math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 9.3s

Alternatives: 22

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: 95.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;im \leq -2.35 \cdot 10^{+103}:\\ \;\;\;\;\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{elif}\;im \leq -0.0225:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 4.7 \cdot 10^{-5}:\\ \;\;\;\;\sin re \cdot \left(1 + im \cdot \left(0.5 \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 2.8 \cdot 10^{+70}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* re 0.5) (+ (exp (- im)) (exp im)))))
   (if (<= im -2.35e+103)
     (*
      (sin re)
      (+ 1.0 (* (* im im) (+ 0.5 (* (* im im) 0.041666666666666664)))))
     (if (<= im -0.0225)
       t_0
       (if (<= im 4.7e-5)
         (* (sin re) (+ 1.0 (* im (* 0.5 im))))
         (if (<= im 2.8e+70)
           t_0
           (* 0.041666666666666664 (* (sin re) (pow im 4.0)))))))))

C

double code(double re, double im) {
	double t_0 = (re * 0.5) * (exp(-im) + exp(im));
	double tmp;
	if (im <= -2.35e+103) {
		tmp = sin(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	} else if (im <= -0.0225) {
		tmp = t_0;
	} else if (im <= 4.7e-5) {
		tmp = sin(re) * (1.0 + (im * (0.5 * im)));
	} else if (im <= 2.8e+70) {
		tmp = t_0;
	} else {
		tmp = 0.041666666666666664 * (sin(re) * pow(im, 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (re * 0.5d0) * (exp(-im) + exp(im))
    if (im <= (-2.35d+103)) then
        tmp = sin(re) * (1.0d0 + ((im * im) * (0.5d0 + ((im * im) * 0.041666666666666664d0))))
    else if (im <= (-0.0225d0)) then
        tmp = t_0
    else if (im <= 4.7d-5) then
        tmp = sin(re) * (1.0d0 + (im * (0.5d0 * im)))
    else if (im <= 2.8d+70) then
        tmp = t_0
    else
        tmp = 0.041666666666666664d0 * (sin(re) * (im ** 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (re * 0.5) * (Math.exp(-im) + Math.exp(im));
	double tmp;
	if (im <= -2.35e+103) {
		tmp = Math.sin(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	} else if (im <= -0.0225) {
		tmp = t_0;
	} else if (im <= 4.7e-5) {
		tmp = Math.sin(re) * (1.0 + (im * (0.5 * im)));
	} else if (im <= 2.8e+70) {
		tmp = t_0;
	} else {
		tmp = 0.041666666666666664 * (Math.sin(re) * Math.pow(im, 4.0));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (re * 0.5) * (math.exp(-im) + math.exp(im))
	tmp = 0
	if im <= -2.35e+103:
		tmp = math.sin(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))))
	elif im <= -0.0225:
		tmp = t_0
	elif im <= 4.7e-5:
		tmp = math.sin(re) * (1.0 + (im * (0.5 * im)))
	elif im <= 2.8e+70:
		tmp = t_0
	else:
		tmp = 0.041666666666666664 * (math.sin(re) * math.pow(im, 4.0))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(re * 0.5) * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (im <= -2.35e+103)
		tmp = Float64(sin(re) * Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(Float64(im * im) * 0.041666666666666664)))));
	elseif (im <= -0.0225)
		tmp = t_0;
	elseif (im <= 4.7e-5)
		tmp = Float64(sin(re) * Float64(1.0 + Float64(im * Float64(0.5 * im))));
	elseif (im <= 2.8e+70)
		tmp = t_0;
	else
		tmp = Float64(0.041666666666666664 * Float64(sin(re) * (im ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (re * 0.5) * (exp(-im) + exp(im));
	tmp = 0.0;
	if (im <= -2.35e+103)
		tmp = sin(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	elseif (im <= -0.0225)
		tmp = t_0;
	elseif (im <= 4.7e-5)
		tmp = sin(re) * (1.0 + (im * (0.5 * im)));
	elseif (im <= 2.8e+70)
		tmp = t_0;
	else
		tmp = 0.041666666666666664 * (sin(re) * (im ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(re * 0.5), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -2.35e+103], 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], If[LessEqual[im, -0.0225], t$95$0, If[LessEqual[im, 4.7e-5], N[(N[Sin[re], $MachinePrecision] * N[(1.0 + N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.8e+70], t$95$0, N[(0.041666666666666664 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -2.35000000000000016e103

   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%

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

   if -2.35000000000000016e103 < im < -0.022499999999999999 or 4.69999999999999972e-5 < 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%

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

   4. Taylor expanded in re around 0 82.1%

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

      1. associate-*r* 82.1%

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

   6. Simplified 82.1%

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

   if -0.022499999999999999 < 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 0 100.0%

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

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

   if 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 98.1%

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

      1. *-rgt-identity 98.1%

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

      2. *-commutative 98.1%

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

      3. associate-*r* 98.1%

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

      4. *-commutative 98.1%

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

      5. associate-*r* 98.1%

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

      6. distribute-rgt-out 98.1%

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

      7. distribute-lft-out 98.1%

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

      8. metadata-eval 98.1%

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

      9. pow-sqr 98.1%

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

      10. associate-*r* 98.1%

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

      11. distribute-rgt-out 98.1%

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

      12. unpow2 98.1%

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

      13. unpow2 98.1%

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

   6. Simplified 98.1%

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

   7. Taylor expanded in im around inf 98.1%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.35 \cdot 10^{+103}:\\ \;\;\;\;\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{elif}\;im \leq -0.0225:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{elif}\;im \leq 4.7 \cdot 10^{-5}:\\ \;\;\;\;\sin re \cdot \left(1 + im \cdot \left(0.5 \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 2.8 \cdot 10^{+70}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\\ \end{array} \]

Alternative 3: 86.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(\sin re \cdot \left(im \cdot im\right)\right)\\ t_1 := 0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)\\ t_2 := re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{if}\;im \leq -5.8 \cdot 10^{+174}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -2.4 \cdot 10^{+68}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq -210000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 450:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 3.8 \cdot 10^{+55}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 1.8 \cdot 10^{+150}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (* (sin re) (* im im))))
        (t_1
         (+
          0.08333333333333333
          (+ (/ 0.25 (* re re)) (* (* re re) 0.016666666666666666))))
        (t_2
         (*
          re
          (+ 1.0 (* (* im im) (+ 0.5 (* (* im im) 0.041666666666666664)))))))
   (if (<= im -5.8e+174)
     t_0
     (if (<= im -2.4e+68)
       t_2
       (if (<= im -210000000000.0)
         t_1
         (if (<= im 450.0)
           (sin re)
           (if (<= im 3.8e+55) t_1 (if (<= im 1.8e+150) t_2 t_0))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (sin(re) * (im * im));
	double t_1 = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	double t_2 = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	double tmp;
	if (im <= -5.8e+174) {
		tmp = t_0;
	} else if (im <= -2.4e+68) {
		tmp = t_2;
	} else if (im <= -210000000000.0) {
		tmp = t_1;
	} else if (im <= 450.0) {
		tmp = sin(re);
	} else if (im <= 3.8e+55) {
		tmp = t_1;
	} else if (im <= 1.8e+150) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 0.5d0 * (sin(re) * (im * im))
    t_1 = 0.08333333333333333d0 + ((0.25d0 / (re * re)) + ((re * re) * 0.016666666666666666d0))
    t_2 = re * (1.0d0 + ((im * im) * (0.5d0 + ((im * im) * 0.041666666666666664d0))))
    if (im <= (-5.8d+174)) then
        tmp = t_0
    else if (im <= (-2.4d+68)) then
        tmp = t_2
    else if (im <= (-210000000000.0d0)) then
        tmp = t_1
    else if (im <= 450.0d0) then
        tmp = sin(re)
    else if (im <= 3.8d+55) then
        tmp = t_1
    else if (im <= 1.8d+150) then
        tmp = t_2
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * (Math.sin(re) * (im * im));
	double t_1 = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	double t_2 = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	double tmp;
	if (im <= -5.8e+174) {
		tmp = t_0;
	} else if (im <= -2.4e+68) {
		tmp = t_2;
	} else if (im <= -210000000000.0) {
		tmp = t_1;
	} else if (im <= 450.0) {
		tmp = Math.sin(re);
	} else if (im <= 3.8e+55) {
		tmp = t_1;
	} else if (im <= 1.8e+150) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * (math.sin(re) * (im * im))
	t_1 = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666))
	t_2 = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))))
	tmp = 0
	if im <= -5.8e+174:
		tmp = t_0
	elif im <= -2.4e+68:
		tmp = t_2
	elif im <= -210000000000.0:
		tmp = t_1
	elif im <= 450.0:
		tmp = math.sin(re)
	elif im <= 3.8e+55:
		tmp = t_1
	elif im <= 1.8e+150:
		tmp = t_2
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(sin(re) * Float64(im * im)))
	t_1 = Float64(0.08333333333333333 + Float64(Float64(0.25 / Float64(re * re)) + Float64(Float64(re * re) * 0.016666666666666666)))
	t_2 = Float64(re * Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(Float64(im * im) * 0.041666666666666664)))))
	tmp = 0.0
	if (im <= -5.8e+174)
		tmp = t_0;
	elseif (im <= -2.4e+68)
		tmp = t_2;
	elseif (im <= -210000000000.0)
		tmp = t_1;
	elseif (im <= 450.0)
		tmp = sin(re);
	elseif (im <= 3.8e+55)
		tmp = t_1;
	elseif (im <= 1.8e+150)
		tmp = t_2;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (sin(re) * (im * im));
	t_1 = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	t_2 = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	tmp = 0.0;
	if (im <= -5.8e+174)
		tmp = t_0;
	elseif (im <= -2.4e+68)
		tmp = t_2;
	elseif (im <= -210000000000.0)
		tmp = t_1;
	elseif (im <= 450.0)
		tmp = sin(re);
	elseif (im <= 3.8e+55)
		tmp = t_1;
	elseif (im <= 1.8e+150)
		tmp = t_2;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(N[Sin[re], $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.08333333333333333 + N[(N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision] + N[(N[(re * re), $MachinePrecision] * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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, -5.8e+174], t$95$0, If[LessEqual[im, -2.4e+68], t$95$2, If[LessEqual[im, -210000000000.0], t$95$1, If[LessEqual[im, 450.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 3.8e+55], t$95$1, If[LessEqual[im, 1.8e+150], t$95$2, t$95$0]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -5.7999999999999999e174 or 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 98.3%

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

      1. *-commutative 98.3%

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

      2. associate-*r* 98.3%

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

      3. distribute-rgt1-in 98.3%

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

      4. *-commutative 98.3%

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

      5. unpow2 98.3%

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

      6. associate-*l* 98.3%

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

   6. Simplified 98.3%

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

   7. Taylor expanded in im around inf 98.3%

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

      1. *-commutative 98.3%

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

      2. unpow2 98.3%

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

   9. Simplified 98.3%

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

   if -5.7999999999999999e174 < im < -2.40000000000000008e68 or 3.8e55 < 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 92.5%

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

      1. *-rgt-identity 92.5%

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

      2. *-commutative 92.5%

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

      3. associate-*r* 92.5%

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

      4. *-commutative 92.5%

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

      5. associate-*r* 92.5%

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

      6. distribute-rgt-out 92.5%

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

      7. distribute-lft-out 92.5%

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

      8. metadata-eval 92.5%

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

      9. pow-sqr 92.5%

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

      10. associate-*r* 92.5%

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

      11. distribute-rgt-out 92.5%

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

      12. unpow2 92.5%

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

      13. unpow2 92.5%

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

   6. Simplified 92.5%

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

   7. Taylor expanded in re around 0 65.4%

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

      1. *-commutative 65.4%

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

      2. unpow2 65.4%

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

      3. unpow2 65.4%

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

   9. Simplified 65.4%

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

   if -2.40000000000000008e68 < im < -2.1e11 or 450 < im < 3.8e55

   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 32.1%

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

   6. Taylor expanded in re around 0 47.4%

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

      1. associate-*r/ 47.4%

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

      2. metadata-eval 47.4%

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

      3. unpow2 47.4%

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

      4. *-commutative 47.4%

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

      5. unpow2 47.4%

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

   8. Simplified 47.4%

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

   if -2.1e11 < im < 450

   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 97.8%

      \[\leadsto \color{blue}{\sin re} \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -5.8 \cdot 10^{+174}:\\ \;\;\;\;0.5 \cdot \left(\sin re \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq -2.4 \cdot 10^{+68}:\\ \;\;\;\;re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{elif}\;im \leq -210000000000:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)\\ \mathbf{elif}\;im \leq 450:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 3.8 \cdot 10^{+55}:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)\\ \mathbf{elif}\;im \leq 1.8 \cdot 10^{+150}:\\ \;\;\;\;re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sin re \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 4: 86.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(\sin re \cdot \left(im \cdot im\right)\right)\\ t_1 := 0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)\\ t_2 := re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{if}\;im \leq -5.8 \cdot 10^{+174}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -1.2 \cdot 10^{+71}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq -2.7 \cdot 10^{+21}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 700:\\ \;\;\;\;\sin re \cdot \left(1 + im \cdot \left(0.5 \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 4.8 \cdot 10^{+54}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 1.8 \cdot 10^{+150}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (* (sin re) (* im im))))
        (t_1
         (+
          0.08333333333333333
          (+ (/ 0.25 (* re re)) (* (* re re) 0.016666666666666666))))
        (t_2
         (*
          re
          (+ 1.0 (* (* im im) (+ 0.5 (* (* im im) 0.041666666666666664)))))))
   (if (<= im -5.8e+174)
     t_0
     (if (<= im -1.2e+71)
       t_2
       (if (<= im -2.7e+21)
         t_1
         (if (<= im 700.0)
           (* (sin re) (+ 1.0 (* im (* 0.5 im))))
           (if (<= im 4.8e+54) t_1 (if (<= im 1.8e+150) t_2 t_0))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (sin(re) * (im * im));
	double t_1 = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	double t_2 = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	double tmp;
	if (im <= -5.8e+174) {
		tmp = t_0;
	} else if (im <= -1.2e+71) {
		tmp = t_2;
	} else if (im <= -2.7e+21) {
		tmp = t_1;
	} else if (im <= 700.0) {
		tmp = sin(re) * (1.0 + (im * (0.5 * im)));
	} else if (im <= 4.8e+54) {
		tmp = t_1;
	} else if (im <= 1.8e+150) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 0.5d0 * (sin(re) * (im * im))
    t_1 = 0.08333333333333333d0 + ((0.25d0 / (re * re)) + ((re * re) * 0.016666666666666666d0))
    t_2 = re * (1.0d0 + ((im * im) * (0.5d0 + ((im * im) * 0.041666666666666664d0))))
    if (im <= (-5.8d+174)) then
        tmp = t_0
    else if (im <= (-1.2d+71)) then
        tmp = t_2
    else if (im <= (-2.7d+21)) then
        tmp = t_1
    else if (im <= 700.0d0) then
        tmp = sin(re) * (1.0d0 + (im * (0.5d0 * im)))
    else if (im <= 4.8d+54) then
        tmp = t_1
    else if (im <= 1.8d+150) then
        tmp = t_2
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * (Math.sin(re) * (im * im));
	double t_1 = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	double t_2 = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	double tmp;
	if (im <= -5.8e+174) {
		tmp = t_0;
	} else if (im <= -1.2e+71) {
		tmp = t_2;
	} else if (im <= -2.7e+21) {
		tmp = t_1;
	} else if (im <= 700.0) {
		tmp = Math.sin(re) * (1.0 + (im * (0.5 * im)));
	} else if (im <= 4.8e+54) {
		tmp = t_1;
	} else if (im <= 1.8e+150) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * (math.sin(re) * (im * im))
	t_1 = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666))
	t_2 = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))))
	tmp = 0
	if im <= -5.8e+174:
		tmp = t_0
	elif im <= -1.2e+71:
		tmp = t_2
	elif im <= -2.7e+21:
		tmp = t_1
	elif im <= 700.0:
		tmp = math.sin(re) * (1.0 + (im * (0.5 * im)))
	elif im <= 4.8e+54:
		tmp = t_1
	elif im <= 1.8e+150:
		tmp = t_2
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(sin(re) * Float64(im * im)))
	t_1 = Float64(0.08333333333333333 + Float64(Float64(0.25 / Float64(re * re)) + Float64(Float64(re * re) * 0.016666666666666666)))
	t_2 = Float64(re * Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(Float64(im * im) * 0.041666666666666664)))))
	tmp = 0.0
	if (im <= -5.8e+174)
		tmp = t_0;
	elseif (im <= -1.2e+71)
		tmp = t_2;
	elseif (im <= -2.7e+21)
		tmp = t_1;
	elseif (im <= 700.0)
		tmp = Float64(sin(re) * Float64(1.0 + Float64(im * Float64(0.5 * im))));
	elseif (im <= 4.8e+54)
		tmp = t_1;
	elseif (im <= 1.8e+150)
		tmp = t_2;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (sin(re) * (im * im));
	t_1 = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	t_2 = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	tmp = 0.0;
	if (im <= -5.8e+174)
		tmp = t_0;
	elseif (im <= -1.2e+71)
		tmp = t_2;
	elseif (im <= -2.7e+21)
		tmp = t_1;
	elseif (im <= 700.0)
		tmp = sin(re) * (1.0 + (im * (0.5 * im)));
	elseif (im <= 4.8e+54)
		tmp = t_1;
	elseif (im <= 1.8e+150)
		tmp = t_2;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(N[Sin[re], $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.08333333333333333 + N[(N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision] + N[(N[(re * re), $MachinePrecision] * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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, -5.8e+174], t$95$0, If[LessEqual[im, -1.2e+71], t$95$2, If[LessEqual[im, -2.7e+21], t$95$1, If[LessEqual[im, 700.0], N[(N[Sin[re], $MachinePrecision] * N[(1.0 + N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 4.8e+54], t$95$1, If[LessEqual[im, 1.8e+150], t$95$2, t$95$0]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -5.7999999999999999e174 or 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 98.3%

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

      1. *-commutative 98.3%

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

      2. associate-*r* 98.3%

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

      3. distribute-rgt1-in 98.3%

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

      4. *-commutative 98.3%

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

      5. unpow2 98.3%

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

      6. associate-*l* 98.3%

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

   6. Simplified 98.3%

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

   7. Taylor expanded in im around inf 98.3%

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

      1. *-commutative 98.3%

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

      2. unpow2 98.3%

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

   9. Simplified 98.3%

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

   if -5.7999999999999999e174 < im < -1.1999999999999999e71 or 4.79999999999999997e54 < 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 92.5%

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

      1. *-rgt-identity 92.5%

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

      2. *-commutative 92.5%

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

      3. associate-*r* 92.5%

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

      4. *-commutative 92.5%

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

      5. associate-*r* 92.5%

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

      6. distribute-rgt-out 92.5%

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

      7. distribute-lft-out 92.5%

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

      8. metadata-eval 92.5%

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

      9. pow-sqr 92.5%

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

      10. associate-*r* 92.5%

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

      11. distribute-rgt-out 92.5%

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

      12. unpow2 92.5%

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

      13. unpow2 92.5%

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

   6. Simplified 92.5%

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

   7. Taylor expanded in re around 0 65.4%

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

      1. *-commutative 65.4%

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

      2. unpow2 65.4%

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

      3. unpow2 65.4%

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

   9. Simplified 65.4%

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

   if -1.1999999999999999e71 < im < -2.7e21 or 700 < im < 4.79999999999999997e54

   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 39.2%

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

   6. Taylor expanded in re around 0 58.0%

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

      1. associate-*r/ 58.0%

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

      2. metadata-eval 58.0%

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

      3. unpow2 58.0%

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

      4. *-commutative 58.0%

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

      5. unpow2 58.0%

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

   8. Simplified 58.0%

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

   if -2.7e21 < im < 700

   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 94.9%

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

      1. *-commutative 94.9%

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

      2. associate-*r* 94.9%

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

      3. distribute-rgt1-in 94.9%

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

      4. *-commutative 94.9%

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

      5. unpow2 94.9%

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

      6. associate-*l* 94.9%

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

   6. Simplified 94.9%

      \[\leadsto \color{blue}{\left(im \cdot \left(im \cdot 0.5\right) + 1\right) \cdot \sin re} \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -5.8 \cdot 10^{+174}:\\ \;\;\;\;0.5 \cdot \left(\sin re \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq -1.2 \cdot 10^{+71}:\\ \;\;\;\;re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{elif}\;im \leq -2.7 \cdot 10^{+21}:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)\\ \mathbf{elif}\;im \leq 700:\\ \;\;\;\;\sin re \cdot \left(1 + im \cdot \left(0.5 \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 4.8 \cdot 10^{+54}:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)\\ \mathbf{elif}\;im \leq 1.8 \cdot 10^{+150}:\\ \;\;\;\;re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sin re \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 5: 82.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)\\ t_1 := re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{if}\;im \leq -3.8 \cdot 10^{+71}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -210000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 600:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.65 \cdot 10^{+54}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{+193}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(\sin re \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (+
          0.08333333333333333
          (+ (/ 0.25 (* re re)) (* (* re re) 0.016666666666666666))))
        (t_1
         (*
          re
          (+ 1.0 (* (* im im) (+ 0.5 (* (* im im) 0.041666666666666664)))))))
   (if (<= im -3.8e+71)
     t_1
     (if (<= im -210000000000.0)
       t_0
       (if (<= im 600.0)
         (sin re)
         (if (<= im 1.65e+54)
           t_0
           (if (<= im 2.5e+193) t_1 (* 0.5 (* im (* (sin re) im))))))))))

C

double code(double re, double im) {
	double t_0 = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	double t_1 = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	double tmp;
	if (im <= -3.8e+71) {
		tmp = t_1;
	} else if (im <= -210000000000.0) {
		tmp = t_0;
	} else if (im <= 600.0) {
		tmp = sin(re);
	} else if (im <= 1.65e+54) {
		tmp = t_0;
	} else if (im <= 2.5e+193) {
		tmp = t_1;
	} 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.08333333333333333d0 + ((0.25d0 / (re * re)) + ((re * re) * 0.016666666666666666d0))
    t_1 = re * (1.0d0 + ((im * im) * (0.5d0 + ((im * im) * 0.041666666666666664d0))))
    if (im <= (-3.8d+71)) then
        tmp = t_1
    else if (im <= (-210000000000.0d0)) then
        tmp = t_0
    else if (im <= 600.0d0) then
        tmp = sin(re)
    else if (im <= 1.65d+54) then
        tmp = t_0
    else if (im <= 2.5d+193) then
        tmp = t_1
    else
        tmp = 0.5d0 * (im * (sin(re) * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	double t_1 = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	double tmp;
	if (im <= -3.8e+71) {
		tmp = t_1;
	} else if (im <= -210000000000.0) {
		tmp = t_0;
	} else if (im <= 600.0) {
		tmp = Math.sin(re);
	} else if (im <= 1.65e+54) {
		tmp = t_0;
	} else if (im <= 2.5e+193) {
		tmp = t_1;
	} else {
		tmp = 0.5 * (im * (Math.sin(re) * im));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666))
	t_1 = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))))
	tmp = 0
	if im <= -3.8e+71:
		tmp = t_1
	elif im <= -210000000000.0:
		tmp = t_0
	elif im <= 600.0:
		tmp = math.sin(re)
	elif im <= 1.65e+54:
		tmp = t_0
	elif im <= 2.5e+193:
		tmp = t_1
	else:
		tmp = 0.5 * (im * (math.sin(re) * im))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.08333333333333333 + Float64(Float64(0.25 / Float64(re * re)) + Float64(Float64(re * re) * 0.016666666666666666)))
	t_1 = Float64(re * Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(Float64(im * im) * 0.041666666666666664)))))
	tmp = 0.0
	if (im <= -3.8e+71)
		tmp = t_1;
	elseif (im <= -210000000000.0)
		tmp = t_0;
	elseif (im <= 600.0)
		tmp = sin(re);
	elseif (im <= 1.65e+54)
		tmp = t_0;
	elseif (im <= 2.5e+193)
		tmp = t_1;
	else
		tmp = Float64(0.5 * Float64(im * Float64(sin(re) * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	t_1 = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	tmp = 0.0;
	if (im <= -3.8e+71)
		tmp = t_1;
	elseif (im <= -210000000000.0)
		tmp = t_0;
	elseif (im <= 600.0)
		tmp = sin(re);
	elseif (im <= 1.65e+54)
		tmp = t_0;
	elseif (im <= 2.5e+193)
		tmp = t_1;
	else
		tmp = 0.5 * (im * (sin(re) * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.08333333333333333 + N[(N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision] + N[(N[(re * re), $MachinePrecision] * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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, -3.8e+71], t$95$1, If[LessEqual[im, -210000000000.0], t$95$0, If[LessEqual[im, 600.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.65e+54], t$95$0, If[LessEqual[im, 2.5e+193], t$95$1, N[(0.5 * N[(im * N[(N[Sin[re], $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -3.8000000000000001e71 or 1.65e54 < im < 2.49999999999999986e193

   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 95.3%

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

      1. *-rgt-identity 95.3%

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

      2. *-commutative 95.3%

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

      3. associate-*r* 95.3%

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

      4. *-commutative 95.3%

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

      5. associate-*r* 95.3%

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

      6. distribute-rgt-out 95.3%

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

      7. distribute-lft-out 95.3%

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

      8. metadata-eval 95.3%

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

      9. pow-sqr 95.3%

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

      10. associate-*r* 95.3%

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

      11. distribute-rgt-out 95.3%

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

      12. unpow2 95.3%

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

      13. unpow2 95.3%

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

   6. Simplified 95.3%

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

   7. Taylor expanded in re around 0 69.7%

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

      1. *-commutative 69.7%

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

      2. unpow2 69.7%

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

      3. unpow2 69.7%

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

   9. Simplified 69.7%

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

   if -3.8000000000000001e71 < im < -2.1e11 or 600 < im < 1.65e54

   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 32.1%

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

   6. Taylor expanded in re around 0 47.4%

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

      1. associate-*r/ 47.4%

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

      2. metadata-eval 47.4%

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

      3. unpow2 47.4%

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

      4. *-commutative 47.4%

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

      5. unpow2 47.4%

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

   8. Simplified 47.4%

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

   if -2.1e11 < 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 97.8%

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

   if 2.49999999999999986e193 < 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 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3.8 \cdot 10^{+71}:\\ \;\;\;\;re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{elif}\;im \leq -210000000000:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)\\ \mathbf{elif}\;im \leq 600:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.65 \cdot 10^{+54}:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{+193}:\\ \;\;\;\;re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(\sin re \cdot im\right)\right)\\ \end{array} \]

Alternative 6: 88.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (sin re) (+ 1.0 (* (* im im) (+ 0.5 (* (* im im) 0.041666666666666664))))))

C

double code(double re, double im) {
	return sin(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = sin(re) * (1.0d0 + ((im * im) * (0.5d0 + ((im * im) * 0.041666666666666664d0))))
end function

Java

public static double code(double re, double im) {
	return Math.sin(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
}

Python

def code(re, im):
	return math.sin(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))))

Julia

function code(re, im)
	return Float64(sin(re) * Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(Float64(im * im) * 0.041666666666666664)))))
end

MATLAB

function tmp = code(re, im)
	tmp = sin(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
end

Wolfram

code[re_, im_] := 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. 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%

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

7. Final simplification 88.1%

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

Alternative 7: 81.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)\\ t_1 := re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{if}\;im \leq -5.6 \cdot 10^{+69}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -210000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 660:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.45 \cdot 10^{+57}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (+
          0.08333333333333333
          (+ (/ 0.25 (* re re)) (* (* re re) 0.016666666666666666))))
        (t_1
         (*
          re
          (+ 1.0 (* (* im im) (+ 0.5 (* (* im im) 0.041666666666666664)))))))
   (if (<= im -5.6e+69)
     t_1
     (if (<= im -210000000000.0)
       t_0
       (if (<= im 660.0) (sin re) (if (<= im 1.45e+57) t_0 t_1))))))

C

double code(double re, double im) {
	double t_0 = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	double t_1 = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	double tmp;
	if (im <= -5.6e+69) {
		tmp = t_1;
	} else if (im <= -210000000000.0) {
		tmp = t_0;
	} else if (im <= 660.0) {
		tmp = sin(re);
	} else if (im <= 1.45e+57) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.08333333333333333d0 + ((0.25d0 / (re * re)) + ((re * re) * 0.016666666666666666d0))
    t_1 = re * (1.0d0 + ((im * im) * (0.5d0 + ((im * im) * 0.041666666666666664d0))))
    if (im <= (-5.6d+69)) then
        tmp = t_1
    else if (im <= (-210000000000.0d0)) then
        tmp = t_0
    else if (im <= 660.0d0) then
        tmp = sin(re)
    else if (im <= 1.45d+57) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	double t_1 = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	double tmp;
	if (im <= -5.6e+69) {
		tmp = t_1;
	} else if (im <= -210000000000.0) {
		tmp = t_0;
	} else if (im <= 660.0) {
		tmp = Math.sin(re);
	} else if (im <= 1.45e+57) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666))
	t_1 = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))))
	tmp = 0
	if im <= -5.6e+69:
		tmp = t_1
	elif im <= -210000000000.0:
		tmp = t_0
	elif im <= 660.0:
		tmp = math.sin(re)
	elif im <= 1.45e+57:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.08333333333333333 + Float64(Float64(0.25 / Float64(re * re)) + Float64(Float64(re * re) * 0.016666666666666666)))
	t_1 = Float64(re * Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(Float64(im * im) * 0.041666666666666664)))))
	tmp = 0.0
	if (im <= -5.6e+69)
		tmp = t_1;
	elseif (im <= -210000000000.0)
		tmp = t_0;
	elseif (im <= 660.0)
		tmp = sin(re);
	elseif (im <= 1.45e+57)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	t_1 = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	tmp = 0.0;
	if (im <= -5.6e+69)
		tmp = t_1;
	elseif (im <= -210000000000.0)
		tmp = t_0;
	elseif (im <= 660.0)
		tmp = sin(re);
	elseif (im <= 1.45e+57)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.08333333333333333 + N[(N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision] + N[(N[(re * re), $MachinePrecision] * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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, -5.6e+69], t$95$1, If[LessEqual[im, -210000000000.0], t$95$0, If[LessEqual[im, 660.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.45e+57], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -5.59999999999999964e69 or 1.4500000000000001e57 < 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 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 -5.59999999999999964e69 < im < -2.1e11 or 660 < im < 1.4500000000000001e57

   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 32.1%

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

   6. Taylor expanded in re around 0 47.4%

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

      1. associate-*r/ 47.4%

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

      2. metadata-eval 47.4%

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

      3. unpow2 47.4%

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

      4. *-commutative 47.4%

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

      5. unpow2 47.4%

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

   8. Simplified 47.4%

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

   if -2.1e11 < 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 97.8%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -5.6 \cdot 10^{+69}:\\ \;\;\;\;re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{elif}\;im \leq -210000000000:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)\\ \mathbf{elif}\;im \leq 660:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.45 \cdot 10^{+57}:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \end{array} \]

Alternative 8: 50.6% accurate, 14.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -2.8 \cdot 10^{+138}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq -5 \cdot 10^{+41} \lor \neg \left(im \leq 390\right) \land im \leq 5.2 \cdot 10^{+146}:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -2.8e+138)
   (* 0.5 (* re (* im im)))
   (if (or (<= im -5e+41) (and (not (<= im 390.0)) (<= im 5.2e+146)))
     (+
      0.08333333333333333
      (+ (/ 0.25 (* re re)) (* (* re re) 0.016666666666666666)))
     (* re (+ 1.0 (* 0.5 (* im im)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -2.8e+138) {
		tmp = 0.5 * (re * (im * im));
	} else if ((im <= -5e+41) || (!(im <= 390.0) && (im <= 5.2e+146))) {
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	} else {
		tmp = re * (1.0 + (0.5 * (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 <= (-2.8d+138)) then
        tmp = 0.5d0 * (re * (im * im))
    else if ((im <= (-5d+41)) .or. (.not. (im <= 390.0d0)) .and. (im <= 5.2d+146)) then
        tmp = 0.08333333333333333d0 + ((0.25d0 / (re * re)) + ((re * re) * 0.016666666666666666d0))
    else
        tmp = re * (1.0d0 + (0.5d0 * (im * im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -2.8e+138) {
		tmp = 0.5 * (re * (im * im));
	} else if ((im <= -5e+41) || (!(im <= 390.0) && (im <= 5.2e+146))) {
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	} else {
		tmp = re * (1.0 + (0.5 * (im * im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -2.8e+138:
		tmp = 0.5 * (re * (im * im))
	elif (im <= -5e+41) or (not (im <= 390.0) and (im <= 5.2e+146)):
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666))
	else:
		tmp = re * (1.0 + (0.5 * (im * im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -2.8e+138)
		tmp = Float64(0.5 * Float64(re * Float64(im * im)));
	elseif ((im <= -5e+41) || (!(im <= 390.0) && (im <= 5.2e+146)))
		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(0.5 * Float64(im * im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -2.8e+138)
		tmp = 0.5 * (re * (im * im));
	elseif ((im <= -5e+41) || (~((im <= 390.0)) && (im <= 5.2e+146)))
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	else
		tmp = re * (1.0 + (0.5 * (im * im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -2.8e+138], N[(0.5 * N[(re * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[im, -5e+41], And[N[Not[LessEqual[im, 390.0]], $MachinePrecision], LessEqual[im, 5.2e+146]]], 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[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < -2.8000000000000001e138

   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 90.0%

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

      1. *-commutative 90.0%

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

      2. associate-*r* 90.0%

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

      3. distribute-rgt1-in 90.0%

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

      4. *-commutative 90.0%

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

      5. unpow2 90.0%

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

      6. associate-*l* 90.0%

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

   6. Simplified 90.0%

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

   7. Taylor expanded in im around inf 90.0%

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

      1. *-commutative 90.0%

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

      2. unpow2 90.0%

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

      3. associate-*l* 62.9%

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

   9. Simplified 62.9%

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

   10. Taylor expanded in re around 0 72.1%

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

       1. unpow2 72.1%

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

   12. Simplified 72.1%

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

   if -2.8000000000000001e138 < im < -5.00000000000000022e41 or 390 < im < 5.20000000000000028e146

   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 17.8%

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

   6. Taylor expanded in re around 0 35.1%

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

      1. associate-*r/ 35.1%

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

      2. metadata-eval 35.1%

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

      3. unpow2 35.1%

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

      4. *-commutative 35.1%

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

      5. unpow2 35.1%

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

   8. Simplified 35.1%

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

   if -5.00000000000000022e41 < im < 390 or 5.20000000000000028e146 < 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%

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

      1. *-commutative 91.5%

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

      2. associate-*r* 91.5%

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

      3. distribute-rgt1-in 91.5%

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

      4. *-commutative 91.5%

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

      5. unpow2 91.5%

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

      6. associate-*l* 91.5%

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

   6. Simplified 91.5%

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

   7. Taylor expanded in re around 0 57.1%

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

      1. *-commutative 57.1%

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

      2. unpow2 57.1%

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

   9. Simplified 57.1%

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

4. Final simplification 53.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.8 \cdot 10^{+138}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq -5 \cdot 10^{+41} \lor \neg \left(im \leq 390\right) \land im \leq 5.2 \cdot 10^{+146}:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 9: 55.8% accurate, 16.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -7.2 \cdot 10^{+276}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq -4.4 \cdot 10^{+154}:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -7.2e+276)
   (* 0.5 (* re (* im im)))
   (if (<= re -4.4e+154)
     (+
      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 (re <= -7.2e+276) {
		tmp = 0.5 * (re * (im * im));
	} else if (re <= -4.4e+154) {
		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 (re <= (-7.2d+276)) then
        tmp = 0.5d0 * (re * (im * im))
    else if (re <= (-4.4d+154)) 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 (re <= -7.2e+276) {
		tmp = 0.5 * (re * (im * im));
	} else if (re <= -4.4e+154) {
		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 re <= -7.2e+276:
		tmp = 0.5 * (re * (im * im))
	elif re <= -4.4e+154:
		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 (re <= -7.2e+276)
		tmp = Float64(0.5 * Float64(re * Float64(im * im)));
	elseif (re <= -4.4e+154)
		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 (re <= -7.2e+276)
		tmp = 0.5 * (re * (im * im));
	elseif (re <= -4.4e+154)
		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[re, -7.2e+276], N[(0.5 * N[(re * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -4.4e+154], 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 re < -7.19999999999999959e276

   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 89.3%

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

      1. *-commutative 89.3%

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

      2. associate-*r* 89.3%

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

      3. distribute-rgt1-in 89.3%

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

      4. *-commutative 89.3%

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

      5. unpow2 89.3%

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

      6. associate-*l* 89.3%

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

   6. Simplified 89.3%

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

   7. Taylor expanded in im around inf 47.0%

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

      1. *-commutative 47.0%

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

      2. unpow2 47.0%

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

      3. associate-*l* 47.0%

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

   9. Simplified 47.0%

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

   10. Taylor expanded in re around 0 56.9%

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

       1. unpow2 56.9%

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

   12. Simplified 56.9%

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

   if -7.19999999999999959e276 < re < -4.4000000000000002e154

   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 4.8%

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

   6. Taylor expanded in re around 0 57.8%

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

      1. associate-*r/ 57.8%

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

      2. metadata-eval 57.8%

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

      3. unpow2 57.8%

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

      4. *-commutative 57.8%

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

      5. unpow2 57.8%

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

   8. Simplified 57.8%

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

   if -4.4000000000000002e154 < 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 87.7%

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

      1. *-rgt-identity 87.7%

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

      2. *-commutative 87.7%

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

      3. associate-*r* 87.7%

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

      4. *-commutative 87.7%

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

      5. associate-*r* 87.7%

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

      6. distribute-rgt-out 87.7%

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

      7. distribute-lft-out 87.7%

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

      8. metadata-eval 87.7%

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

      9. pow-sqr 87.7%

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

      10. associate-*r* 87.7%

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

      11. distribute-rgt-out 87.7%

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

      12. unpow2 87.7%

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

      13. unpow2 87.7%

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

   6. Simplified 87.7%

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

   7. Taylor expanded in re around 0 60.8%

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

      1. *-commutative 60.8%

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

      2. unpow2 60.8%

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

      3. unpow2 60.8%

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

   9. Simplified 60.8%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -7.2 \cdot 10^{+276}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq -4.4 \cdot 10^{+154}:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \end{array} \]

Alternative 10: 41.3% accurate, 20.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(im \cdot \left(re \cdot im\right)\right)\\ \mathbf{if}\;im \leq -3.5 \cdot 10^{+184}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -2.6 \cdot 10^{+39}:\\ \;\;\;\;\frac{0.25}{re \cdot re}\\ \mathbf{elif}\;im \leq -1.4 \lor \neg \left(im \leq 1.4\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (* im (* re im)))))
   (if (<= im -3.5e+184)
     t_0
     (if (<= im -2.6e+39)
       (/ 0.25 (* re re))
       (if (or (<= im -1.4) (not (<= im 1.4))) t_0 re)))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (im * (re * im));
	double tmp;
	if (im <= -3.5e+184) {
		tmp = t_0;
	} else if (im <= -2.6e+39) {
		tmp = 0.25 / (re * re);
	} else if ((im <= -1.4) || !(im <= 1.4)) {
		tmp = t_0;
	} else {
		tmp = re;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (im * (re * im))
    if (im <= (-3.5d+184)) then
        tmp = t_0
    else if (im <= (-2.6d+39)) then
        tmp = 0.25d0 / (re * re)
    else if ((im <= (-1.4d0)) .or. (.not. (im <= 1.4d0))) then
        tmp = t_0
    else
        tmp = re
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * (im * (re * im));
	double tmp;
	if (im <= -3.5e+184) {
		tmp = t_0;
	} else if (im <= -2.6e+39) {
		tmp = 0.25 / (re * re);
	} else if ((im <= -1.4) || !(im <= 1.4)) {
		tmp = t_0;
	} else {
		tmp = re;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * (im * (re * im))
	tmp = 0
	if im <= -3.5e+184:
		tmp = t_0
	elif im <= -2.6e+39:
		tmp = 0.25 / (re * re)
	elif (im <= -1.4) or not (im <= 1.4):
		tmp = t_0
	else:
		tmp = re
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(im * Float64(re * im)))
	tmp = 0.0
	if (im <= -3.5e+184)
		tmp = t_0;
	elseif (im <= -2.6e+39)
		tmp = Float64(0.25 / Float64(re * re));
	elseif ((im <= -1.4) || !(im <= 1.4))
		tmp = t_0;
	else
		tmp = re;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (im * (re * im));
	tmp = 0.0;
	if (im <= -3.5e+184)
		tmp = t_0;
	elseif (im <= -2.6e+39)
		tmp = 0.25 / (re * re);
	elseif ((im <= -1.4) || ~((im <= 1.4)))
		tmp = t_0;
	else
		tmp = re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(im * N[(re * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -3.5e+184], t$95$0, If[LessEqual[im, -2.6e+39], N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[im, -1.4], N[Not[LessEqual[im, 1.4]], $MachinePrecision]], t$95$0, re]]]]

Derivation

1. Split input into 3 regimes

2. if im < -3.49999999999999978e184 or -2.6e39 < im < -1.3999999999999999 or 1.3999999999999999 < 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 54.4%

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

      1. *-commutative 54.4%

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

      2. associate-*r* 54.4%

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

      3. distribute-rgt1-in 54.4%

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

      4. *-commutative 54.4%

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

      5. unpow2 54.4%

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

      6. associate-*l* 54.4%

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

   6. Simplified 54.4%

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

   7. Taylor expanded in im around inf 54.4%

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

      1. *-commutative 54.4%

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

      2. unpow2 54.4%

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

      3. associate-*l* 45.7%

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

   9. Simplified 45.7%

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

   10. Taylor expanded in re around 0 36.7%

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

   if -3.49999999999999978e184 < im < -2.6e39

   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 21.2%

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

   6. Taylor expanded in re around 0 21.2%

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

      1. unpow2 21.2%

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

   8. Simplified 21.2%

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

   if -1.3999999999999999 < im < 1.3999999999999999

   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 58.9%

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

      1. associate-*r* 58.9%

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

   6. Simplified 58.9%

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

   7. Taylor expanded in im around 0 57.5%

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

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3.5 \cdot 10^{+184}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(re \cdot im\right)\right)\\ \mathbf{elif}\;im \leq -2.6 \cdot 10^{+39}:\\ \;\;\;\;\frac{0.25}{re \cdot re}\\ \mathbf{elif}\;im \leq -1.4 \lor \neg \left(im \leq 1.4\right):\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(re \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array} \]

Alternative 11: 48.0% accurate, 27.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1.4 \lor \neg \left(im \leq 1.4\right):\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -1.4) (not (<= im 1.4))) (* 0.5 (* re (* im im))) re))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -1.4) || !(im <= 1.4)) {
		tmp = 0.5 * (re * (im * im));
	} else {
		tmp = 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 <= (-1.4d0)) .or. (.not. (im <= 1.4d0))) then
        tmp = 0.5d0 * (re * (im * im))
    else
        tmp = re
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -1.4) || !(im <= 1.4)) {
		tmp = 0.5 * (re * (im * im));
	} else {
		tmp = re;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -1.4) or not (im <= 1.4):
		tmp = 0.5 * (re * (im * im))
	else:
		tmp = re
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -1.4) || !(im <= 1.4))
		tmp = Float64(0.5 * Float64(re * Float64(im * im)));
	else
		tmp = re;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -1.4) || ~((im <= 1.4)))
		tmp = 0.5 * (re * (im * im));
	else
		tmp = re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -1.4], N[Not[LessEqual[im, 1.4]], $MachinePrecision]], N[(0.5 * N[(re * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], re]

Derivation

1. Split input into 2 regimes

2. if im < -1.3999999999999999 or 1.3999999999999999 < 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 45.2%

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

      1. *-commutative 45.2%

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

      2. associate-*r* 45.2%

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

      3. distribute-rgt1-in 45.2%

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

      4. *-commutative 45.2%

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

      5. unpow2 45.2%

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

      6. associate-*l* 45.2%

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

   6. Simplified 45.2%

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

   7. Taylor expanded in im around inf 45.2%

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

      1. *-commutative 45.2%

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

      2. unpow2 45.2%

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

      3. associate-*l* 34.9%

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

   9. Simplified 34.9%

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

   10. Taylor expanded in re around 0 38.5%

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

       1. unpow2 38.5%

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

   12. Simplified 38.5%

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

   if -1.3999999999999999 < im < 1.3999999999999999

   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 58.9%

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

      1. associate-*r* 58.9%

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

   6. Simplified 58.9%

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

   7. Taylor expanded in im around 0 57.5%

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

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.4 \lor \neg \left(im \leq 1.4\right):\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array} \]

Alternative 12: 33.1% accurate, 33.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -460000000000 \lor \neg \left(im \leq 760000000000\right):\\ \;\;\;\;\frac{0.25}{re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -460000000000.0) (not (<= im 760000000000.0)))
   (/ 0.25 (* re re))
   re))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -460000000000.0) || !(im <= 760000000000.0)) {
		tmp = 0.25 / (re * re);
	} else {
		tmp = 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 <= (-460000000000.0d0)) .or. (.not. (im <= 760000000000.0d0))) then
        tmp = 0.25d0 / (re * re)
    else
        tmp = re
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -460000000000.0) || !(im <= 760000000000.0)) {
		tmp = 0.25 / (re * re);
	} else {
		tmp = re;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -460000000000.0) or not (im <= 760000000000.0):
		tmp = 0.25 / (re * re)
	else:
		tmp = re
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -460000000000.0) || !(im <= 760000000000.0))
		tmp = Float64(0.25 / Float64(re * re));
	else
		tmp = re;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -460000000000.0) || ~((im <= 760000000000.0)))
		tmp = 0.25 / (re * re);
	else
		tmp = re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -460000000000.0], N[Not[LessEqual[im, 760000000000.0]], $MachinePrecision]], N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision], re]

Derivation

1. Split input into 2 regimes

2. if im < -4.6e11 or 7.6e11 < 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 18.0%

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

   6. Taylor expanded in re around 0 17.8%

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

      1. unpow2 17.8%

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

   8. Simplified 17.8%

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

   if -4.6e11 < im < 7.6e11

   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 59.8%

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

      1. associate-*r* 59.8%

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

   6. Simplified 59.8%

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

   7. Taylor expanded in im around 0 56.3%

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

4. Final simplification 37.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -460000000000 \lor \neg \left(im \leq 760000000000\right):\\ \;\;\;\;\frac{0.25}{re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array} \]

Alternative 13: 48.2% accurate, 34.3× speedup

Math

\[\begin{array}{l} \\ re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* re (+ 1.0 (* 0.5 (* im im)))))

C

double code(double re, double im) {
	return re * (1.0 + (0.5 * (im * im)));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = re * (1.0d0 + (0.5d0 * (im * im)))
end function

Java

public static double code(double re, double im) {
	return re * (1.0 + (0.5 * (im * im)));
}

Python

def code(re, im):
	return re * (1.0 + (0.5 * (im * im)))

Julia

function code(re, im)
	return Float64(re * Float64(1.0 + Float64(0.5 * Float64(im * im))))
end

MATLAB

function tmp = code(re, im)
	tmp = re * (1.0 + (0.5 * (im * im)));
end

Wolfram

code[re_, im_] := N[(re * N[(1.0 + N[(0.5 * N[(im * 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. Taylor expanded in im around 0 72.4%

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

   1. *-commutative 72.4%

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

   2. associate-*r* 72.4%

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

   3. distribute-rgt1-in 72.4%

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

   4. *-commutative 72.4%

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

   5. unpow2 72.4%

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

   6. associate-*l* 72.4%

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

6. Simplified 72.4%

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

7. Taylor expanded in re around 0 48.3%

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

   1. *-commutative 48.3%

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

   2. unpow2 48.3%

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

9. Simplified 48.3%

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

10. Final simplification 48.3%

    \[\leadsto re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right) \]

Alternative 14: 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 15: 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 16: 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 17: 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 18: 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 19: 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 20: 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 21: 4.8% accurate, 309.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return 1.0

Julia

function code(re, im)
	return 1.0
end

MATLAB

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

Wolfram

code[re_, im_] := 1.0

Derivation

1. Initial program 100.0%

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

   1. *-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 22: 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))))