math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 9.7s

Alternatives: 25

Speedup: 1.0×

• 47.8% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. *-commutative 100.0%

      \[\leadsto \color{blue}{\left(\sin re \cdot 0.5\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

   2. associate-*l* 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

   3. sub0-neg 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 86.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 4.7 \cdot 10^{-5}:\\ \;\;\;\;\sin re \cdot \left(im \cdot \left(0.5 \cdot im\right) + 1\right)\\ \mathbf{elif}\;im \leq 2.55 \cdot 10^{+77}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 4.7e-5)
   (* (sin re) (+ (* im (* 0.5 im)) 1.0))
   (if (<= im 2.55e+77)
     (* (* re 0.5) (+ (exp (- im)) (exp im)))
     (*
      (sin re)
      (+ 1.0 (* (* im im) (+ 0.5 (* (* im im) 0.041666666666666664))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 4.7e-5) {
		tmp = sin(re) * ((im * (0.5 * im)) + 1.0);
	} else if (im <= 2.55e+77) {
		tmp = (re * 0.5) * (exp(-im) + exp(im));
	} else {
		tmp = sin(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 4.7d-5) then
        tmp = sin(re) * ((im * (0.5d0 * im)) + 1.0d0)
    else if (im <= 2.55d+77) then
        tmp = (re * 0.5d0) * (exp(-im) + exp(im))
    else
        tmp = sin(re) * (1.0d0 + ((im * im) * (0.5d0 + ((im * im) * 0.041666666666666664d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 4.7e-5) {
		tmp = Math.sin(re) * ((im * (0.5 * im)) + 1.0);
	} else if (im <= 2.55e+77) {
		tmp = (re * 0.5) * (Math.exp(-im) + Math.exp(im));
	} else {
		tmp = Math.sin(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 4.7e-5:
		tmp = math.sin(re) * ((im * (0.5 * im)) + 1.0)
	elif im <= 2.55e+77:
		tmp = (re * 0.5) * (math.exp(-im) + math.exp(im))
	else:
		tmp = math.sin(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 4.7e-5)
		tmp = Float64(sin(re) * Float64(Float64(im * Float64(0.5 * im)) + 1.0));
	elseif (im <= 2.55e+77)
		tmp = Float64(Float64(re * 0.5) * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64(sin(re) * Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(Float64(im * im) * 0.041666666666666664)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 4.7e-5)
		tmp = sin(re) * ((im * (0.5 * im)) + 1.0);
	elseif (im <= 2.55e+77)
		tmp = (re * 0.5) * (exp(-im) + exp(im));
	else
		tmp = sin(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 4.7e-5], N[(N[Sin[re], $MachinePrecision] * N[(N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.55e+77], N[(N[(re * 0.5), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 4.69999999999999972e-5

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\sin re \cdot 0.5\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. associate-*l* 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 78.7%

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

      1. *-commutative 78.7%

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

      2. associate-*r* 78.7%

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

      3. distribute-rgt1-in 78.7%

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

      4. *-commutative 78.7%

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

      5. unpow2 78.7%

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

      6. associate-*l* 78.7%

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

   6. Simplified 78.7%

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

   if 4.69999999999999972e-5 < im < 2.54999999999999985e77

   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. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\sin re \cdot 0.5\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. associate-*l* 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 84.6%

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

      1. associate-*r* 84.6%

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

   6. Simplified 84.6%

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

   if 2.54999999999999985e77 < 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. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\sin re \cdot 0.5\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. associate-*l* 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{\sin re + \left(0.5 \cdot \left(\sin re \cdot {im}^{2}\right) + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right)} \]
   5. Step-by-step derivation

      1. *-rgt-identity 100.0%

         \[\leadsto \color{blue}{\sin re \cdot 1} + \left(0.5 \cdot \left(\sin re \cdot {im}^{2}\right) + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right) \]

      2. *-commutative 100.0%

         \[\leadsto \sin re \cdot 1 + \left(0.5 \cdot \color{blue}{\left({im}^{2} \cdot \sin re\right)} + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right) \]

      3. associate-*r* 100.0%

         \[\leadsto \sin re \cdot 1 + \left(\color{blue}{\left(0.5 \cdot {im}^{2}\right) \cdot \sin re} + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right) \]

      4. *-commutative 100.0%

         \[\leadsto \sin re \cdot 1 + \left(\left(0.5 \cdot {im}^{2}\right) \cdot \sin re + 0.041666666666666664 \cdot \color{blue}{\left({im}^{4} \cdot \sin re\right)}\right) \]

      5. associate-*r* 100.0%

         \[\leadsto \sin re \cdot 1 + \left(\left(0.5 \cdot {im}^{2}\right) \cdot \sin re + \color{blue}{\left(0.041666666666666664 \cdot {im}^{4}\right) \cdot \sin re}\right) \]

      6. distribute-rgt-out 100.0%

         \[\leadsto \sin re \cdot 1 + \color{blue}{\sin re \cdot \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot {im}^{4}\right)} \]

      7. distribute-lft-out 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot {im}^{4}\right)\right)} \]

      8. metadata-eval 100.0%

         \[\leadsto \sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot {im}^{\color{blue}{\left(2 \cdot 2\right)}}\right)\right) \]

      9. pow-sqr 100.0%

         \[\leadsto \sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)}\right)\right) \]

      10. associate-*r* 100.0%

          \[\leadsto \sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + \color{blue}{\left(0.041666666666666664 \cdot {im}^{2}\right) \cdot {im}^{2}}\right)\right) \]

      11. distribute-rgt-out 100.0%

          \[\leadsto \sin re \cdot \left(1 + \color{blue}{{im}^{2} \cdot \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)}\right) \]

      12. unpow2 100.0%

          \[\leadsto \sin re \cdot \left(1 + \color{blue}{\left(im \cdot im\right)} \cdot \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)\right) \]

      13. unpow2 100.0%

          \[\leadsto \sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.7 \cdot 10^{-5}:\\ \;\;\;\;\sin re \cdot \left(im \cdot \left(0.5 \cdot im\right) + 1\right)\\ \mathbf{elif}\;im \leq 2.55 \cdot 10^{+77}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \end{array} \]

Alternative 3: 89.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 210000000000 \lor \neg \left(im \leq 2.55 \cdot 10^{+77}\right):\\ \;\;\;\;\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{re \cdot re} + \frac{{re}^{4} \cdot 0.0002777777777777778 - 0.006944444444444444}{re \cdot \left(re \cdot 0.016666666666666666\right) - 0.08333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 210000000000.0) (not (<= im 2.55e+77)))
   (*
    (sin re)
    (+ 1.0 (* (* im im) (+ 0.5 (* (* im im) 0.041666666666666664)))))
   (+
    (/ 0.25 (* re re))
    (/
     (- (* (pow re 4.0) 0.0002777777777777778) 0.006944444444444444)
     (- (* re (* re 0.016666666666666666)) 0.08333333333333333)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 210000000000.0) || !(im <= 2.55e+77)) {
		tmp = sin(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	} else {
		tmp = (0.25 / (re * re)) + (((pow(re, 4.0) * 0.0002777777777777778) - 0.006944444444444444) / ((re * (re * 0.016666666666666666)) - 0.08333333333333333));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= 210000000000.0d0) .or. (.not. (im <= 2.55d+77))) then
        tmp = sin(re) * (1.0d0 + ((im * im) * (0.5d0 + ((im * im) * 0.041666666666666664d0))))
    else
        tmp = (0.25d0 / (re * re)) + ((((re ** 4.0d0) * 0.0002777777777777778d0) - 0.006944444444444444d0) / ((re * (re * 0.016666666666666666d0)) - 0.08333333333333333d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= 210000000000.0) || !(im <= 2.55e+77)) {
		tmp = Math.sin(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	} else {
		tmp = (0.25 / (re * re)) + (((Math.pow(re, 4.0) * 0.0002777777777777778) - 0.006944444444444444) / ((re * (re * 0.016666666666666666)) - 0.08333333333333333));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= 210000000000.0) or not (im <= 2.55e+77):
		tmp = math.sin(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))))
	else:
		tmp = (0.25 / (re * re)) + (((math.pow(re, 4.0) * 0.0002777777777777778) - 0.006944444444444444) / ((re * (re * 0.016666666666666666)) - 0.08333333333333333))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 210000000000.0) || !(im <= 2.55e+77))
		tmp = Float64(sin(re) * Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(Float64(im * im) * 0.041666666666666664)))));
	else
		tmp = Float64(Float64(0.25 / Float64(re * re)) + Float64(Float64(Float64((re ^ 4.0) * 0.0002777777777777778) - 0.006944444444444444) / Float64(Float64(re * Float64(re * 0.016666666666666666)) - 0.08333333333333333)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= 210000000000.0) || ~((im <= 2.55e+77)))
		tmp = sin(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	else
		tmp = (0.25 / (re * re)) + ((((re ^ 4.0) * 0.0002777777777777778) - 0.006944444444444444) / ((re * (re * 0.016666666666666666)) - 0.08333333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 210000000000.0], N[Not[LessEqual[im, 2.55e+77]], $MachinePrecision]], N[(N[Sin[re], $MachinePrecision] * N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[Power[re, 4.0], $MachinePrecision] * 0.0002777777777777778), $MachinePrecision] - 0.006944444444444444), $MachinePrecision] / N[(N[(re * N[(re * 0.016666666666666666), $MachinePrecision]), $MachinePrecision] - 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2.1e11 or 2.54999999999999985e77 < 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. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\sin re \cdot 0.5\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. associate-*l* 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 91.5%

      \[\leadsto \color{blue}{\sin re + \left(0.5 \cdot \left(\sin re \cdot {im}^{2}\right) + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right)} \]
   5. Step-by-step derivation

      1. *-rgt-identity 91.5%

         \[\leadsto \color{blue}{\sin re \cdot 1} + \left(0.5 \cdot \left(\sin re \cdot {im}^{2}\right) + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right) \]

      2. *-commutative 91.5%

         \[\leadsto \sin re \cdot 1 + \left(0.5 \cdot \color{blue}{\left({im}^{2} \cdot \sin re\right)} + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right) \]

      3. associate-*r* 91.5%

         \[\leadsto \sin re \cdot 1 + \left(\color{blue}{\left(0.5 \cdot {im}^{2}\right) \cdot \sin re} + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right) \]

      4. *-commutative 91.5%

         \[\leadsto \sin re \cdot 1 + \left(\left(0.5 \cdot {im}^{2}\right) \cdot \sin re + 0.041666666666666664 \cdot \color{blue}{\left({im}^{4} \cdot \sin re\right)}\right) \]

      5. associate-*r* 91.5%

         \[\leadsto \sin re \cdot 1 + \left(\left(0.5 \cdot {im}^{2}\right) \cdot \sin re + \color{blue}{\left(0.041666666666666664 \cdot {im}^{4}\right) \cdot \sin re}\right) \]

      6. distribute-rgt-out 91.5%

         \[\leadsto \sin re \cdot 1 + \color{blue}{\sin re \cdot \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot {im}^{4}\right)} \]

      7. distribute-lft-out 91.5%

         \[\leadsto \color{blue}{\sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot {im}^{4}\right)\right)} \]

      8. metadata-eval 91.5%

         \[\leadsto \sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot {im}^{\color{blue}{\left(2 \cdot 2\right)}}\right)\right) \]

      9. pow-sqr 91.5%

         \[\leadsto \sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)}\right)\right) \]

      10. associate-*r* 91.5%

          \[\leadsto \sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + \color{blue}{\left(0.041666666666666664 \cdot {im}^{2}\right) \cdot {im}^{2}}\right)\right) \]

      11. distribute-rgt-out 91.5%

          \[\leadsto \sin re \cdot \left(1 + \color{blue}{{im}^{2} \cdot \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)}\right) \]

      12. unpow2 91.5%

          \[\leadsto \sin re \cdot \left(1 + \color{blue}{\left(im \cdot im\right)} \cdot \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)\right) \]

      13. unpow2 91.5%

          \[\leadsto \sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \]

   6. Simplified 91.5%

      \[\leadsto \color{blue}{\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)} \]

   if 2.1e11 < im < 2.54999999999999985e77

   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. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\sin re \cdot 0.5\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. associate-*l* 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 31.2%

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

   6. Taylor expanded in re around 0 40.8%

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

      1. +-commutative 40.8%

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

      2. associate-+l+ 40.8%

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

      3. associate-*r/ 40.8%

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

      4. metadata-eval 40.8%

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

      5. unpow2 40.8%

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

      6. *-commutative 40.8%

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

      7. unpow2 40.8%

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

      8. associate-*l* 40.8%

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

      9. fma-def 40.8%

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

   8. Simplified 40.8%

      \[\leadsto \color{blue}{\frac{0.25}{re \cdot re} + \mathsf{fma}\left(re, re \cdot 0.016666666666666666, 0.08333333333333333\right)} \]
   9. Step-by-step derivation

      1. fma-udef 40.8%

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

      2. associate-*l* 40.8%

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

      3. flip-+ 30.8%

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

      4. swap-sqr 30.8%

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

      5. pow2 30.8%

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

      6. metadata-eval 30.8%

         \[\leadsto \frac{0.25}{re \cdot re} + \frac{{\left(re \cdot re\right)}^{\color{blue}{\left(\sqrt{4}\right)}} \cdot \left(0.016666666666666666 \cdot 0.016666666666666666\right) - 0.08333333333333333 \cdot 0.08333333333333333}{\left(re \cdot re\right) \cdot 0.016666666666666666 - 0.08333333333333333} \]

      7. pow-prod-down 30.8%

         \[\leadsto \frac{0.25}{re \cdot re} + \frac{\color{blue}{\left({re}^{\left(\sqrt{4}\right)} \cdot {re}^{\left(\sqrt{4}\right)}\right)} \cdot \left(0.016666666666666666 \cdot 0.016666666666666666\right) - 0.08333333333333333 \cdot 0.08333333333333333}{\left(re \cdot re\right) \cdot 0.016666666666666666 - 0.08333333333333333} \]

      8. pow-prod-up 30.8%

         \[\leadsto \frac{0.25}{re \cdot re} + \frac{\color{blue}{{re}^{\left(\sqrt{4} + \sqrt{4}\right)}} \cdot \left(0.016666666666666666 \cdot 0.016666666666666666\right) - 0.08333333333333333 \cdot 0.08333333333333333}{\left(re \cdot re\right) \cdot 0.016666666666666666 - 0.08333333333333333} \]

      9. metadata-eval 30.8%

         \[\leadsto \frac{0.25}{re \cdot re} + \frac{{re}^{\left(\color{blue}{2} + \sqrt{4}\right)} \cdot \left(0.016666666666666666 \cdot 0.016666666666666666\right) - 0.08333333333333333 \cdot 0.08333333333333333}{\left(re \cdot re\right) \cdot 0.016666666666666666 - 0.08333333333333333} \]

      10. metadata-eval 30.8%

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

      11. metadata-eval 30.8%

          \[\leadsto \frac{0.25}{re \cdot re} + \frac{{re}^{\color{blue}{4}} \cdot \left(0.016666666666666666 \cdot 0.016666666666666666\right) - 0.08333333333333333 \cdot 0.08333333333333333}{\left(re \cdot re\right) \cdot 0.016666666666666666 - 0.08333333333333333} \]

      12. metadata-eval 30.8%

          \[\leadsto \frac{0.25}{re \cdot re} + \frac{{re}^{4} \cdot \color{blue}{0.0002777777777777778} - 0.08333333333333333 \cdot 0.08333333333333333}{\left(re \cdot re\right) \cdot 0.016666666666666666 - 0.08333333333333333} \]

      13. metadata-eval 30.8%

          \[\leadsto \frac{0.25}{re \cdot re} + \frac{{re}^{4} \cdot 0.0002777777777777778 - \color{blue}{0.006944444444444444}}{\left(re \cdot re\right) \cdot 0.016666666666666666 - 0.08333333333333333} \]

      14. associate-*l* 30.8%

          \[\leadsto \frac{0.25}{re \cdot re} + \frac{{re}^{4} \cdot 0.0002777777777777778 - 0.006944444444444444}{\color{blue}{re \cdot \left(re \cdot 0.016666666666666666\right)} - 0.08333333333333333} \]

   10. Applied egg-rr 30.8%

       \[\leadsto \frac{0.25}{re \cdot re} + \color{blue}{\frac{{re}^{4} \cdot 0.0002777777777777778 - 0.006944444444444444}{re \cdot \left(re \cdot 0.016666666666666666\right) - 0.08333333333333333}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 210000000000 \lor \neg \left(im \leq 2.55 \cdot 10^{+77}\right):\\ \;\;\;\;\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{re \cdot re} + \frac{{re}^{4} \cdot 0.0002777777777777778 - 0.006944444444444444}{re \cdot \left(re \cdot 0.016666666666666666\right) - 0.08333333333333333}\\ \end{array} \]

Alternative 4: 81.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{if}\;im \leq 600:\\ \;\;\;\;\sin re \cdot \left(im \cdot \left(0.5 \cdot im\right) + 1\right)\\ \mathbf{elif}\;im \leq 8.7 \cdot 10^{+61}:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)\\ \mathbf{elif}\;im \leq 2.15 \cdot 10^{+112}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 2.4 \cdot 10^{+118}:\\ \;\;\;\;0.5 \cdot \left(\left(im \cdot im\right) \cdot \left(re + -0.16666666666666666 \cdot {re}^{3}\right)\right)\\ \mathbf{elif}\;im \leq 1.8 \cdot 10^{+150}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sin re \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (*
          re
          (+ 1.0 (* (* im im) (+ 0.5 (* (* im im) 0.041666666666666664)))))))
   (if (<= im 600.0)
     (* (sin re) (+ (* im (* 0.5 im)) 1.0))
     (if (<= im 8.7e+61)
       (+
        0.08333333333333333
        (+ (/ 0.25 (* re re)) (* (* re re) 0.016666666666666666)))
       (if (<= im 2.15e+112)
         t_0
         (if (<= im 2.4e+118)
           (* 0.5 (* (* im im) (+ re (* -0.16666666666666666 (pow re 3.0)))))
           (if (<= im 1.8e+150) t_0 (* 0.5 (* (sin re) (* im im))))))))))

C

double code(double re, double im) {
	double t_0 = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	double tmp;
	if (im <= 600.0) {
		tmp = sin(re) * ((im * (0.5 * im)) + 1.0);
	} else if (im <= 8.7e+61) {
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	} else if (im <= 2.15e+112) {
		tmp = t_0;
	} else if (im <= 2.4e+118) {
		tmp = 0.5 * ((im * im) * (re + (-0.16666666666666666 * pow(re, 3.0))));
	} else if (im <= 1.8e+150) {
		tmp = t_0;
	} else {
		tmp = 0.5 * (sin(re) * (im * im));
	}
	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 = re * (1.0d0 + ((im * im) * (0.5d0 + ((im * im) * 0.041666666666666664d0))))
    if (im <= 600.0d0) then
        tmp = sin(re) * ((im * (0.5d0 * im)) + 1.0d0)
    else if (im <= 8.7d+61) then
        tmp = 0.08333333333333333d0 + ((0.25d0 / (re * re)) + ((re * re) * 0.016666666666666666d0))
    else if (im <= 2.15d+112) then
        tmp = t_0
    else if (im <= 2.4d+118) then
        tmp = 0.5d0 * ((im * im) * (re + ((-0.16666666666666666d0) * (re ** 3.0d0))))
    else if (im <= 1.8d+150) then
        tmp = t_0
    else
        tmp = 0.5d0 * (sin(re) * (im * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	double tmp;
	if (im <= 600.0) {
		tmp = Math.sin(re) * ((im * (0.5 * im)) + 1.0);
	} else if (im <= 8.7e+61) {
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	} else if (im <= 2.15e+112) {
		tmp = t_0;
	} else if (im <= 2.4e+118) {
		tmp = 0.5 * ((im * im) * (re + (-0.16666666666666666 * Math.pow(re, 3.0))));
	} else if (im <= 1.8e+150) {
		tmp = t_0;
	} else {
		tmp = 0.5 * (Math.sin(re) * (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))))
	tmp = 0
	if im <= 600.0:
		tmp = math.sin(re) * ((im * (0.5 * im)) + 1.0)
	elif im <= 8.7e+61:
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666))
	elif im <= 2.15e+112:
		tmp = t_0
	elif im <= 2.4e+118:
		tmp = 0.5 * ((im * im) * (re + (-0.16666666666666666 * math.pow(re, 3.0))))
	elif im <= 1.8e+150:
		tmp = t_0
	else:
		tmp = 0.5 * (math.sin(re) * (im * im))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(Float64(im * im) * 0.041666666666666664)))))
	tmp = 0.0
	if (im <= 600.0)
		tmp = Float64(sin(re) * Float64(Float64(im * Float64(0.5 * im)) + 1.0));
	elseif (im <= 8.7e+61)
		tmp = Float64(0.08333333333333333 + Float64(Float64(0.25 / Float64(re * re)) + Float64(Float64(re * re) * 0.016666666666666666)));
	elseif (im <= 2.15e+112)
		tmp = t_0;
	elseif (im <= 2.4e+118)
		tmp = Float64(0.5 * Float64(Float64(im * im) * Float64(re + Float64(-0.16666666666666666 * (re ^ 3.0)))));
	elseif (im <= 1.8e+150)
		tmp = t_0;
	else
		tmp = Float64(0.5 * Float64(sin(re) * Float64(im * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	tmp = 0.0;
	if (im <= 600.0)
		tmp = sin(re) * ((im * (0.5 * im)) + 1.0);
	elseif (im <= 8.7e+61)
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	elseif (im <= 2.15e+112)
		tmp = t_0;
	elseif (im <= 2.4e+118)
		tmp = 0.5 * ((im * im) * (re + (-0.16666666666666666 * (re ^ 3.0))));
	elseif (im <= 1.8e+150)
		tmp = t_0;
	else
		tmp = 0.5 * (sin(re) * (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 600.0], N[(N[Sin[re], $MachinePrecision] * N[(N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 8.7e+61], 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, 2.15e+112], t$95$0, If[LessEqual[im, 2.4e+118], N[(0.5 * N[(N[(im * im), $MachinePrecision] * N[(re + N[(-0.16666666666666666 * N[Power[re, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.8e+150], t$95$0, N[(0.5 * N[(N[Sin[re], $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if im < 600

   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. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\sin re \cdot 0.5\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. associate-*l* 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 78.6%

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

      1. *-commutative 78.6%

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

      2. associate-*r* 78.6%

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

      3. distribute-rgt1-in 78.6%

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

      4. *-commutative 78.6%

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

      5. unpow2 78.6%

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

      6. associate-*l* 78.6%

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

   6. Simplified 78.6%

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

   if 600 < im < 8.7000000000000002e61

   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. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\sin re \cdot 0.5\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. associate-*l* 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 34.9%

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

   6. Taylor expanded in re around 0 56.4%

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

      1. associate-*r/ 56.4%

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

      2. metadata-eval 56.4%

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

      3. unpow2 56.4%

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

      4. *-commutative 56.4%

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

      5. unpow2 56.4%

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

   8. Simplified 56.4%

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

   if 8.7000000000000002e61 < im < 2.14999999999999991e112 or 2.4e118 < im < 1.79999999999999993e150

   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. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\sin re \cdot 0.5\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. associate-*l* 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 86.8%

      \[\leadsto \color{blue}{\sin re + \left(0.5 \cdot \left(\sin re \cdot {im}^{2}\right) + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right)} \]
   5. Step-by-step derivation

      1. *-rgt-identity 86.8%

         \[\leadsto \color{blue}{\sin re \cdot 1} + \left(0.5 \cdot \left(\sin re \cdot {im}^{2}\right) + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right) \]

      2. *-commutative 86.8%

         \[\leadsto \sin re \cdot 1 + \left(0.5 \cdot \color{blue}{\left({im}^{2} \cdot \sin re\right)} + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right) \]

      3. associate-*r* 86.8%

         \[\leadsto \sin re \cdot 1 + \left(\color{blue}{\left(0.5 \cdot {im}^{2}\right) \cdot \sin re} + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right) \]

      4. *-commutative 86.8%

         \[\leadsto \sin re \cdot 1 + \left(\left(0.5 \cdot {im}^{2}\right) \cdot \sin re + 0.041666666666666664 \cdot \color{blue}{\left({im}^{4} \cdot \sin re\right)}\right) \]

      5. associate-*r* 86.8%

         \[\leadsto \sin re \cdot 1 + \left(\left(0.5 \cdot {im}^{2}\right) \cdot \sin re + \color{blue}{\left(0.041666666666666664 \cdot {im}^{4}\right) \cdot \sin re}\right) \]

      6. distribute-rgt-out 86.8%

         \[\leadsto \sin re \cdot 1 + \color{blue}{\sin re \cdot \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot {im}^{4}\right)} \]

      7. distribute-lft-out 86.8%

         \[\leadsto \color{blue}{\sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot {im}^{4}\right)\right)} \]

      8. metadata-eval 86.8%

         \[\leadsto \sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot {im}^{\color{blue}{\left(2 \cdot 2\right)}}\right)\right) \]

      9. pow-sqr 86.8%

         \[\leadsto \sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)}\right)\right) \]

      10. associate-*r* 86.8%

          \[\leadsto \sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + \color{blue}{\left(0.041666666666666664 \cdot {im}^{2}\right) \cdot {im}^{2}}\right)\right) \]

      11. distribute-rgt-out 86.8%

          \[\leadsto \sin re \cdot \left(1 + \color{blue}{{im}^{2} \cdot \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)}\right) \]

      12. unpow2 86.8%

          \[\leadsto \sin re \cdot \left(1 + \color{blue}{\left(im \cdot im\right)} \cdot \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)\right) \]

      13. unpow2 86.8%

          \[\leadsto \sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \]

   6. Simplified 86.8%

      \[\leadsto \color{blue}{\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)} \]

   7. Taylor expanded in re around 0 78.6%

      \[\leadsto \color{blue}{\left(1 + \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right) \cdot {im}^{2}\right) \cdot re} \]
   8. Step-by-step derivation

      1. *-commutative 78.6%

         \[\leadsto \left(1 + \color{blue}{{im}^{2} \cdot \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)}\right) \cdot re \]

      2. unpow2 78.6%

         \[\leadsto \left(1 + \color{blue}{\left(im \cdot im\right)} \cdot \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)\right) \cdot re \]

      3. unpow2 78.6%

         \[\leadsto \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \cdot re \]

   9. Simplified 78.6%

      \[\leadsto \color{blue}{\left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right) \cdot re} \]

   if 2.14999999999999991e112 < im < 2.4e118

   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. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\sin re \cdot 0.5\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. associate-*l* 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 5.3%

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

      1. *-commutative 5.3%

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

      2. associate-*r* 5.3%

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

      3. distribute-rgt1-in 5.3%

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

      4. *-commutative 5.3%

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

      5. unpow2 5.3%

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

      6. associate-*l* 5.3%

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

   6. Simplified 5.3%

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

   7. Taylor expanded in im around inf 5.3%

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

      1. *-commutative 5.3%

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

      2. unpow2 5.3%

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

      3. associate-*l* 5.3%

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

   9. Simplified 5.3%

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

   10. Taylor expanded in re around 0 1.2%

       \[\leadsto 0.5 \cdot \color{blue}{\left(-0.16666666666666666 \cdot \left({re}^{3} \cdot {im}^{2}\right) + re \cdot {im}^{2}\right)} \]
   11. Step-by-step derivation

       1. +-commutative 1.2%

          \[\leadsto 0.5 \cdot \color{blue}{\left(re \cdot {im}^{2} + -0.16666666666666666 \cdot \left({re}^{3} \cdot {im}^{2}\right)\right)} \]

       2. associate-*r* 1.2%

          \[\leadsto 0.5 \cdot \left(re \cdot {im}^{2} + \color{blue}{\left(-0.16666666666666666 \cdot {re}^{3}\right) \cdot {im}^{2}}\right) \]

       3. distribute-rgt-out 67.8%

          \[\leadsto 0.5 \cdot \color{blue}{\left({im}^{2} \cdot \left(re + -0.16666666666666666 \cdot {re}^{3}\right)\right)} \]

       4. unpow2 67.8%

          \[\leadsto 0.5 \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \left(re + -0.16666666666666666 \cdot {re}^{3}\right)\right) \]

   12. Simplified 67.8%

       \[\leadsto 0.5 \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot \left(re + -0.16666666666666666 \cdot {re}^{3}\right)\right)} \]

   if 1.79999999999999993e150 < 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. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\sin re \cdot 0.5\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. associate-*l* 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 97.0%

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

      1. *-commutative 97.0%

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

      2. associate-*r* 97.0%

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

      3. distribute-rgt1-in 97.0%

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

      4. *-commutative 97.0%

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

      5. unpow2 97.0%

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

      6. associate-*l* 97.0%

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

   6. Simplified 97.0%

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

   7. Taylor expanded in im around inf 97.0%

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

      1. *-commutative 97.0%

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

      2. unpow2 97.0%

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

   9. Simplified 97.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(im \cdot im\right) \cdot \sin re\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 600:\\ \;\;\;\;\sin re \cdot \left(im \cdot \left(0.5 \cdot im\right) + 1\right)\\ \mathbf{elif}\;im \leq 8.7 \cdot 10^{+61}:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)\\ \mathbf{elif}\;im \leq 2.15 \cdot 10^{+112}:\\ \;\;\;\;re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{elif}\;im \leq 2.4 \cdot 10^{+118}:\\ \;\;\;\;0.5 \cdot \left(\left(im \cdot im\right) \cdot \left(re + -0.16666666666666666 \cdot {re}^{3}\right)\right)\\ \mathbf{elif}\;im \leq 1.8 \cdot 10^{+150}:\\ \;\;\;\;re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sin re \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 5: 89.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 580 \lor \neg \left(im \leq 2.8 \cdot 10^{+70}\right):\\ \;\;\;\;\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 580.0) (not (<= im 2.8e+70)))
   (*
    (sin re)
    (+ 1.0 (* (* im im) (+ 0.5 (* (* im im) 0.041666666666666664)))))
   (+
    0.08333333333333333
    (+ (/ 0.25 (* re re)) (* (* re re) 0.016666666666666666)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 580.0) || !(im <= 2.8e+70)) {
		tmp = sin(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	} else {
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= 580.0d0) .or. (.not. (im <= 2.8d+70))) then
        tmp = sin(re) * (1.0d0 + ((im * im) * (0.5d0 + ((im * im) * 0.041666666666666664d0))))
    else
        tmp = 0.08333333333333333d0 + ((0.25d0 / (re * re)) + ((re * re) * 0.016666666666666666d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= 580.0) || !(im <= 2.8e+70)) {
		tmp = Math.sin(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	} else {
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= 580.0) or not (im <= 2.8e+70):
		tmp = math.sin(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))))
	else:
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 580.0) || !(im <= 2.8e+70))
		tmp = Float64(sin(re) * Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(Float64(im * im) * 0.041666666666666664)))));
	else
		tmp = Float64(0.08333333333333333 + Float64(Float64(0.25 / Float64(re * re)) + Float64(Float64(re * re) * 0.016666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= 580.0) || ~((im <= 2.8e+70)))
		tmp = sin(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	else
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 580.0], N[Not[LessEqual[im, 2.8e+70]], $MachinePrecision]], N[(N[Sin[re], $MachinePrecision] * N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.08333333333333333 + N[(N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision] + N[(N[(re * re), $MachinePrecision] * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 580 or 2.7999999999999999e70 < 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. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\sin re \cdot 0.5\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. associate-*l* 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 91.5%

      \[\leadsto \color{blue}{\sin re + \left(0.5 \cdot \left(\sin re \cdot {im}^{2}\right) + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right)} \]
   5. Step-by-step derivation

      1. *-rgt-identity 91.5%

         \[\leadsto \color{blue}{\sin re \cdot 1} + \left(0.5 \cdot \left(\sin re \cdot {im}^{2}\right) + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right) \]

      2. *-commutative 91.5%

         \[\leadsto \sin re \cdot 1 + \left(0.5 \cdot \color{blue}{\left({im}^{2} \cdot \sin re\right)} + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right) \]

      3. associate-*r* 91.5%

         \[\leadsto \sin re \cdot 1 + \left(\color{blue}{\left(0.5 \cdot {im}^{2}\right) \cdot \sin re} + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right) \]

      4. *-commutative 91.5%

         \[\leadsto \sin re \cdot 1 + \left(\left(0.5 \cdot {im}^{2}\right) \cdot \sin re + 0.041666666666666664 \cdot \color{blue}{\left({im}^{4} \cdot \sin re\right)}\right) \]

      5. associate-*r* 91.5%

         \[\leadsto \sin re \cdot 1 + \left(\left(0.5 \cdot {im}^{2}\right) \cdot \sin re + \color{blue}{\left(0.041666666666666664 \cdot {im}^{4}\right) \cdot \sin re}\right) \]

      6. distribute-rgt-out 91.5%

         \[\leadsto \sin re \cdot 1 + \color{blue}{\sin re \cdot \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot {im}^{4}\right)} \]

      7. distribute-lft-out 91.5%

         \[\leadsto \color{blue}{\sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot {im}^{4}\right)\right)} \]

      8. metadata-eval 91.5%

         \[\leadsto \sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot {im}^{\color{blue}{\left(2 \cdot 2\right)}}\right)\right) \]

      9. pow-sqr 91.5%

         \[\leadsto \sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)}\right)\right) \]

      10. associate-*r* 91.5%

          \[\leadsto \sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + \color{blue}{\left(0.041666666666666664 \cdot {im}^{2}\right) \cdot {im}^{2}}\right)\right) \]

      11. distribute-rgt-out 91.5%

          \[\leadsto \sin re \cdot \left(1 + \color{blue}{{im}^{2} \cdot \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)}\right) \]

      12. unpow2 91.5%

          \[\leadsto \sin re \cdot \left(1 + \color{blue}{\left(im \cdot im\right)} \cdot \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)\right) \]

      13. unpow2 91.5%

          \[\leadsto \sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \]

   6. Simplified 91.5%

      \[\leadsto \color{blue}{\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)} \]

   if 580 < im < 2.7999999999999999e70

   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. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\sin re \cdot 0.5\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. associate-*l* 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 31.5%

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

   6. Taylor expanded in re around 0 50.8%

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

      1. associate-*r/ 50.8%

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

      2. metadata-eval 50.8%

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

      3. unpow2 50.8%

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

      4. *-commutative 50.8%

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

      5. unpow2 50.8%

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

   8. Simplified 50.8%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 580 \lor \neg \left(im \leq 2.8 \cdot 10^{+70}\right):\\ \;\;\;\;\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)\\ \end{array} \]

Alternative 6: 67.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 660:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.02 \cdot 10^{+62}:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)\\ \mathbf{elif}\;im \leq 3.55 \cdot 10^{+193}:\\ \;\;\;\;re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(\sin re \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 660.0)
   (sin re)
   (if (<= im 1.02e+62)
     (+
      0.08333333333333333
      (+ (/ 0.25 (* re re)) (* (* re re) 0.016666666666666666)))
     (if (<= im 3.55e+193)
       (* re (+ 1.0 (* (* im im) (+ 0.5 (* (* im im) 0.041666666666666664)))))
       (* 0.5 (* im (* (sin re) im)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 660.0) {
		tmp = sin(re);
	} else if (im <= 1.02e+62) {
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	} else if (im <= 3.55e+193) {
		tmp = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	} else {
		tmp = 0.5 * (im * (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 <= 660.0d0) then
        tmp = sin(re)
    else if (im <= 1.02d+62) then
        tmp = 0.08333333333333333d0 + ((0.25d0 / (re * re)) + ((re * re) * 0.016666666666666666d0))
    else if (im <= 3.55d+193) then
        tmp = re * (1.0d0 + ((im * im) * (0.5d0 + ((im * im) * 0.041666666666666664d0))))
    else
        tmp = 0.5d0 * (im * (sin(re) * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 660.0) {
		tmp = Math.sin(re);
	} else if (im <= 1.02e+62) {
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	} else if (im <= 3.55e+193) {
		tmp = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	} else {
		tmp = 0.5 * (im * (Math.sin(re) * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 660.0:
		tmp = math.sin(re)
	elif im <= 1.02e+62:
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666))
	elif im <= 3.55e+193:
		tmp = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))))
	else:
		tmp = 0.5 * (im * (math.sin(re) * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 660.0)
		tmp = sin(re);
	elseif (im <= 1.02e+62)
		tmp = Float64(0.08333333333333333 + Float64(Float64(0.25 / Float64(re * re)) + Float64(Float64(re * re) * 0.016666666666666666)));
	elseif (im <= 3.55e+193)
		tmp = Float64(re * Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(Float64(im * im) * 0.041666666666666664)))));
	else
		tmp = Float64(0.5 * Float64(im * Float64(sin(re) * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 660.0)
		tmp = sin(re);
	elseif (im <= 1.02e+62)
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	elseif (im <= 3.55e+193)
		tmp = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	else
		tmp = 0.5 * (im * (sin(re) * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 660.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.02e+62], 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, 3.55e+193], N[(re * N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[(N[Sin[re], $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < 660

   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. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\sin re \cdot 0.5\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. associate-*l* 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 65.5%

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

   if 660 < im < 1.02000000000000002e62

   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. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\sin re \cdot 0.5\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. associate-*l* 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 34.9%

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

   6. Taylor expanded in re around 0 56.4%

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

      1. associate-*r/ 56.4%

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

      2. metadata-eval 56.4%

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

      3. unpow2 56.4%

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

      4. *-commutative 56.4%

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

      5. unpow2 56.4%

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

   8. Simplified 56.4%

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

   if 1.02000000000000002e62 < im < 3.5499999999999999e193

   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. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\sin re \cdot 0.5\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. associate-*l* 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 93.6%

      \[\leadsto \color{blue}{\sin re + \left(0.5 \cdot \left(\sin re \cdot {im}^{2}\right) + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right)} \]
   5. Step-by-step derivation

      1. *-rgt-identity 93.6%

         \[\leadsto \color{blue}{\sin re \cdot 1} + \left(0.5 \cdot \left(\sin re \cdot {im}^{2}\right) + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right) \]

      2. *-commutative 93.6%

         \[\leadsto \sin re \cdot 1 + \left(0.5 \cdot \color{blue}{\left({im}^{2} \cdot \sin re\right)} + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right) \]

      3. associate-*r* 93.6%

         \[\leadsto \sin re \cdot 1 + \left(\color{blue}{\left(0.5 \cdot {im}^{2}\right) \cdot \sin re} + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right) \]

      4. *-commutative 93.6%

         \[\leadsto \sin re \cdot 1 + \left(\left(0.5 \cdot {im}^{2}\right) \cdot \sin re + 0.041666666666666664 \cdot \color{blue}{\left({im}^{4} \cdot \sin re\right)}\right) \]

      5. associate-*r* 93.6%

         \[\leadsto \sin re \cdot 1 + \left(\left(0.5 \cdot {im}^{2}\right) \cdot \sin re + \color{blue}{\left(0.041666666666666664 \cdot {im}^{4}\right) \cdot \sin re}\right) \]

      6. distribute-rgt-out 93.6%

         \[\leadsto \sin re \cdot 1 + \color{blue}{\sin re \cdot \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot {im}^{4}\right)} \]

      7. distribute-lft-out 93.6%

         \[\leadsto \color{blue}{\sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot {im}^{4}\right)\right)} \]

      8. metadata-eval 93.6%

         \[\leadsto \sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot {im}^{\color{blue}{\left(2 \cdot 2\right)}}\right)\right) \]

      9. pow-sqr 93.6%

         \[\leadsto \sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)}\right)\right) \]

      10. associate-*r* 93.6%

          \[\leadsto \sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + \color{blue}{\left(0.041666666666666664 \cdot {im}^{2}\right) \cdot {im}^{2}}\right)\right) \]

      11. distribute-rgt-out 93.6%

          \[\leadsto \sin re \cdot \left(1 + \color{blue}{{im}^{2} \cdot \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)}\right) \]

      12. unpow2 93.6%

          \[\leadsto \sin re \cdot \left(1 + \color{blue}{\left(im \cdot im\right)} \cdot \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)\right) \]

      13. unpow2 93.6%

          \[\leadsto \sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \]

   6. Simplified 93.6%

      \[\leadsto \color{blue}{\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)} \]

   7. Taylor expanded in re around 0 69.0%

      \[\leadsto \color{blue}{\left(1 + \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right) \cdot {im}^{2}\right) \cdot re} \]
   8. Step-by-step derivation

      1. *-commutative 69.0%

         \[\leadsto \left(1 + \color{blue}{{im}^{2} \cdot \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)}\right) \cdot re \]

      2. unpow2 69.0%

         \[\leadsto \left(1 + \color{blue}{\left(im \cdot im\right)} \cdot \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)\right) \cdot re \]

      3. unpow2 69.0%

         \[\leadsto \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \cdot re \]

   9. Simplified 69.0%

      \[\leadsto \color{blue}{\left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right) \cdot re} \]

   if 3.5499999999999999e193 < 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. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\sin re \cdot 0.5\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. associate-*l* 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

      3. distribute-rgt1-in 100.0%

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

      4. *-commutative 100.0%

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

      5. unpow2 100.0%

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

      6. associate-*l* 100.0%

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

   6. Simplified 100.0%

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

   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. *-commutative 100.0%

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

      2. unpow2 100.0%

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

      3. associate-*l* 86.9%

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

   9. Simplified 86.9%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 660:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.02 \cdot 10^{+62}:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)\\ \mathbf{elif}\;im \leq 3.55 \cdot 10^{+193}:\\ \;\;\;\;re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(\sin re \cdot im\right)\right)\\ \end{array} \]

Alternative 7: 69.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 520:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2.7 \cdot 10^{+61}:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)\\ \mathbf{elif}\;im \leq 1.8 \cdot 10^{+150}:\\ \;\;\;\;re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sin re \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 520.0)
   (sin re)
   (if (<= im 2.7e+61)
     (+
      0.08333333333333333
      (+ (/ 0.25 (* re re)) (* (* re re) 0.016666666666666666)))
     (if (<= im 1.8e+150)
       (* re (+ 1.0 (* (* im im) (+ 0.5 (* (* im im) 0.041666666666666664)))))
       (* 0.5 (* (sin re) (* im im)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 520.0) {
		tmp = sin(re);
	} else if (im <= 2.7e+61) {
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	} else if (im <= 1.8e+150) {
		tmp = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	} else {
		tmp = 0.5 * (sin(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 <= 520.0d0) then
        tmp = sin(re)
    else if (im <= 2.7d+61) then
        tmp = 0.08333333333333333d0 + ((0.25d0 / (re * re)) + ((re * re) * 0.016666666666666666d0))
    else if (im <= 1.8d+150) then
        tmp = re * (1.0d0 + ((im * im) * (0.5d0 + ((im * im) * 0.041666666666666664d0))))
    else
        tmp = 0.5d0 * (sin(re) * (im * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 520.0) {
		tmp = Math.sin(re);
	} else if (im <= 2.7e+61) {
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	} else if (im <= 1.8e+150) {
		tmp = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	} else {
		tmp = 0.5 * (Math.sin(re) * (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 520.0:
		tmp = math.sin(re)
	elif im <= 2.7e+61:
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666))
	elif im <= 1.8e+150:
		tmp = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))))
	else:
		tmp = 0.5 * (math.sin(re) * (im * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 520.0)
		tmp = sin(re);
	elseif (im <= 2.7e+61)
		tmp = Float64(0.08333333333333333 + Float64(Float64(0.25 / Float64(re * re)) + Float64(Float64(re * re) * 0.016666666666666666)));
	elseif (im <= 1.8e+150)
		tmp = Float64(re * Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(Float64(im * im) * 0.041666666666666664)))));
	else
		tmp = Float64(0.5 * Float64(sin(re) * Float64(im * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 520.0)
		tmp = sin(re);
	elseif (im <= 2.7e+61)
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	elseif (im <= 1.8e+150)
		tmp = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	else
		tmp = 0.5 * (sin(re) * (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 520.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 2.7e+61], 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, 1.8e+150], N[(re * N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Sin[re], $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < 520

   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. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\sin re \cdot 0.5\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. associate-*l* 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 65.5%

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

   if 520 < im < 2.7000000000000002e61

   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. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\sin re \cdot 0.5\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. associate-*l* 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 34.9%

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

   6. Taylor expanded in re around 0 56.4%

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

      1. associate-*r/ 56.4%

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

      2. metadata-eval 56.4%

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

      3. unpow2 56.4%

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

      4. *-commutative 56.4%

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

      5. unpow2 56.4%

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

   8. Simplified 56.4%

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

   if 2.7000000000000002e61 < im < 1.79999999999999993e150

   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. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\sin re \cdot 0.5\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. associate-*l* 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 90.8%

      \[\leadsto \color{blue}{\sin re + \left(0.5 \cdot \left(\sin re \cdot {im}^{2}\right) + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right)} \]
   5. Step-by-step derivation

      1. *-rgt-identity 90.8%

         \[\leadsto \color{blue}{\sin re \cdot 1} + \left(0.5 \cdot \left(\sin re \cdot {im}^{2}\right) + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right) \]

      2. *-commutative 90.8%

         \[\leadsto \sin re \cdot 1 + \left(0.5 \cdot \color{blue}{\left({im}^{2} \cdot \sin re\right)} + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right) \]

      3. associate-*r* 90.8%

         \[\leadsto \sin re \cdot 1 + \left(\color{blue}{\left(0.5 \cdot {im}^{2}\right) \cdot \sin re} + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right) \]

      4. *-commutative 90.8%

         \[\leadsto \sin re \cdot 1 + \left(\left(0.5 \cdot {im}^{2}\right) \cdot \sin re + 0.041666666666666664 \cdot \color{blue}{\left({im}^{4} \cdot \sin re\right)}\right) \]

      5. associate-*r* 90.8%

         \[\leadsto \sin re \cdot 1 + \left(\left(0.5 \cdot {im}^{2}\right) \cdot \sin re + \color{blue}{\left(0.041666666666666664 \cdot {im}^{4}\right) \cdot \sin re}\right) \]

      6. distribute-rgt-out 90.8%

         \[\leadsto \sin re \cdot 1 + \color{blue}{\sin re \cdot \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot {im}^{4}\right)} \]

      7. distribute-lft-out 90.8%

         \[\leadsto \color{blue}{\sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot {im}^{4}\right)\right)} \]

      8. metadata-eval 90.8%

         \[\leadsto \sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot {im}^{\color{blue}{\left(2 \cdot 2\right)}}\right)\right) \]

      9. pow-sqr 90.8%

         \[\leadsto \sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)}\right)\right) \]

      10. associate-*r* 90.8%

          \[\leadsto \sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + \color{blue}{\left(0.041666666666666664 \cdot {im}^{2}\right) \cdot {im}^{2}}\right)\right) \]

      11. distribute-rgt-out 90.8%

          \[\leadsto \sin re \cdot \left(1 + \color{blue}{{im}^{2} \cdot \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)}\right) \]

      12. unpow2 90.8%

          \[\leadsto \sin re \cdot \left(1 + \color{blue}{\left(im \cdot im\right)} \cdot \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)\right) \]

      13. unpow2 90.8%

          \[\leadsto \sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \]

   6. Simplified 90.8%

      \[\leadsto \color{blue}{\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)} \]

   7. Taylor expanded in re around 0 65.0%

      \[\leadsto \color{blue}{\left(1 + \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right) \cdot {im}^{2}\right) \cdot re} \]
   8. Step-by-step derivation

      1. *-commutative 65.0%

         \[\leadsto \left(1 + \color{blue}{{im}^{2} \cdot \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)}\right) \cdot re \]

      2. unpow2 65.0%

         \[\leadsto \left(1 + \color{blue}{\left(im \cdot im\right)} \cdot \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)\right) \cdot re \]

      3. unpow2 65.0%

         \[\leadsto \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \cdot re \]

   9. Simplified 65.0%

      \[\leadsto \color{blue}{\left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right) \cdot re} \]

   if 1.79999999999999993e150 < 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. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\sin re \cdot 0.5\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. associate-*l* 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 97.0%

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

      1. *-commutative 97.0%

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

      2. associate-*r* 97.0%

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

      3. distribute-rgt1-in 97.0%

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

      4. *-commutative 97.0%

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

      5. unpow2 97.0%

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

      6. associate-*l* 97.0%

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

   6. Simplified 97.0%

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

   7. Taylor expanded in im around inf 97.0%

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

      1. *-commutative 97.0%

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

      2. unpow2 97.0%

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

   9. Simplified 97.0%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 520:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2.7 \cdot 10^{+61}:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)\\ \mathbf{elif}\;im \leq 1.8 \cdot 10^{+150}:\\ \;\;\;\;re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sin re \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 8: 82.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 550:\\ \;\;\;\;\sin re \cdot \left(im \cdot \left(0.5 \cdot im\right) + 1\right)\\ \mathbf{elif}\;im \leq 2.7 \cdot 10^{+60}:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)\\ \mathbf{elif}\;im \leq 1.8 \cdot 10^{+150}:\\ \;\;\;\;re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sin re \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 550.0)
   (* (sin re) (+ (* im (* 0.5 im)) 1.0))
   (if (<= im 2.7e+60)
     (+
      0.08333333333333333
      (+ (/ 0.25 (* re re)) (* (* re re) 0.016666666666666666)))
     (if (<= im 1.8e+150)
       (* re (+ 1.0 (* (* im im) (+ 0.5 (* (* im im) 0.041666666666666664)))))
       (* 0.5 (* (sin re) (* im im)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 550.0) {
		tmp = sin(re) * ((im * (0.5 * im)) + 1.0);
	} else if (im <= 2.7e+60) {
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	} else if (im <= 1.8e+150) {
		tmp = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	} else {
		tmp = 0.5 * (sin(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 <= 550.0d0) then
        tmp = sin(re) * ((im * (0.5d0 * im)) + 1.0d0)
    else if (im <= 2.7d+60) then
        tmp = 0.08333333333333333d0 + ((0.25d0 / (re * re)) + ((re * re) * 0.016666666666666666d0))
    else if (im <= 1.8d+150) then
        tmp = re * (1.0d0 + ((im * im) * (0.5d0 + ((im * im) * 0.041666666666666664d0))))
    else
        tmp = 0.5d0 * (sin(re) * (im * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 550.0) {
		tmp = Math.sin(re) * ((im * (0.5 * im)) + 1.0);
	} else if (im <= 2.7e+60) {
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	} else if (im <= 1.8e+150) {
		tmp = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	} else {
		tmp = 0.5 * (Math.sin(re) * (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 550.0:
		tmp = math.sin(re) * ((im * (0.5 * im)) + 1.0)
	elif im <= 2.7e+60:
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666))
	elif im <= 1.8e+150:
		tmp = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))))
	else:
		tmp = 0.5 * (math.sin(re) * (im * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 550.0)
		tmp = Float64(sin(re) * Float64(Float64(im * Float64(0.5 * im)) + 1.0));
	elseif (im <= 2.7e+60)
		tmp = Float64(0.08333333333333333 + Float64(Float64(0.25 / Float64(re * re)) + Float64(Float64(re * re) * 0.016666666666666666)));
	elseif (im <= 1.8e+150)
		tmp = Float64(re * Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(Float64(im * im) * 0.041666666666666664)))));
	else
		tmp = Float64(0.5 * Float64(sin(re) * Float64(im * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 550.0)
		tmp = sin(re) * ((im * (0.5 * im)) + 1.0);
	elseif (im <= 2.7e+60)
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	elseif (im <= 1.8e+150)
		tmp = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	else
		tmp = 0.5 * (sin(re) * (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 550.0], N[(N[Sin[re], $MachinePrecision] * N[(N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.7e+60], 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, 1.8e+150], N[(re * N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Sin[re], $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < 550

   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. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\sin re \cdot 0.5\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. associate-*l* 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 78.6%

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

      1. *-commutative 78.6%

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

      2. associate-*r* 78.6%

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

      3. distribute-rgt1-in 78.6%

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

      4. *-commutative 78.6%

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

      5. unpow2 78.6%

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

      6. associate-*l* 78.6%

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

   6. Simplified 78.6%

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

   if 550 < im < 2.6999999999999999e60

   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. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\sin re \cdot 0.5\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. associate-*l* 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 34.9%

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

   6. Taylor expanded in re around 0 56.4%

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

      1. associate-*r/ 56.4%

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

      2. metadata-eval 56.4%

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

      3. unpow2 56.4%

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

      4. *-commutative 56.4%

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

      5. unpow2 56.4%

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

   8. Simplified 56.4%

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

   if 2.6999999999999999e60 < im < 1.79999999999999993e150

   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. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\sin re \cdot 0.5\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. associate-*l* 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 90.8%

      \[\leadsto \color{blue}{\sin re + \left(0.5 \cdot \left(\sin re \cdot {im}^{2}\right) + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right)} \]
   5. Step-by-step derivation

      1. *-rgt-identity 90.8%

         \[\leadsto \color{blue}{\sin re \cdot 1} + \left(0.5 \cdot \left(\sin re \cdot {im}^{2}\right) + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right) \]

      2. *-commutative 90.8%

         \[\leadsto \sin re \cdot 1 + \left(0.5 \cdot \color{blue}{\left({im}^{2} \cdot \sin re\right)} + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right) \]

      3. associate-*r* 90.8%

         \[\leadsto \sin re \cdot 1 + \left(\color{blue}{\left(0.5 \cdot {im}^{2}\right) \cdot \sin re} + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right) \]

      4. *-commutative 90.8%

         \[\leadsto \sin re \cdot 1 + \left(\left(0.5 \cdot {im}^{2}\right) \cdot \sin re + 0.041666666666666664 \cdot \color{blue}{\left({im}^{4} \cdot \sin re\right)}\right) \]

      5. associate-*r* 90.8%

         \[\leadsto \sin re \cdot 1 + \left(\left(0.5 \cdot {im}^{2}\right) \cdot \sin re + \color{blue}{\left(0.041666666666666664 \cdot {im}^{4}\right) \cdot \sin re}\right) \]

      6. distribute-rgt-out 90.8%

         \[\leadsto \sin re \cdot 1 + \color{blue}{\sin re \cdot \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot {im}^{4}\right)} \]

      7. distribute-lft-out 90.8%

         \[\leadsto \color{blue}{\sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot {im}^{4}\right)\right)} \]

      8. metadata-eval 90.8%

         \[\leadsto \sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot {im}^{\color{blue}{\left(2 \cdot 2\right)}}\right)\right) \]

      9. pow-sqr 90.8%

         \[\leadsto \sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)}\right)\right) \]

      10. associate-*r* 90.8%

          \[\leadsto \sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + \color{blue}{\left(0.041666666666666664 \cdot {im}^{2}\right) \cdot {im}^{2}}\right)\right) \]

      11. distribute-rgt-out 90.8%

          \[\leadsto \sin re \cdot \left(1 + \color{blue}{{im}^{2} \cdot \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)}\right) \]

      12. unpow2 90.8%

          \[\leadsto \sin re \cdot \left(1 + \color{blue}{\left(im \cdot im\right)} \cdot \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)\right) \]

      13. unpow2 90.8%

          \[\leadsto \sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \]

   6. Simplified 90.8%

      \[\leadsto \color{blue}{\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)} \]

   7. Taylor expanded in re around 0 65.0%

      \[\leadsto \color{blue}{\left(1 + \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right) \cdot {im}^{2}\right) \cdot re} \]
   8. Step-by-step derivation

      1. *-commutative 65.0%

         \[\leadsto \left(1 + \color{blue}{{im}^{2} \cdot \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)}\right) \cdot re \]

      2. unpow2 65.0%

         \[\leadsto \left(1 + \color{blue}{\left(im \cdot im\right)} \cdot \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)\right) \cdot re \]

      3. unpow2 65.0%

         \[\leadsto \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \cdot re \]

   9. Simplified 65.0%

      \[\leadsto \color{blue}{\left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right) \cdot re} \]

   if 1.79999999999999993e150 < 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. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\sin re \cdot 0.5\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. associate-*l* 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 97.0%

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

      1. *-commutative 97.0%

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

      2. associate-*r* 97.0%

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

      3. distribute-rgt1-in 97.0%

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

      4. *-commutative 97.0%

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

      5. unpow2 97.0%

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

      6. associate-*l* 97.0%

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

   6. Simplified 97.0%

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

   7. Taylor expanded in im around inf 97.0%

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

      1. *-commutative 97.0%

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

      2. unpow2 97.0%

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

   9. Simplified 97.0%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 550:\\ \;\;\;\;\sin re \cdot \left(im \cdot \left(0.5 \cdot im\right) + 1\right)\\ \mathbf{elif}\;im \leq 2.7 \cdot 10^{+60}:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)\\ \mathbf{elif}\;im \leq 1.8 \cdot 10^{+150}:\\ \;\;\;\;re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sin re \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 9: 66.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 820:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 3.5 \cdot 10^{+61}:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 820.0)
   (sin re)
   (if (<= im 3.5e+61)
     (+
      0.08333333333333333
      (+ (/ 0.25 (* re re)) (* (* re re) 0.016666666666666666)))
     (* re (+ 1.0 (* (* im im) (+ 0.5 (* (* im im) 0.041666666666666664))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 820.0) {
		tmp = sin(re);
	} else if (im <= 3.5e+61) {
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	} else {
		tmp = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 820.0d0) then
        tmp = sin(re)
    else if (im <= 3.5d+61) then
        tmp = 0.08333333333333333d0 + ((0.25d0 / (re * re)) + ((re * re) * 0.016666666666666666d0))
    else
        tmp = re * (1.0d0 + ((im * im) * (0.5d0 + ((im * im) * 0.041666666666666664d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 820.0) {
		tmp = Math.sin(re);
	} else if (im <= 3.5e+61) {
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	} else {
		tmp = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 820.0:
		tmp = math.sin(re)
	elif im <= 3.5e+61:
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666))
	else:
		tmp = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 820.0)
		tmp = sin(re);
	elseif (im <= 3.5e+61)
		tmp = Float64(0.08333333333333333 + Float64(Float64(0.25 / Float64(re * re)) + Float64(Float64(re * re) * 0.016666666666666666)));
	else
		tmp = Float64(re * Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(Float64(im * im) * 0.041666666666666664)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 820.0)
		tmp = sin(re);
	elseif (im <= 3.5e+61)
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	else
		tmp = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 820.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 3.5e+61], N[(0.08333333333333333 + N[(N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision] + N[(N[(re * re), $MachinePrecision] * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 820

   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. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\sin re \cdot 0.5\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. associate-*l* 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 65.5%

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

   if 820 < im < 3.50000000000000018e61

   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. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\sin re \cdot 0.5\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. associate-*l* 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 34.9%

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

   6. Taylor expanded in re around 0 56.4%

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

      1. associate-*r/ 56.4%

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

      2. metadata-eval 56.4%

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

      3. unpow2 56.4%

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

      4. *-commutative 56.4%

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

      5. unpow2 56.4%

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

   8. Simplified 56.4%

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

   if 3.50000000000000018e61 < 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. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\sin re \cdot 0.5\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. associate-*l* 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 96.3%

      \[\leadsto \color{blue}{\sin re + \left(0.5 \cdot \left(\sin re \cdot {im}^{2}\right) + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right)} \]
   5. Step-by-step derivation

      1. *-rgt-identity 96.3%

         \[\leadsto \color{blue}{\sin re \cdot 1} + \left(0.5 \cdot \left(\sin re \cdot {im}^{2}\right) + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right) \]

      2. *-commutative 96.3%

         \[\leadsto \sin re \cdot 1 + \left(0.5 \cdot \color{blue}{\left({im}^{2} \cdot \sin re\right)} + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right) \]

      3. associate-*r* 96.3%

         \[\leadsto \sin re \cdot 1 + \left(\color{blue}{\left(0.5 \cdot {im}^{2}\right) \cdot \sin re} + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right) \]

      4. *-commutative 96.3%

         \[\leadsto \sin re \cdot 1 + \left(\left(0.5 \cdot {im}^{2}\right) \cdot \sin re + 0.041666666666666664 \cdot \color{blue}{\left({im}^{4} \cdot \sin re\right)}\right) \]

      5. associate-*r* 96.3%

         \[\leadsto \sin re \cdot 1 + \left(\left(0.5 \cdot {im}^{2}\right) \cdot \sin re + \color{blue}{\left(0.041666666666666664 \cdot {im}^{4}\right) \cdot \sin re}\right) \]

      6. distribute-rgt-out 96.3%

         \[\leadsto \sin re \cdot 1 + \color{blue}{\sin re \cdot \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot {im}^{4}\right)} \]

      7. distribute-lft-out 96.3%

         \[\leadsto \color{blue}{\sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot {im}^{4}\right)\right)} \]

      8. metadata-eval 96.3%

         \[\leadsto \sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot {im}^{\color{blue}{\left(2 \cdot 2\right)}}\right)\right) \]

      9. pow-sqr 96.3%

         \[\leadsto \sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)}\right)\right) \]

      10. associate-*r* 96.3%

          \[\leadsto \sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + \color{blue}{\left(0.041666666666666664 \cdot {im}^{2}\right) \cdot {im}^{2}}\right)\right) \]

      11. distribute-rgt-out 96.3%

          \[\leadsto \sin re \cdot \left(1 + \color{blue}{{im}^{2} \cdot \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)}\right) \]

      12. unpow2 96.3%

          \[\leadsto \sin re \cdot \left(1 + \color{blue}{\left(im \cdot im\right)} \cdot \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)\right) \]

      13. unpow2 96.3%

          \[\leadsto \sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \]

   6. Simplified 96.3%

      \[\leadsto \color{blue}{\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)} \]

   7. Taylor expanded in re around 0 68.0%

      \[\leadsto \color{blue}{\left(1 + \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right) \cdot {im}^{2}\right) \cdot re} \]
   8. Step-by-step derivation

      1. *-commutative 68.0%

         \[\leadsto \left(1 + \color{blue}{{im}^{2} \cdot \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)}\right) \cdot re \]

      2. unpow2 68.0%

         \[\leadsto \left(1 + \color{blue}{\left(im \cdot im\right)} \cdot \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)\right) \cdot re \]

      3. unpow2 68.0%

         \[\leadsto \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \cdot re \]

   9. Simplified 68.0%

      \[\leadsto \color{blue}{\left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right) \cdot re} \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 820:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 3.5 \cdot 10^{+61}:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \end{array} \]

Alternative 10: 56.3% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 340 \lor \neg \left(im \leq 1.85 \cdot 10^{+61}\right):\\ \;\;\;\;re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 340.0) (not (<= im 1.85e+61)))
   (* re (+ 1.0 (* (* im im) (+ 0.5 (* (* im im) 0.041666666666666664)))))
   (+
    0.08333333333333333
    (+ (/ 0.25 (* re re)) (* (* re re) 0.016666666666666666)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 340.0) || !(im <= 1.85e+61)) {
		tmp = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	} else {
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= 340.0d0) .or. (.not. (im <= 1.85d+61))) then
        tmp = re * (1.0d0 + ((im * im) * (0.5d0 + ((im * im) * 0.041666666666666664d0))))
    else
        tmp = 0.08333333333333333d0 + ((0.25d0 / (re * re)) + ((re * re) * 0.016666666666666666d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= 340.0) || !(im <= 1.85e+61)) {
		tmp = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	} else {
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= 340.0) or not (im <= 1.85e+61):
		tmp = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))))
	else:
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 340.0) || !(im <= 1.85e+61))
		tmp = Float64(re * Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(Float64(im * im) * 0.041666666666666664)))));
	else
		tmp = Float64(0.08333333333333333 + Float64(Float64(0.25 / Float64(re * re)) + Float64(Float64(re * re) * 0.016666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= 340.0) || ~((im <= 1.85e+61)))
		tmp = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	else
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 340.0], N[Not[LessEqual[im, 1.85e+61]], $MachinePrecision]], N[(re * N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.08333333333333333 + N[(N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision] + N[(N[(re * re), $MachinePrecision] * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 340 or 1.85000000000000001e61 < 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. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\sin re \cdot 0.5\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. associate-*l* 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 91.2%

      \[\leadsto \color{blue}{\sin re + \left(0.5 \cdot \left(\sin re \cdot {im}^{2}\right) + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right)} \]
   5. Step-by-step derivation

      1. *-rgt-identity 91.2%

         \[\leadsto \color{blue}{\sin re \cdot 1} + \left(0.5 \cdot \left(\sin re \cdot {im}^{2}\right) + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right) \]

      2. *-commutative 91.2%

         \[\leadsto \sin re \cdot 1 + \left(0.5 \cdot \color{blue}{\left({im}^{2} \cdot \sin re\right)} + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right) \]

      3. associate-*r* 91.2%

         \[\leadsto \sin re \cdot 1 + \left(\color{blue}{\left(0.5 \cdot {im}^{2}\right) \cdot \sin re} + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right) \]

      4. *-commutative 91.2%

         \[\leadsto \sin re \cdot 1 + \left(\left(0.5 \cdot {im}^{2}\right) \cdot \sin re + 0.041666666666666664 \cdot \color{blue}{\left({im}^{4} \cdot \sin re\right)}\right) \]

      5. associate-*r* 91.2%

         \[\leadsto \sin re \cdot 1 + \left(\left(0.5 \cdot {im}^{2}\right) \cdot \sin re + \color{blue}{\left(0.041666666666666664 \cdot {im}^{4}\right) \cdot \sin re}\right) \]

      6. distribute-rgt-out 91.2%

         \[\leadsto \sin re \cdot 1 + \color{blue}{\sin re \cdot \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot {im}^{4}\right)} \]

      7. distribute-lft-out 91.2%

         \[\leadsto \color{blue}{\sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot {im}^{4}\right)\right)} \]

      8. metadata-eval 91.2%

         \[\leadsto \sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot {im}^{\color{blue}{\left(2 \cdot 2\right)}}\right)\right) \]

      9. pow-sqr 91.2%

         \[\leadsto \sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)}\right)\right) \]

      10. associate-*r* 91.2%

          \[\leadsto \sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + \color{blue}{\left(0.041666666666666664 \cdot {im}^{2}\right) \cdot {im}^{2}}\right)\right) \]

      11. distribute-rgt-out 91.2%

          \[\leadsto \sin re \cdot \left(1 + \color{blue}{{im}^{2} \cdot \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)}\right) \]

      12. unpow2 91.2%

          \[\leadsto \sin re \cdot \left(1 + \color{blue}{\left(im \cdot im\right)} \cdot \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)\right) \]

      13. unpow2 91.2%

          \[\leadsto \sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \]

   6. Simplified 91.2%

      \[\leadsto \color{blue}{\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)} \]

   7. Taylor expanded in re around 0 59.3%

      \[\leadsto \color{blue}{\left(1 + \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right) \cdot {im}^{2}\right) \cdot re} \]
   8. Step-by-step derivation

      1. *-commutative 59.3%

         \[\leadsto \left(1 + \color{blue}{{im}^{2} \cdot \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)}\right) \cdot re \]

      2. unpow2 59.3%

         \[\leadsto \left(1 + \color{blue}{\left(im \cdot im\right)} \cdot \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)\right) \cdot re \]

      3. unpow2 59.3%

         \[\leadsto \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \cdot re \]

   9. Simplified 59.3%

      \[\leadsto \color{blue}{\left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right) \cdot re} \]

   if 340 < im < 1.85000000000000001e61

   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. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\sin re \cdot 0.5\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. associate-*l* 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 34.9%

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

   6. Taylor expanded in re around 0 56.4%

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

      1. associate-*r/ 56.4%

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

      2. metadata-eval 56.4%

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

      3. unpow2 56.4%

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

      4. *-commutative 56.4%

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

      5. unpow2 56.4%

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

   8. Simplified 56.4%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 340 \lor \neg \left(im \leq 1.85 \cdot 10^{+61}\right):\\ \;\;\;\;re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)\\ \end{array} \]

Alternative 11: 49.8% accurate, 18.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 650 \lor \neg \left(im \leq 1.92 \cdot 10^{+146}\right):\\ \;\;\;\;re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 650.0) (not (<= im 1.92e+146)))
   (* re (+ 1.0 (* 0.5 (* im im))))
   (+
    0.08333333333333333
    (+ (/ 0.25 (* re re)) (* (* re re) 0.016666666666666666)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 650.0) || !(im <= 1.92e+146)) {
		tmp = re * (1.0 + (0.5 * (im * im)));
	} else {
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= 650.0d0) .or. (.not. (im <= 1.92d+146))) then
        tmp = re * (1.0d0 + (0.5d0 * (im * im)))
    else
        tmp = 0.08333333333333333d0 + ((0.25d0 / (re * re)) + ((re * re) * 0.016666666666666666d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= 650.0) || !(im <= 1.92e+146)) {
		tmp = re * (1.0 + (0.5 * (im * im)));
	} else {
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= 650.0) or not (im <= 1.92e+146):
		tmp = re * (1.0 + (0.5 * (im * im)))
	else:
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 650.0) || !(im <= 1.92e+146))
		tmp = Float64(re * Float64(1.0 + Float64(0.5 * Float64(im * im))));
	else
		tmp = Float64(0.08333333333333333 + Float64(Float64(0.25 / Float64(re * re)) + Float64(Float64(re * re) * 0.016666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= 650.0) || ~((im <= 1.92e+146)))
		tmp = re * (1.0 + (0.5 * (im * im)));
	else
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 650.0], N[Not[LessEqual[im, 1.92e+146]], $MachinePrecision]], N[(re * N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.08333333333333333 + N[(N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision] + N[(N[(re * re), $MachinePrecision] * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 650 or 1.91999999999999993e146 < 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. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\sin re \cdot 0.5\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. associate-*l* 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 80.4%

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

      1. *-commutative 80.4%

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

      2. associate-*r* 80.4%

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

      3. distribute-rgt1-in 80.4%

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

      4. *-commutative 80.4%

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

      5. unpow2 80.4%

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

      6. associate-*l* 80.4%

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

   6. Simplified 80.4%

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

   7. Taylor expanded in re around 0 52.9%

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

      1. *-commutative 52.9%

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

      2. unpow2 52.9%

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

   9. Simplified 52.9%

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

   if 650 < im < 1.91999999999999993e146

   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. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\sin re \cdot 0.5\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. associate-*l* 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 16.7%

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

   6. Taylor expanded in re around 0 38.1%

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

      1. associate-*r/ 38.1%

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

      2. metadata-eval 38.1%

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

      3. unpow2 38.1%

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

      4. *-commutative 38.1%

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

      5. unpow2 38.1%

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

   8. Simplified 38.1%

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

4. Final simplification 51.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 650 \lor \neg \left(im \leq 1.92 \cdot 10^{+146}\right):\\ \;\;\;\;re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)\\ \end{array} \]

Alternative 12: 45.8% accurate, 23.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.75 \cdot 10^{-286} \lor \neg \left(re \leq 1.4 \cdot 10^{-193}\right):\\ \;\;\;\;re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{re \cdot re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re 1.75e-286) (not (<= re 1.4e-193)))
   (* re (+ 1.0 (* 0.5 (* im im))))
   (/ 0.25 (* re re))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= 1.75e-286) || !(re <= 1.4e-193)) {
		tmp = re * (1.0 + (0.5 * (im * im)));
	} 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 ((re <= 1.75d-286) .or. (.not. (re <= 1.4d-193))) then
        tmp = re * (1.0d0 + (0.5d0 * (im * im)))
    else
        tmp = 0.25d0 / (re * re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= 1.75e-286) || !(re <= 1.4e-193)) {
		tmp = re * (1.0 + (0.5 * (im * im)));
	} else {
		tmp = 0.25 / (re * re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= 1.75e-286) or not (re <= 1.4e-193):
		tmp = re * (1.0 + (0.5 * (im * im)))
	else:
		tmp = 0.25 / (re * re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= 1.75e-286) || !(re <= 1.4e-193))
		tmp = Float64(re * Float64(1.0 + Float64(0.5 * Float64(im * im))));
	else
		tmp = Float64(0.25 / Float64(re * re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= 1.75e-286) || ~((re <= 1.4e-193)))
		tmp = re * (1.0 + (0.5 * (im * im)));
	else
		tmp = 0.25 / (re * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, 1.75e-286], N[Not[LessEqual[re, 1.4e-193]], $MachinePrecision]], N[(re * N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.74999999999999994e-286 or 1.4000000000000001e-193 < re

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\sin re \cdot 0.5\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. associate-*l* 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 74.1%

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

      1. *-commutative 74.1%

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

      2. associate-*r* 74.1%

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

      3. distribute-rgt1-in 74.1%

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

      4. *-commutative 74.1%

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

      5. unpow2 74.1%

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

      6. associate-*l* 74.1%

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

   6. Simplified 74.1%

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

   7. Taylor expanded in re around 0 47.9%

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

      1. *-commutative 47.9%

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

      2. unpow2 47.9%

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

   9. Simplified 47.9%

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

   if 1.74999999999999994e-286 < re < 1.4000000000000001e-193

   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. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\sin re \cdot 0.5\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. associate-*l* 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 81.3%

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

   6. Taylor expanded in re around 0 81.3%

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

      1. unpow2 81.3%

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

   8. Simplified 81.3%

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

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.75 \cdot 10^{-286} \lor \neg \left(re \leq 1.4 \cdot 10^{-193}\right):\\ \;\;\;\;re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{re \cdot re}\\ \end{array} \]

Alternative 13: 34.9% accurate, 27.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 780000000000:\\ \;\;\;\;re\\ \mathbf{elif}\;im \leq 1.16 \cdot 10^{+73}:\\ \;\;\;\;\frac{0.25}{re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(re \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 780000000000.0)
   re
   (if (<= im 1.16e+73) (/ 0.25 (* re re)) (* 0.5 (* im (* re im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 780000000000.0) {
		tmp = re;
	} else if (im <= 1.16e+73) {
		tmp = 0.25 / (re * re);
	} else {
		tmp = 0.5 * (im * (re * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 780000000000.0d0) then
        tmp = re
    else if (im <= 1.16d+73) then
        tmp = 0.25d0 / (re * re)
    else
        tmp = 0.5d0 * (im * (re * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 780000000000.0) {
		tmp = re;
	} else if (im <= 1.16e+73) {
		tmp = 0.25 / (re * re);
	} else {
		tmp = 0.5 * (im * (re * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 780000000000.0:
		tmp = re
	elif im <= 1.16e+73:
		tmp = 0.25 / (re * re)
	else:
		tmp = 0.5 * (im * (re * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 780000000000.0)
		tmp = re;
	elseif (im <= 1.16e+73)
		tmp = Float64(0.25 / Float64(re * re));
	else
		tmp = Float64(0.5 * Float64(im * Float64(re * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 780000000000.0)
		tmp = re;
	elseif (im <= 1.16e+73)
		tmp = 0.25 / (re * re);
	else
		tmp = 0.5 * (im * (re * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 780000000000.0], re, If[LessEqual[im, 1.16e+73], N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[(re * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 7.8e11

   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. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\sin re \cdot 0.5\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. associate-*l* 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 63.3%

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

      1. associate-*r* 63.3%

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

   6. Simplified 63.3%

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

   7. Taylor expanded in im around 0 38.1%

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

   if 7.8e11 < im < 1.16000000000000007e73

   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. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\sin re \cdot 0.5\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. associate-*l* 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 34.4%

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

   6. Taylor expanded in re around 0 34.4%

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

      1. unpow2 34.4%

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

   8. Simplified 34.4%

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

   if 1.16000000000000007e73 < 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. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\sin re \cdot 0.5\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. associate-*l* 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 62.7%

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

      1. *-commutative 62.7%

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

      2. associate-*r* 62.7%

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

      3. distribute-rgt1-in 62.7%

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

      4. *-commutative 62.7%

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

      5. unpow2 62.7%

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

      6. associate-*l* 62.7%

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

   6. Simplified 62.7%

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

   7. Taylor expanded in im around inf 62.7%

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

      1. *-commutative 62.7%

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

      2. unpow2 62.7%

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

      3. associate-*l* 51.1%

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

   9. Simplified 51.1%

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

   10. Taylor expanded in re around 0 37.2%

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

4. Final simplification 37.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 780000000000:\\ \;\;\;\;re\\ \mathbf{elif}\;im \leq 1.16 \cdot 10^{+73}:\\ \;\;\;\;\frac{0.25}{re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(re \cdot im\right)\right)\\ \end{array} \]

Alternative 14: 37.9% accurate, 27.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 780000000000:\\ \;\;\;\;re\\ \mathbf{elif}\;im \leq 5.4 \cdot 10^{+71}:\\ \;\;\;\;\frac{0.25}{re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 780000000000.0)
   re
   (if (<= im 5.4e+71) (/ 0.25 (* re re)) (* 0.5 (* re (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 780000000000.0) {
		tmp = re;
	} else if (im <= 5.4e+71) {
		tmp = 0.25 / (re * 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 <= 780000000000.0d0) then
        tmp = re
    else if (im <= 5.4d+71) then
        tmp = 0.25d0 / (re * 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 <= 780000000000.0) {
		tmp = re;
	} else if (im <= 5.4e+71) {
		tmp = 0.25 / (re * re);
	} else {
		tmp = 0.5 * (re * (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 780000000000.0:
		tmp = re
	elif im <= 5.4e+71:
		tmp = 0.25 / (re * re)
	else:
		tmp = 0.5 * (re * (im * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 780000000000.0)
		tmp = re;
	elseif (im <= 5.4e+71)
		tmp = Float64(0.25 / Float64(re * 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 <= 780000000000.0)
		tmp = re;
	elseif (im <= 5.4e+71)
		tmp = 0.25 / (re * re);
	else
		tmp = 0.5 * (re * (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 780000000000.0], re, If[LessEqual[im, 5.4e+71], N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(re * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 7.8e11

   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. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\sin re \cdot 0.5\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. associate-*l* 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 63.3%

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

      1. associate-*r* 63.3%

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

   6. Simplified 63.3%

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

   7. Taylor expanded in im around 0 38.1%

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

   if 7.8e11 < im < 5.39999999999999993e71

   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. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\sin re \cdot 0.5\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. associate-*l* 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 34.4%

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

   6. Taylor expanded in re around 0 34.4%

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

      1. unpow2 34.4%

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

   8. Simplified 34.4%

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

   if 5.39999999999999993e71 < 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. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\sin re \cdot 0.5\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. associate-*l* 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 62.7%

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

      1. *-commutative 62.7%

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

      2. associate-*r* 62.7%

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

      3. distribute-rgt1-in 62.7%

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

      4. *-commutative 62.7%

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

      5. unpow2 62.7%

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

      6. associate-*l* 62.7%

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

   6. Simplified 62.7%

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

   7. Taylor expanded in im around inf 62.7%

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

      1. *-commutative 62.7%

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

      2. unpow2 62.7%

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

      3. associate-*l* 51.1%

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

   9. Simplified 51.1%

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

   10. Taylor expanded in re around 0 48.8%

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

       1. unpow2 48.8%

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

   12. Simplified 48.8%

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

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 780000000000:\\ \;\;\;\;re\\ \mathbf{elif}\;im \leq 5.4 \cdot 10^{+71}:\\ \;\;\;\;\frac{0.25}{re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 15: 37.9% accurate, 27.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 700000000000:\\ \;\;\;\;re\\ \mathbf{elif}\;im \leq 1.8 \cdot 10^{+71}:\\ \;\;\;\;\frac{0.5}{re} \cdot \frac{0.5}{re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 700000000000.0)
   re
   (if (<= im 1.8e+71) (* (/ 0.5 re) (/ 0.5 re)) (* 0.5 (* re (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 700000000000.0) {
		tmp = re;
	} else if (im <= 1.8e+71) {
		tmp = (0.5 / re) * (0.5 / 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 <= 700000000000.0d0) then
        tmp = re
    else if (im <= 1.8d+71) then
        tmp = (0.5d0 / re) * (0.5d0 / 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 <= 700000000000.0) {
		tmp = re;
	} else if (im <= 1.8e+71) {
		tmp = (0.5 / re) * (0.5 / re);
	} else {
		tmp = 0.5 * (re * (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 700000000000.0:
		tmp = re
	elif im <= 1.8e+71:
		tmp = (0.5 / re) * (0.5 / re)
	else:
		tmp = 0.5 * (re * (im * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 700000000000.0)
		tmp = re;
	elseif (im <= 1.8e+71)
		tmp = Float64(Float64(0.5 / re) * Float64(0.5 / 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 <= 700000000000.0)
		tmp = re;
	elseif (im <= 1.8e+71)
		tmp = (0.5 / re) * (0.5 / re);
	else
		tmp = 0.5 * (re * (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 700000000000.0], re, If[LessEqual[im, 1.8e+71], N[(N[(0.5 / re), $MachinePrecision] * N[(0.5 / re), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(re * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 7e11

   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. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\sin re \cdot 0.5\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. associate-*l* 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 63.3%

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

      1. associate-*r* 63.3%

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

   6. Simplified 63.3%

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

   7. Taylor expanded in im around 0 38.1%

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

   if 7e11 < im < 1.8e71

   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. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\sin re \cdot 0.5\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. associate-*l* 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 34.4%

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

   6. Taylor expanded in re around 0 34.4%

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

      1. unpow2 34.4%

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

   8. Simplified 34.4%

      \[\leadsto \color{blue}{\frac{0.25}{re \cdot re}} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 34.4%

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

      2. sqrt-div 34.4%

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

      3. metadata-eval 34.4%

         \[\leadsto \frac{\color{blue}{0.5}}{\sqrt{re \cdot re}} \cdot \sqrt{\frac{0.25}{re \cdot re}} \]

      4. sqrt-prod 34.3%

         \[\leadsto \frac{0.5}{\color{blue}{\sqrt{re} \cdot \sqrt{re}}} \cdot \sqrt{\frac{0.25}{re \cdot re}} \]

      5. add-sqr-sqrt 45.6%

         \[\leadsto \frac{0.5}{\color{blue}{re}} \cdot \sqrt{\frac{0.25}{re \cdot re}} \]

      6. sqrt-div 45.6%

         \[\leadsto \frac{0.5}{re} \cdot \color{blue}{\frac{\sqrt{0.25}}{\sqrt{re \cdot re}}} \]

      7. metadata-eval 45.6%

         \[\leadsto \frac{0.5}{re} \cdot \frac{\color{blue}{0.5}}{\sqrt{re \cdot re}} \]

      8. sqrt-prod 34.3%

         \[\leadsto \frac{0.5}{re} \cdot \frac{0.5}{\color{blue}{\sqrt{re} \cdot \sqrt{re}}} \]

      9. add-sqr-sqrt 34.4%

         \[\leadsto \frac{0.5}{re} \cdot \frac{0.5}{\color{blue}{re}} \]

   10. Applied egg-rr 34.4%

       \[\leadsto \color{blue}{\frac{0.5}{re} \cdot \frac{0.5}{re}} \]

   if 1.8e71 < 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. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\sin re \cdot 0.5\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. associate-*l* 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 62.7%

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

      1. *-commutative 62.7%

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

      2. associate-*r* 62.7%

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

      3. distribute-rgt1-in 62.7%

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

      4. *-commutative 62.7%

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

      5. unpow2 62.7%

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

      6. associate-*l* 62.7%

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

   6. Simplified 62.7%

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

   7. Taylor expanded in im around inf 62.7%

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

      1. *-commutative 62.7%

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

      2. unpow2 62.7%

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

      3. associate-*l* 51.1%

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

   9. Simplified 51.1%

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

   10. Taylor expanded in re around 0 48.8%

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

       1. unpow2 48.8%

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

   12. Simplified 48.8%

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

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 700000000000:\\ \;\;\;\;re\\ \mathbf{elif}\;im \leq 1.8 \cdot 10^{+71}:\\ \;\;\;\;\frac{0.5}{re} \cdot \frac{0.5}{re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 16: 30.4% accurate, 43.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 700000000000:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{re \cdot re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 700000000000.0) re (/ 0.25 (* re re))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 700000000000.0) {
		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 <= 700000000000.0d0) 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 <= 700000000000.0) {
		tmp = re;
	} else {
		tmp = 0.25 / (re * re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 700000000000.0:
		tmp = re
	else:
		tmp = 0.25 / (re * re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 700000000000.0)
		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 <= 700000000000.0)
		tmp = re;
	else
		tmp = 0.25 / (re * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 700000000000.0], re, N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 7e11

   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. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\sin re \cdot 0.5\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. associate-*l* 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 63.3%

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

      1. associate-*r* 63.3%

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

   6. Simplified 63.3%

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

   7. Taylor expanded in im around 0 38.1%

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

   if 7e11 < 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. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\sin re \cdot 0.5\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. associate-*l* 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 16.0%

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

   6. Taylor expanded in re around 0 15.8%

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

      1. unpow2 15.8%

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

   8. Simplified 15.8%

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

4. Final simplification 33.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 700000000000:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{re \cdot re}\\ \end{array} \]

Alternative 17: 3.9% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ -512 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 -512.0)

C

double code(double re, double im) {
	return -512.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = -512.0d0
end function

Java

public static double code(double re, double im) {
	return -512.0;
}

Python

def code(re, im):
	return -512.0

Julia

function code(re, im)
	return -512.0
end

MATLAB

function tmp = code(re, im)
	tmp = -512.0;
end

Wolfram

code[re_, im_] := -512.0

Derivation

1. Initial program 100.0%

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

   1. *-commutative 100.0%

      \[\leadsto \color{blue}{\left(\sin re \cdot 0.5\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

   2. associate-*l* 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

   3. sub0-neg 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in im around 0 88.1%

   \[\leadsto \color{blue}{\sin re + \left(0.5 \cdot \left(\sin re \cdot {im}^{2}\right) + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right)} \]
5. Step-by-step derivation

   1. *-rgt-identity 88.1%

      \[\leadsto \color{blue}{\sin re \cdot 1} + \left(0.5 \cdot \left(\sin re \cdot {im}^{2}\right) + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right) \]

   2. *-commutative 88.1%

      \[\leadsto \sin re \cdot 1 + \left(0.5 \cdot \color{blue}{\left({im}^{2} \cdot \sin re\right)} + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right) \]

   3. associate-*r* 88.1%

      \[\leadsto \sin re \cdot 1 + \left(\color{blue}{\left(0.5 \cdot {im}^{2}\right) \cdot \sin re} + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right) \]

   4. *-commutative 88.1%

      \[\leadsto \sin re \cdot 1 + \left(\left(0.5 \cdot {im}^{2}\right) \cdot \sin re + 0.041666666666666664 \cdot \color{blue}{\left({im}^{4} \cdot \sin re\right)}\right) \]

   5. associate-*r* 88.1%

      \[\leadsto \sin re \cdot 1 + \left(\left(0.5 \cdot {im}^{2}\right) \cdot \sin re + \color{blue}{\left(0.041666666666666664 \cdot {im}^{4}\right) \cdot \sin re}\right) \]

   6. distribute-rgt-out 88.1%

      \[\leadsto \sin re \cdot 1 + \color{blue}{\sin re \cdot \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot {im}^{4}\right)} \]

   7. distribute-lft-out 88.1%

      \[\leadsto \color{blue}{\sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot {im}^{4}\right)\right)} \]

   8. metadata-eval 88.1%

      \[\leadsto \sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot {im}^{\color{blue}{\left(2 \cdot 2\right)}}\right)\right) \]

   9. pow-sqr 88.1%

      \[\leadsto \sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)}\right)\right) \]

   10. associate-*r* 88.1%

       \[\leadsto \sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + \color{blue}{\left(0.041666666666666664 \cdot {im}^{2}\right) \cdot {im}^{2}}\right)\right) \]

   11. distribute-rgt-out 88.1%

       \[\leadsto \sin re \cdot \left(1 + \color{blue}{{im}^{2} \cdot \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)}\right) \]

   12. unpow2 88.1%

       \[\leadsto \sin re \cdot \left(1 + \color{blue}{\left(im \cdot im\right)} \cdot \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)\right) \]

   13. unpow2 88.1%

       \[\leadsto \sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \]

6. Simplified 88.1%

   \[\leadsto \color{blue}{\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)} \]

8. Applied egg-rr 3.4%

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

9. Final simplification 3.4%

   \[\leadsto -512 \]

Alternative 18: 4.9% accurate, 309.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 -1.0)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return -1.0

Julia

function code(re, im)
	return -1.0
end

MATLAB

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

Wolfram

code[re_, im_] := -1.0

Derivation

1. Initial program 100.0%

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

   1. *-commutative 100.0%

      \[\leadsto \color{blue}{\left(\sin re \cdot 0.5\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

   2. associate-*l* 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

   3. sub0-neg 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in im around 0 88.1%

   \[\leadsto \color{blue}{\sin re + \left(0.5 \cdot \left(\sin re \cdot {im}^{2}\right) + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right)} \]
5. Step-by-step derivation

   1. *-rgt-identity 88.1%

      \[\leadsto \color{blue}{\sin re \cdot 1} + \left(0.5 \cdot \left(\sin re \cdot {im}^{2}\right) + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right) \]

   2. *-commutative 88.1%

      \[\leadsto \sin re \cdot 1 + \left(0.5 \cdot \color{blue}{\left({im}^{2} \cdot \sin re\right)} + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right) \]

   3. associate-*r* 88.1%

      \[\leadsto \sin re \cdot 1 + \left(\color{blue}{\left(0.5 \cdot {im}^{2}\right) \cdot \sin re} + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right) \]

   4. *-commutative 88.1%

      \[\leadsto \sin re \cdot 1 + \left(\left(0.5 \cdot {im}^{2}\right) \cdot \sin re + 0.041666666666666664 \cdot \color{blue}{\left({im}^{4} \cdot \sin re\right)}\right) \]

   5. associate-*r* 88.1%

      \[\leadsto \sin re \cdot 1 + \left(\left(0.5 \cdot {im}^{2}\right) \cdot \sin re + \color{blue}{\left(0.041666666666666664 \cdot {im}^{4}\right) \cdot \sin re}\right) \]

   6. distribute-rgt-out 88.1%

      \[\leadsto \sin re \cdot 1 + \color{blue}{\sin re \cdot \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot {im}^{4}\right)} \]

   7. distribute-lft-out 88.1%

      \[\leadsto \color{blue}{\sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot {im}^{4}\right)\right)} \]

   8. metadata-eval 88.1%

      \[\leadsto \sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot {im}^{\color{blue}{\left(2 \cdot 2\right)}}\right)\right) \]

   9. pow-sqr 88.1%

      \[\leadsto \sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)}\right)\right) \]

   10. associate-*r* 88.1%

       \[\leadsto \sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + \color{blue}{\left(0.041666666666666664 \cdot {im}^{2}\right) \cdot {im}^{2}}\right)\right) \]

   11. distribute-rgt-out 88.1%

       \[\leadsto \sin re \cdot \left(1 + \color{blue}{{im}^{2} \cdot \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)}\right) \]

   12. unpow2 88.1%

       \[\leadsto \sin re \cdot \left(1 + \color{blue}{\left(im \cdot im\right)} \cdot \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)\right) \]

   13. unpow2 88.1%

       \[\leadsto \sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \]

6. Simplified 88.1%

   \[\leadsto \color{blue}{\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)} \]

8. Applied egg-rr 4.1%

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

9. Final simplification 4.1%

   \[\leadsto -1 \]

Alternative 19: 4.7% accurate, 309.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 -0.5)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return -0.5

Julia

function code(re, im)
	return -0.5
end

MATLAB

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

Wolfram

code[re_, im_] := -0.5

Derivation

1. Initial program 100.0%

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

   1. *-commutative 100.0%

      \[\leadsto \color{blue}{\left(\sin re \cdot 0.5\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

   2. associate-*l* 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

   3. sub0-neg 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in im around 0 88.1%

   \[\leadsto \color{blue}{\sin re + \left(0.5 \cdot \left(\sin re \cdot {im}^{2}\right) + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right)} \]
5. Step-by-step derivation

   1. *-rgt-identity 88.1%

      \[\leadsto \color{blue}{\sin re \cdot 1} + \left(0.5 \cdot \left(\sin re \cdot {im}^{2}\right) + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right) \]

   2. *-commutative 88.1%

      \[\leadsto \sin re \cdot 1 + \left(0.5 \cdot \color{blue}{\left({im}^{2} \cdot \sin re\right)} + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right) \]

   3. associate-*r* 88.1%

      \[\leadsto \sin re \cdot 1 + \left(\color{blue}{\left(0.5 \cdot {im}^{2}\right) \cdot \sin re} + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right) \]

   4. *-commutative 88.1%

      \[\leadsto \sin re \cdot 1 + \left(\left(0.5 \cdot {im}^{2}\right) \cdot \sin re + 0.041666666666666664 \cdot \color{blue}{\left({im}^{4} \cdot \sin re\right)}\right) \]

   5. associate-*r* 88.1%

      \[\leadsto \sin re \cdot 1 + \left(\left(0.5 \cdot {im}^{2}\right) \cdot \sin re + \color{blue}{\left(0.041666666666666664 \cdot {im}^{4}\right) \cdot \sin re}\right) \]

   6. distribute-rgt-out 88.1%

      \[\leadsto \sin re \cdot 1 + \color{blue}{\sin re \cdot \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot {im}^{4}\right)} \]

   7. distribute-lft-out 88.1%

      \[\leadsto \color{blue}{\sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot {im}^{4}\right)\right)} \]

   8. metadata-eval 88.1%

      \[\leadsto \sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot {im}^{\color{blue}{\left(2 \cdot 2\right)}}\right)\right) \]

   9. pow-sqr 88.1%

      \[\leadsto \sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)}\right)\right) \]

   10. associate-*r* 88.1%

       \[\leadsto \sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + \color{blue}{\left(0.041666666666666664 \cdot {im}^{2}\right) \cdot {im}^{2}}\right)\right) \]

   11. distribute-rgt-out 88.1%

       \[\leadsto \sin re \cdot \left(1 + \color{blue}{{im}^{2} \cdot \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)}\right) \]

   12. unpow2 88.1%

       \[\leadsto \sin re \cdot \left(1 + \color{blue}{\left(im \cdot im\right)} \cdot \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)\right) \]

   13. unpow2 88.1%

       \[\leadsto \sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \]

6. Simplified 88.1%

   \[\leadsto \color{blue}{\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)} \]

8. Applied egg-rr 4.1%

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

9. Final simplification 4.1%

   \[\leadsto -0.5 \]

Alternative 20: 4.1% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ 0.015625 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.015625)

C

double code(double re, double im) {
	return 0.015625;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.015625d0
end function

Java

public static double code(double re, double im) {
	return 0.015625;
}

Python

def code(re, im):
	return 0.015625

Julia

function code(re, im)
	return 0.015625
end

MATLAB

function tmp = code(re, im)
	tmp = 0.015625;
end

Wolfram

code[re_, im_] := 0.015625

Derivation

1. Initial program 100.0%

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

   1. *-commutative 100.0%

      \[\leadsto \color{blue}{\left(\sin re \cdot 0.5\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

   2. associate-*l* 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

   3. sub0-neg 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in im around 0 88.1%

   \[\leadsto \color{blue}{\sin re + \left(0.5 \cdot \left(\sin re \cdot {im}^{2}\right) + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right)} \]
5. Step-by-step derivation

   1. *-rgt-identity 88.1%

      \[\leadsto \color{blue}{\sin re \cdot 1} + \left(0.5 \cdot \left(\sin re \cdot {im}^{2}\right) + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right) \]

   2. *-commutative 88.1%

      \[\leadsto \sin re \cdot 1 + \left(0.5 \cdot \color{blue}{\left({im}^{2} \cdot \sin re\right)} + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right) \]

   3. associate-*r* 88.1%

      \[\leadsto \sin re \cdot 1 + \left(\color{blue}{\left(0.5 \cdot {im}^{2}\right) \cdot \sin re} + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right) \]

   4. *-commutative 88.1%

      \[\leadsto \sin re \cdot 1 + \left(\left(0.5 \cdot {im}^{2}\right) \cdot \sin re + 0.041666666666666664 \cdot \color{blue}{\left({im}^{4} \cdot \sin re\right)}\right) \]

   5. associate-*r* 88.1%

      \[\leadsto \sin re \cdot 1 + \left(\left(0.5 \cdot {im}^{2}\right) \cdot \sin re + \color{blue}{\left(0.041666666666666664 \cdot {im}^{4}\right) \cdot \sin re}\right) \]

   6. distribute-rgt-out 88.1%

      \[\leadsto \sin re \cdot 1 + \color{blue}{\sin re \cdot \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot {im}^{4}\right)} \]

   7. distribute-lft-out 88.1%

      \[\leadsto \color{blue}{\sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot {im}^{4}\right)\right)} \]

   8. metadata-eval 88.1%

      \[\leadsto \sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot {im}^{\color{blue}{\left(2 \cdot 2\right)}}\right)\right) \]

   9. pow-sqr 88.1%

      \[\leadsto \sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)}\right)\right) \]

   10. associate-*r* 88.1%

       \[\leadsto \sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + \color{blue}{\left(0.041666666666666664 \cdot {im}^{2}\right) \cdot {im}^{2}}\right)\right) \]

   11. distribute-rgt-out 88.1%

       \[\leadsto \sin re \cdot \left(1 + \color{blue}{{im}^{2} \cdot \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)}\right) \]

   12. unpow2 88.1%

       \[\leadsto \sin re \cdot \left(1 + \color{blue}{\left(im \cdot im\right)} \cdot \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)\right) \]

   13. unpow2 88.1%

       \[\leadsto \sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \]

6. Simplified 88.1%

   \[\leadsto \color{blue}{\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)} \]

8. Applied egg-rr 4.2%

   \[\leadsto \color{blue}{0.015625} \]

9. Final simplification 4.2%

   \[\leadsto 0.015625 \]

Alternative 21: 4.3% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ 0.125 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.125)

C

double code(double re, double im) {
	return 0.125;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.125d0
end function

Java

public static double code(double re, double im) {
	return 0.125;
}

Python

def code(re, im):
	return 0.125

Julia

function code(re, im)
	return 0.125
end

MATLAB

function tmp = code(re, im)
	tmp = 0.125;
end

Wolfram

code[re_, im_] := 0.125

Derivation

1. Initial program 100.0%

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

   1. *-commutative 100.0%

      \[\leadsto \color{blue}{\left(\sin re \cdot 0.5\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

   2. associate-*l* 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

   3. sub0-neg 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in im around 0 88.1%

   \[\leadsto \color{blue}{\sin re + \left(0.5 \cdot \left(\sin re \cdot {im}^{2}\right) + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right)} \]
5. Step-by-step derivation

   1. *-rgt-identity 88.1%

      \[\leadsto \color{blue}{\sin re \cdot 1} + \left(0.5 \cdot \left(\sin re \cdot {im}^{2}\right) + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right) \]

   2. *-commutative 88.1%

      \[\leadsto \sin re \cdot 1 + \left(0.5 \cdot \color{blue}{\left({im}^{2} \cdot \sin re\right)} + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right) \]

   3. associate-*r* 88.1%

      \[\leadsto \sin re \cdot 1 + \left(\color{blue}{\left(0.5 \cdot {im}^{2}\right) \cdot \sin re} + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right) \]

   4. *-commutative 88.1%

      \[\leadsto \sin re \cdot 1 + \left(\left(0.5 \cdot {im}^{2}\right) \cdot \sin re + 0.041666666666666664 \cdot \color{blue}{\left({im}^{4} \cdot \sin re\right)}\right) \]

   5. associate-*r* 88.1%

      \[\leadsto \sin re \cdot 1 + \left(\left(0.5 \cdot {im}^{2}\right) \cdot \sin re + \color{blue}{\left(0.041666666666666664 \cdot {im}^{4}\right) \cdot \sin re}\right) \]

   6. distribute-rgt-out 88.1%

      \[\leadsto \sin re \cdot 1 + \color{blue}{\sin re \cdot \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot {im}^{4}\right)} \]

   7. distribute-lft-out 88.1%

      \[\leadsto \color{blue}{\sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot {im}^{4}\right)\right)} \]

   8. metadata-eval 88.1%

      \[\leadsto \sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot {im}^{\color{blue}{\left(2 \cdot 2\right)}}\right)\right) \]

   9. pow-sqr 88.1%

      \[\leadsto \sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)}\right)\right) \]

   10. associate-*r* 88.1%

       \[\leadsto \sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + \color{blue}{\left(0.041666666666666664 \cdot {im}^{2}\right) \cdot {im}^{2}}\right)\right) \]

   11. distribute-rgt-out 88.1%

       \[\leadsto \sin re \cdot \left(1 + \color{blue}{{im}^{2} \cdot \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)}\right) \]

   12. unpow2 88.1%

       \[\leadsto \sin re \cdot \left(1 + \color{blue}{\left(im \cdot im\right)} \cdot \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)\right) \]

   13. unpow2 88.1%

       \[\leadsto \sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \]

6. Simplified 88.1%

   \[\leadsto \color{blue}{\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)} \]

8. Applied egg-rr 4.4%

   \[\leadsto \color{blue}{0.125} \]

9. Final simplification 4.4%

   \[\leadsto 0.125 \]

Alternative 22: 4.5% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ 0.25 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.25)

C

double code(double re, double im) {
	return 0.25;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.25d0
end function

Java

public static double code(double re, double im) {
	return 0.25;
}

Python

def code(re, im):
	return 0.25

Julia

function code(re, im)
	return 0.25
end

MATLAB

function tmp = code(re, im)
	tmp = 0.25;
end

Wolfram

code[re_, im_] := 0.25

Derivation

1. Initial program 100.0%

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

   1. *-commutative 100.0%

      \[\leadsto \color{blue}{\left(\sin re \cdot 0.5\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

   2. associate-*l* 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

   3. sub0-neg 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in im around 0 88.1%

   \[\leadsto \color{blue}{\sin re + \left(0.5 \cdot \left(\sin re \cdot {im}^{2}\right) + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right)} \]
5. Step-by-step derivation

   1. *-rgt-identity 88.1%

      \[\leadsto \color{blue}{\sin re \cdot 1} + \left(0.5 \cdot \left(\sin re \cdot {im}^{2}\right) + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right) \]

   2. *-commutative 88.1%

      \[\leadsto \sin re \cdot 1 + \left(0.5 \cdot \color{blue}{\left({im}^{2} \cdot \sin re\right)} + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right) \]

   3. associate-*r* 88.1%

      \[\leadsto \sin re \cdot 1 + \left(\color{blue}{\left(0.5 \cdot {im}^{2}\right) \cdot \sin re} + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right) \]

   4. *-commutative 88.1%

      \[\leadsto \sin re \cdot 1 + \left(\left(0.5 \cdot {im}^{2}\right) \cdot \sin re + 0.041666666666666664 \cdot \color{blue}{\left({im}^{4} \cdot \sin re\right)}\right) \]

   5. associate-*r* 88.1%

      \[\leadsto \sin re \cdot 1 + \left(\left(0.5 \cdot {im}^{2}\right) \cdot \sin re + \color{blue}{\left(0.041666666666666664 \cdot {im}^{4}\right) \cdot \sin re}\right) \]

   6. distribute-rgt-out 88.1%

      \[\leadsto \sin re \cdot 1 + \color{blue}{\sin re \cdot \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot {im}^{4}\right)} \]

   7. distribute-lft-out 88.1%

      \[\leadsto \color{blue}{\sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot {im}^{4}\right)\right)} \]

   8. metadata-eval 88.1%

      \[\leadsto \sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot {im}^{\color{blue}{\left(2 \cdot 2\right)}}\right)\right) \]

   9. pow-sqr 88.1%

      \[\leadsto \sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)}\right)\right) \]

   10. associate-*r* 88.1%

       \[\leadsto \sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + \color{blue}{\left(0.041666666666666664 \cdot {im}^{2}\right) \cdot {im}^{2}}\right)\right) \]

   11. distribute-rgt-out 88.1%

       \[\leadsto \sin re \cdot \left(1 + \color{blue}{{im}^{2} \cdot \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)}\right) \]

   12. unpow2 88.1%

       \[\leadsto \sin re \cdot \left(1 + \color{blue}{\left(im \cdot im\right)} \cdot \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)\right) \]

   13. unpow2 88.1%

       \[\leadsto \sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \]

6. Simplified 88.1%

   \[\leadsto \color{blue}{\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)} \]

8. Applied egg-rr 4.6%

   \[\leadsto \color{blue}{0.25} \]

9. Final simplification 4.6%

   \[\leadsto 0.25 \]

Alternative 23: 4.6% accurate, 309.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 0.5)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return 0.5

Julia

function code(re, im)
	return 0.5
end

MATLAB

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

Wolfram

code[re_, im_] := 0.5

Derivation

1. Initial program 100.0%

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

   1. *-commutative 100.0%

      \[\leadsto \color{blue}{\left(\sin re \cdot 0.5\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

   2. associate-*l* 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

   3. sub0-neg 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in im around 0 88.1%

   \[\leadsto \color{blue}{\sin re + \left(0.5 \cdot \left(\sin re \cdot {im}^{2}\right) + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right)} \]
5. Step-by-step derivation

   1. *-rgt-identity 88.1%

      \[\leadsto \color{blue}{\sin re \cdot 1} + \left(0.5 \cdot \left(\sin re \cdot {im}^{2}\right) + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right) \]

   2. *-commutative 88.1%

      \[\leadsto \sin re \cdot 1 + \left(0.5 \cdot \color{blue}{\left({im}^{2} \cdot \sin re\right)} + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right) \]

   3. associate-*r* 88.1%

      \[\leadsto \sin re \cdot 1 + \left(\color{blue}{\left(0.5 \cdot {im}^{2}\right) \cdot \sin re} + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right) \]

   4. *-commutative 88.1%

      \[\leadsto \sin re \cdot 1 + \left(\left(0.5 \cdot {im}^{2}\right) \cdot \sin re + 0.041666666666666664 \cdot \color{blue}{\left({im}^{4} \cdot \sin re\right)}\right) \]

   5. associate-*r* 88.1%

      \[\leadsto \sin re \cdot 1 + \left(\left(0.5 \cdot {im}^{2}\right) \cdot \sin re + \color{blue}{\left(0.041666666666666664 \cdot {im}^{4}\right) \cdot \sin re}\right) \]

   6. distribute-rgt-out 88.1%

      \[\leadsto \sin re \cdot 1 + \color{blue}{\sin re \cdot \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot {im}^{4}\right)} \]

   7. distribute-lft-out 88.1%

      \[\leadsto \color{blue}{\sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot {im}^{4}\right)\right)} \]

   8. metadata-eval 88.1%

      \[\leadsto \sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot {im}^{\color{blue}{\left(2 \cdot 2\right)}}\right)\right) \]

   9. pow-sqr 88.1%

      \[\leadsto \sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)}\right)\right) \]

   10. associate-*r* 88.1%

       \[\leadsto \sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + \color{blue}{\left(0.041666666666666664 \cdot {im}^{2}\right) \cdot {im}^{2}}\right)\right) \]

   11. distribute-rgt-out 88.1%

       \[\leadsto \sin re \cdot \left(1 + \color{blue}{{im}^{2} \cdot \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)}\right) \]

   12. unpow2 88.1%

       \[\leadsto \sin re \cdot \left(1 + \color{blue}{\left(im \cdot im\right)} \cdot \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)\right) \]

   13. unpow2 88.1%

       \[\leadsto \sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \]

6. Simplified 88.1%

   \[\leadsto \color{blue}{\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)} \]

8. Applied egg-rr 4.7%

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

9. Final simplification 4.7%

   \[\leadsto 0.5 \]

Alternative 24: 4.8% accurate, 309.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return 1.0

Julia

function code(re, im)
	return 1.0
end

MATLAB

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

Wolfram

code[re_, im_] := 1.0

Derivation

1. Initial program 100.0%

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

   1. *-commutative 100.0%

      \[\leadsto \color{blue}{\left(\sin re \cdot 0.5\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

   2. associate-*l* 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

   3. sub0-neg 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in im around 0 88.1%

   \[\leadsto \color{blue}{\sin re + \left(0.5 \cdot \left(\sin re \cdot {im}^{2}\right) + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right)} \]
5. Step-by-step derivation

   1. *-rgt-identity 88.1%

      \[\leadsto \color{blue}{\sin re \cdot 1} + \left(0.5 \cdot \left(\sin re \cdot {im}^{2}\right) + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right) \]

   2. *-commutative 88.1%

      \[\leadsto \sin re \cdot 1 + \left(0.5 \cdot \color{blue}{\left({im}^{2} \cdot \sin re\right)} + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right) \]

   3. associate-*r* 88.1%

      \[\leadsto \sin re \cdot 1 + \left(\color{blue}{\left(0.5 \cdot {im}^{2}\right) \cdot \sin re} + 0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\right) \]

   4. *-commutative 88.1%

      \[\leadsto \sin re \cdot 1 + \left(\left(0.5 \cdot {im}^{2}\right) \cdot \sin re + 0.041666666666666664 \cdot \color{blue}{\left({im}^{4} \cdot \sin re\right)}\right) \]

   5. associate-*r* 88.1%

      \[\leadsto \sin re \cdot 1 + \left(\left(0.5 \cdot {im}^{2}\right) \cdot \sin re + \color{blue}{\left(0.041666666666666664 \cdot {im}^{4}\right) \cdot \sin re}\right) \]

   6. distribute-rgt-out 88.1%

      \[\leadsto \sin re \cdot 1 + \color{blue}{\sin re \cdot \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot {im}^{4}\right)} \]

   7. distribute-lft-out 88.1%

      \[\leadsto \color{blue}{\sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot {im}^{4}\right)\right)} \]

   8. metadata-eval 88.1%

      \[\leadsto \sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot {im}^{\color{blue}{\left(2 \cdot 2\right)}}\right)\right) \]

   9. pow-sqr 88.1%

      \[\leadsto \sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + 0.041666666666666664 \cdot \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)}\right)\right) \]

   10. associate-*r* 88.1%

       \[\leadsto \sin re \cdot \left(1 + \left(0.5 \cdot {im}^{2} + \color{blue}{\left(0.041666666666666664 \cdot {im}^{2}\right) \cdot {im}^{2}}\right)\right) \]

   11. distribute-rgt-out 88.1%

       \[\leadsto \sin re \cdot \left(1 + \color{blue}{{im}^{2} \cdot \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)}\right) \]

   12. unpow2 88.1%

       \[\leadsto \sin re \cdot \left(1 + \color{blue}{\left(im \cdot im\right)} \cdot \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right)\right) \]

   13. unpow2 88.1%

       \[\leadsto \sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \]

6. Simplified 88.1%

   \[\leadsto \color{blue}{\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)} \]

8. Applied egg-rr 4.8%

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

9. Final simplification 4.8%

   \[\leadsto 1 \]

Alternative 25: 27.6% accurate, 309.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 re)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return re

Julia

function code(re, im)
	return re
end

MATLAB

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

Wolfram

code[re_, im_] := re

Derivation

1. Initial program 100.0%

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

   1. *-commutative 100.0%

      \[\leadsto \color{blue}{\left(\sin re \cdot 0.5\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

   2. associate-*l* 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

   3. sub0-neg 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in re around 0 64.8%

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

   1. associate-*r* 64.8%

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

6. Simplified 64.8%

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

7. Taylor expanded in im around 0 30.2%

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

8. Final simplification 30.2%

   \[\leadsto re \]

Reproduce

herbie shell --seed 2023189 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))