.8e+215)
		tmp = Float64(0.5 * Float64(re * Float64(2.0 + Float64(im * im))));
	else
		tmp = Float64(im * Float64(0.5 * Float64(sin(re) * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 120000.0)
		tmp = sin(re);
	elseif (im <= 2.9e+80)
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	elseif (im <= 3.8e+215)
		tmp = 0.5 * (re * (2.0 + (im * im)));
	else
		tmp = im * (0.5 * (sin(re) * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 120000.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 2.9e+80], N[(0.08333333333333333 + N[(N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision] + N[(N[(re * re), $MachinePrecision] * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 3.8e+215], N[(0.5 * N[(re * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(0.5 * N[(N[Sin[re], $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < 1.2e5

   1. Initial program 99.5%

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

      1. sub0-neg 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in im around 0 62.2%

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

   if 1.2e5 < im < 2.89999999999999986e80

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 7.6%

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

   6. Taylor expanded in re around 0 20.2%

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

      1. associate-*r/ 20.2%

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

      2. metadata-eval 20.2%

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

      3. unpow2 20.2%

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

      4. *-commutative 20.2%

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

      5. unpow2 20.2%

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

   8. Simplified 20.2%

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

   if 2.89999999999999986e80 < im < 3.79999999999999968e215

   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 44.5%

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

   6. Simplified 44.5%

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

   7. Taylor expanded in re around 0 53.4%

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

      1. *-commutative 53.4%

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

      2. unpow2 53.4%

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

   9. Simplified 53.4%

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

   if 3.79999999999999968e215 < im

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in im around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. unpow2 100.0%

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

      3. associate-*r* 94.5%

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

      4. *-commutative 94.5%

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

      5. associate-*l* 94.5%

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

      6. *-commutative 94.5%

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

   9. Simplified 94.5%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 120000:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2.9 \cdot 10^{+80}:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)\\ \mathbf{elif}\;im \leq 3.8 \cdot 10^{+215}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(2 + im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(\sin re \cdot im\right)\right)\\ \end{array} \]

Alternative 14: 62.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 115000:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2.9 \cdot 10^{+80}:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(2 + im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 115000.0)
   (sin re)
   (if (<= im 2.9e+80)
     (+
      0.08333333333333333
      (+ (/ 0.25 (* re re)) (* (* re re) 0.016666666666666666)))
     (* 0.5 (* re (+ 2.0 (* im im)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 115000.0) {
		tmp = sin(re);
	} else if (im <= 2.9e+80) {
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	} else {
		tmp = 0.5 * (re * (2.0 + (im * im)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 115000.0d0) then
        tmp = sin(re)
    else if (im <= 2.9d+80) then
        tmp = 0.08333333333333333d0 + ((0.25d0 / (re * re)) + ((re * re) * 0.016666666666666666d0))
    else
        tmp = 0.5d0 * (re * (2.0d0 + (im * im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 115000.0) {
		tmp = Math.sin(re);
	} else if (im <= 2.9e+80) {
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	} else {
		tmp = 0.5 * (re * (2.0 + (im * im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 115000.0:
		tmp = math.sin(re)
	elif im <= 2.9e+80:
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666))
	else:
		tmp = 0.5 * (re * (2.0 + (im * im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 115000.0)
		tmp = sin(re);
	elseif (im <= 2.9e+80)
		tmp = Float64(0.08333333333333333 + Float64(Float64(0.25 / Float64(re * re)) + Float64(Float64(re * re) * 0.016666666666666666)));
	else
		tmp = Float64(0.5 * Float64(re * Float64(2.0 + Float64(im * im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 115000.0)
		tmp = sin(re);
	elseif (im <= 2.9e+80)
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	else
		tmp = 0.5 * (re * (2.0 + (im * im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 115000.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 2.9e+80], N[(0.08333333333333333 + N[(N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision] + N[(N[(re * re), $MachinePrecision] * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(re * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 115000

   1. Initial program 99.5%

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

      1. sub0-neg 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in im around 0 62.2%

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

   if 115000 < im < 2.89999999999999986e80

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 7.6%

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

   6. Taylor expanded in re around 0 20.2%

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

      1. associate-*r/ 20.2%

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

      2. metadata-eval 20.2%

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

      3. unpow2 20.2%

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

      4. *-commutative 20.2%

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

      5. unpow2 20.2%

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

   8. Simplified 20.2%

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

   if 2.89999999999999986e80 < 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 65.0%

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

   6. Simplified 65.0%

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

   7. Taylor expanded in re around 0 57.6%

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

      1. *-commutative 57.6%

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

      2. unpow2 57.6%

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

   9. Simplified 57.6%

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

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 115000:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2.9 \cdot 10^{+80}:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(2 + im \cdot im\right)\right)\\ \end{array} \]

Alternative 15: 34.0% accurate, 34.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.42:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(re \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.42) re (* 0.5 (* im (* re im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.42) {
		tmp = re;
	} else {
		tmp = 0.5 * (im * (re * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.42d0) then
        tmp = re
    else
        tmp = 0.5d0 * (im * (re * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.42) {
		tmp = re;
	} else {
		tmp = 0.5 * (im * (re * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.42:
		tmp = re
	else:
		tmp = 0.5 * (im * (re * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.42)
		tmp = re;
	else
		tmp = Float64(0.5 * Float64(im * Float64(re * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.42)
		tmp = re;
	else
		tmp = 0.5 * (im * (re * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.42], re, N[(0.5 * N[(im * N[(re * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.4199999999999999

   1. Initial program 99.5%

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

      1. sub0-neg 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in re around 0 66.5%

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

   6. Simplified 66.5%

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

   7. Taylor expanded in im around 0 37.5%

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

   if 1.4199999999999999 < 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 48.4%

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

   6. Simplified 48.4%

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

   7. Taylor expanded in re around 0 47.4%

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

      1. *-commutative 47.4%

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

      2. unpow2 47.4%

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

   9. Simplified 47.4%

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

   10. Taylor expanded in im around inf 47.4%

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

       1. unpow2 47.4%

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

       2. *-commutative 47.4%

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

       3. associate-*l* 35.4%

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

   12. Simplified 35.4%

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

4. Final simplification 37.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.42:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(re \cdot im\right)\right)\\ \end{array} \]

Alternative 16: 36.8% accurate, 34.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.42:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.42) re (* 0.5 (* re (* im im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.42) {
		tmp = re;
	} else {
		tmp = 0.5 * (re * (im * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.42d0) then
        tmp = re
    else
        tmp = 0.5d0 * (re * (im * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.42) {
		tmp = re;
	} else {
		tmp = 0.5 * (re * (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.42:
		tmp = re
	else:
		tmp = 0.5 * (re * (im * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.42)
		tmp = re;
	else
		tmp = Float64(0.5 * Float64(re * Float64(im * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.42)
		tmp = re;
	else
		tmp = 0.5 * (re * (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.42], re, N[(0.5 * N[(re * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.4199999999999999

   1. Initial program 99.5%

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

      1. sub0-neg 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in re around 0 66.5%

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

   6. Simplified 66.5%

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

   7. Taylor expanded in im around 0 37.5%

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

   if 1.4199999999999999 < 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 48.4%

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

   6. Simplified 48.4%

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

   7. Taylor expanded in re around 0 47.4%

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

      1. *-commutative 47.4%

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

      2. unpow2 47.4%

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

   9. Simplified 47.4%

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

   10. Taylor expanded in im around inf 47.4%

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

       1. unpow2 47.4%

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

   12. Simplified 47.4%

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

4. Final simplification 39.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.42:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 17: 47.1% accurate, 34.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(0.5 * N[(re * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. sub0-neg 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in im around 0 72.8%

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

6. Simplified 72.8%

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

7. Taylor expanded in re around 0 53.1%

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

   1. *-commutative 53.1%

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

   2. unpow2 53.1%

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

9. Simplified 53.1%

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

10. Final simplification 53.1%

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

Alternative 18: 29.0% accurate, 43.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3700000:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{re \cdot re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3700000.0) re (/ 0.25 (* re re))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3700000.0) {
		tmp = re;
	} else {
		tmp = 0.25 / (re * re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 3700000.0d0) then
        tmp = re
    else
        tmp = 0.25d0 / (re * re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3700000.0) {
		tmp = re;
	} else {
		tmp = 0.25 / (re * re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3700000.0:
		tmp = re
	else:
		tmp = 0.25 / (re * re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3700000.0)
		tmp = re;
	else
		tmp = Float64(0.25 / Float64(re * re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3700000.0)
		tmp = re;
	else
		tmp = 0.25 / (re * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3700000.0], re, N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 3.7e6

   1. Initial program 99.5%

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

      1. sub0-neg 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in re around 0 66.1%

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

   6. Simplified 66.1%

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

   7. Taylor expanded in im around 0 37.3%

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

   if 3.7e6 < im

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 8.1%

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

   6. Taylor expanded in re around 0 8.0%

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

      1. unpow2 8.0%

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

   8. Simplified 8.0%

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

4. Final simplification 30.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3700000:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{re \cdot re}\\ \end{array} \]

Alternative 19: 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 99.6%

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

   1. sub0-neg 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in im around 0 87.0%

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

6. Simplified 87.0%

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

8. Applied egg-rr 3.4%

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

   1. +-inverses 3.4%

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

   2. +-rgt-identity 3.4%

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

   3. associate-*l/ 3.4%

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

   4. *-inverses 3.4%

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

   5. metadata-eval 3.4%

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

10. Simplified 3.4%

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

11. Final simplification 3.4%

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

Alternative 20: 4.8% accurate, 309.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return 1.0

Julia

function code(re, im)
	return 1.0
end

MATLAB

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

Wolfram

code[re_, im_] := 1.0

Derivation

1. Initial program 99.6%

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

   1. sub0-neg 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in im around 0 87.0%

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

6. Simplified 87.0%

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

8. Applied egg-rr 4.3%

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

   1. *-inverses 4.3%

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

10. Simplified 4.3%

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

11. Final simplification 4.3%

    \[\leadsto 1 \]

Alternative 21: 26.2% accurate, 309.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 re)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return re

Julia

function code(re, im)
	return re
end

MATLAB

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

Wolfram

code[re_, im_] := re

Derivation

1. Initial program 99.6%

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

   1. sub0-neg 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in re around 0 69.6%

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

6. Simplified 69.6%

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

7. Taylor expanded in im around 0 29.0%

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

8. Final simplification 29.0%

   \[\leadsto re \]

Reproduce

herbie shell --seed 2023258 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))