math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 7.0s

Alternatives: 10

Speedup: 1.0×

• 50.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. distribute-rgt-in 100.0%

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

   2. cancel-sign-sub 100.0%

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

   3. distribute-rgt-out-- 100.0%

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

   4. sub-neg 100.0%

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

   5. remove-double-neg 100.0%

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

   6. neg-sub0 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 85.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{if}\;im \leq 0.042:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot {im}^{2} + 1\right)\\ \mathbf{elif}\;im \leq 7.8 \cdot 10^{+56}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im \leq 2.15 \cdot 10^{+73}:\\ \;\;\;\;re \cdot \left({im}^{4} \cdot \left({re}^{2} \cdot -0.006944444444444444 + 0.041666666666666664\right)\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (+ (exp (- im)) (exp im)) (* 0.5 re))))
   (if (<= im 0.042)
     (* (sin re) (+ (* 0.5 (pow im 2.0)) 1.0))
     (if (<= im 7.8e+56)
       t_0
       (if (<= im 2.15e+73)
         (*
          re
          (*
           (pow im 4.0)
           (+ (* (pow re 2.0) -0.006944444444444444) 0.041666666666666664)))
         (if (<= im 1.15e+77)
           t_0
           (* 0.041666666666666664 (* (sin re) (pow im 4.0)))))))))

C

double code(double re, double im) {
	double t_0 = (exp(-im) + exp(im)) * (0.5 * re);
	double tmp;
	if (im <= 0.042) {
		tmp = sin(re) * ((0.5 * pow(im, 2.0)) + 1.0);
	} else if (im <= 7.8e+56) {
		tmp = t_0;
	} else if (im <= 2.15e+73) {
		tmp = re * (pow(im, 4.0) * ((pow(re, 2.0) * -0.006944444444444444) + 0.041666666666666664));
	} else if (im <= 1.15e+77) {
		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 = (exp(-im) + exp(im)) * (0.5d0 * re)
    if (im <= 0.042d0) then
        tmp = sin(re) * ((0.5d0 * (im ** 2.0d0)) + 1.0d0)
    else if (im <= 7.8d+56) then
        tmp = t_0
    else if (im <= 2.15d+73) then
        tmp = re * ((im ** 4.0d0) * (((re ** 2.0d0) * (-0.006944444444444444d0)) + 0.041666666666666664d0))
    else if (im <= 1.15d+77) 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 = (Math.exp(-im) + Math.exp(im)) * (0.5 * re);
	double tmp;
	if (im <= 0.042) {
		tmp = Math.sin(re) * ((0.5 * Math.pow(im, 2.0)) + 1.0);
	} else if (im <= 7.8e+56) {
		tmp = t_0;
	} else if (im <= 2.15e+73) {
		tmp = re * (Math.pow(im, 4.0) * ((Math.pow(re, 2.0) * -0.006944444444444444) + 0.041666666666666664));
	} else if (im <= 1.15e+77) {
		tmp = t_0;
	} else {
		tmp = 0.041666666666666664 * (Math.sin(re) * Math.pow(im, 4.0));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (math.exp(-im) + math.exp(im)) * (0.5 * re)
	tmp = 0
	if im <= 0.042:
		tmp = math.sin(re) * ((0.5 * math.pow(im, 2.0)) + 1.0)
	elif im <= 7.8e+56:
		tmp = t_0
	elif im <= 2.15e+73:
		tmp = re * (math.pow(im, 4.0) * ((math.pow(re, 2.0) * -0.006944444444444444) + 0.041666666666666664))
	elif im <= 1.15e+77:
		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(exp(Float64(-im)) + exp(im)) * Float64(0.5 * re))
	tmp = 0.0
	if (im <= 0.042)
		tmp = Float64(sin(re) * Float64(Float64(0.5 * (im ^ 2.0)) + 1.0));
	elseif (im <= 7.8e+56)
		tmp = t_0;
	elseif (im <= 2.15e+73)
		tmp = Float64(re * Float64((im ^ 4.0) * Float64(Float64((re ^ 2.0) * -0.006944444444444444) + 0.041666666666666664)));
	elseif (im <= 1.15e+77)
		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 = (exp(-im) + exp(im)) * (0.5 * re);
	tmp = 0.0;
	if (im <= 0.042)
		tmp = sin(re) * ((0.5 * (im ^ 2.0)) + 1.0);
	elseif (im <= 7.8e+56)
		tmp = t_0;
	elseif (im <= 2.15e+73)
		tmp = re * ((im ^ 4.0) * (((re ^ 2.0) * -0.006944444444444444) + 0.041666666666666664));
	elseif (im <= 1.15e+77)
		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[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 0.042], N[(N[Sin[re], $MachinePrecision] * N[(N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 7.8e+56], t$95$0, If[LessEqual[im, 2.15e+73], N[(re * N[(N[Power[im, 4.0], $MachinePrecision] * N[(N[(N[Power[re, 2.0], $MachinePrecision] * -0.006944444444444444), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.15e+77], 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 < 0.0420000000000000026

   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. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 85.1%

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

   7. Simplified 85.1%

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

   if 0.0420000000000000026 < im < 7.79999999999999989e56 or 2.15000000000000007e73 < im < 1.14999999999999997e77

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in re around 0 73.7%

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

   7. Simplified 73.7%

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

   if 7.79999999999999989e56 < im < 2.15000000000000007e73

   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. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 6.9%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + 0.08333333333333333 \cdot {im}^{2}\right)\right)} \]

   7. Simplified 6.9%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot 0.08333333333333333\right)\right)} \]

   8. Taylor expanded in im around inf 6.9%

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

   9. Taylor expanded in re around 0 72.8%

      \[\leadsto \color{blue}{re \cdot \left(-0.006944444444444444 \cdot \left({im}^{4} \cdot {re}^{2}\right) + 0.041666666666666664 \cdot {im}^{4}\right)} \]
   10. Step-by-step derivation

       1. *-commutative 72.8%

          \[\leadsto re \cdot \left(\color{blue}{\left({im}^{4} \cdot {re}^{2}\right) \cdot -0.006944444444444444} + 0.041666666666666664 \cdot {im}^{4}\right) \]

       2. associate-*l* 72.8%

          \[\leadsto re \cdot \left(\color{blue}{{im}^{4} \cdot \left({re}^{2} \cdot -0.006944444444444444\right)} + 0.041666666666666664 \cdot {im}^{4}\right) \]

       3. *-commutative 72.8%

          \[\leadsto re \cdot \left({im}^{4} \cdot \left({re}^{2} \cdot -0.006944444444444444\right) + \color{blue}{{im}^{4} \cdot 0.041666666666666664}\right) \]

       4. distribute-lft-out 72.8%

          \[\leadsto re \cdot \color{blue}{\left({im}^{4} \cdot \left({re}^{2} \cdot -0.006944444444444444 + 0.041666666666666664\right)\right)} \]

   11. Simplified 72.8%

       \[\leadsto \color{blue}{re \cdot \left({im}^{4} \cdot \left({re}^{2} \cdot -0.006944444444444444 + 0.041666666666666664\right)\right)} \]

   if 1.14999999999999997e77 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + 0.08333333333333333 \cdot {im}^{2}\right)\right)} \]

   7. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot 0.08333333333333333\right)\right)} \]

   8. Taylor expanded in im around inf 100.0%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.042:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot {im}^{2} + 1\right)\\ \mathbf{elif}\;im \leq 7.8 \cdot 10^{+56}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;im \leq 2.15 \cdot 10^{+73}:\\ \;\;\;\;re \cdot \left({im}^{4} \cdot \left({re}^{2} \cdot -0.006944444444444444 + 0.041666666666666664\right)\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.0019:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.0019)
   (sin re)
   (if (<= im 1.15e+77)
     (* (+ (exp (- im)) (exp im)) (* 0.5 re))
     (* 0.041666666666666664 (* (sin re) (pow im 4.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.0019) {
		tmp = sin(re);
	} else if (im <= 1.15e+77) {
		tmp = (exp(-im) + exp(im)) * (0.5 * re);
	} else {
		tmp = 0.041666666666666664 * (sin(re) * pow(im, 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.0019d0) then
        tmp = sin(re)
    else if (im <= 1.15d+77) then
        tmp = (exp(-im) + exp(im)) * (0.5d0 * re)
    else
        tmp = 0.041666666666666664d0 * (sin(re) * (im ** 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.0019) {
		tmp = Math.sin(re);
	} else if (im <= 1.15e+77) {
		tmp = (Math.exp(-im) + Math.exp(im)) * (0.5 * re);
	} else {
		tmp = 0.041666666666666664 * (Math.sin(re) * Math.pow(im, 4.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.0019:
		tmp = math.sin(re)
	elif im <= 1.15e+77:
		tmp = (math.exp(-im) + math.exp(im)) * (0.5 * re)
	else:
		tmp = 0.041666666666666664 * (math.sin(re) * math.pow(im, 4.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.0019)
		tmp = sin(re);
	elseif (im <= 1.15e+77)
		tmp = Float64(Float64(exp(Float64(-im)) + exp(im)) * Float64(0.5 * re));
	else
		tmp = Float64(0.041666666666666664 * Float64(sin(re) * (im ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.0019)
		tmp = sin(re);
	elseif (im <= 1.15e+77)
		tmp = (exp(-im) + exp(im)) * (0.5 * re);
	else
		tmp = 0.041666666666666664 * (sin(re) * (im ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.0019], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.15e+77], N[(N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], N[(0.041666666666666664 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.0019

   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. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 64.0%

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

   if 0.0019 < im < 1.14999999999999997e77

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in re around 0 61.5%

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

   7. Simplified 61.5%

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

   if 1.14999999999999997e77 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + 0.08333333333333333 \cdot {im}^{2}\right)\right)} \]

   7. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot 0.08333333333333333\right)\right)} \]

   8. Taylor expanded in im around inf 100.0%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.0019:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.095:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot {im}^{2} + 1\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.095)
   (* (sin re) (+ (* 0.5 (pow im 2.0)) 1.0))
   (if (<= im 1.15e+77)
     (* (+ (exp (- im)) (exp im)) (* 0.5 re))
     (* 0.041666666666666664 (* (sin re) (pow im 4.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.095) {
		tmp = sin(re) * ((0.5 * pow(im, 2.0)) + 1.0);
	} else if (im <= 1.15e+77) {
		tmp = (exp(-im) + exp(im)) * (0.5 * re);
	} else {
		tmp = 0.041666666666666664 * (sin(re) * pow(im, 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.095d0) then
        tmp = sin(re) * ((0.5d0 * (im ** 2.0d0)) + 1.0d0)
    else if (im <= 1.15d+77) then
        tmp = (exp(-im) + exp(im)) * (0.5d0 * re)
    else
        tmp = 0.041666666666666664d0 * (sin(re) * (im ** 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.095) {
		tmp = Math.sin(re) * ((0.5 * Math.pow(im, 2.0)) + 1.0);
	} else if (im <= 1.15e+77) {
		tmp = (Math.exp(-im) + Math.exp(im)) * (0.5 * re);
	} else {
		tmp = 0.041666666666666664 * (Math.sin(re) * Math.pow(im, 4.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.095:
		tmp = math.sin(re) * ((0.5 * math.pow(im, 2.0)) + 1.0)
	elif im <= 1.15e+77:
		tmp = (math.exp(-im) + math.exp(im)) * (0.5 * re)
	else:
		tmp = 0.041666666666666664 * (math.sin(re) * math.pow(im, 4.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.095)
		tmp = Float64(sin(re) * Float64(Float64(0.5 * (im ^ 2.0)) + 1.0));
	elseif (im <= 1.15e+77)
		tmp = Float64(Float64(exp(Float64(-im)) + exp(im)) * Float64(0.5 * re));
	else
		tmp = Float64(0.041666666666666664 * Float64(sin(re) * (im ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.095)
		tmp = sin(re) * ((0.5 * (im ^ 2.0)) + 1.0);
	elseif (im <= 1.15e+77)
		tmp = (exp(-im) + exp(im)) * (0.5 * re);
	else
		tmp = 0.041666666666666664 * (sin(re) * (im ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.095], N[(N[Sin[re], $MachinePrecision] * N[(N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.15e+77], N[(N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], N[(0.041666666666666664 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.095000000000000001

   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. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 85.1%

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

   7. Simplified 85.1%

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

   if 0.095000000000000001 < im < 1.14999999999999997e77

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in re around 0 61.5%

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

   7. Simplified 61.5%

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

   if 1.14999999999999997e77 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + 0.08333333333333333 \cdot {im}^{2}\right)\right)} \]

   7. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot 0.08333333333333333\right)\right)} \]

   8. Taylor expanded in im around inf 100.0%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.095:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot {im}^{2} + 1\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 620:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 3.2 \cdot 10^{+76}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 620.0)
   (sin re)
   (if (<= im 3.2e+76)
     (log1p (expm1 re))
     (* 0.041666666666666664 (* (sin re) (pow im 4.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 620.0) {
		tmp = sin(re);
	} else if (im <= 3.2e+76) {
		tmp = log1p(expm1(re));
	} else {
		tmp = 0.041666666666666664 * (sin(re) * pow(im, 4.0));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 620.0) {
		tmp = Math.sin(re);
	} else if (im <= 3.2e+76) {
		tmp = Math.log1p(Math.expm1(re));
	} else {
		tmp = 0.041666666666666664 * (Math.sin(re) * Math.pow(im, 4.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 620.0:
		tmp = math.sin(re)
	elif im <= 3.2e+76:
		tmp = math.log1p(math.expm1(re))
	else:
		tmp = 0.041666666666666664 * (math.sin(re) * math.pow(im, 4.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 620.0)
		tmp = sin(re);
	elseif (im <= 3.2e+76)
		tmp = log1p(expm1(re));
	else
		tmp = Float64(0.041666666666666664 * Float64(sin(re) * (im ^ 4.0)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 620.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 3.2e+76], N[Log[1 + N[(Exp[re] - 1), $MachinePrecision]], $MachinePrecision], N[(0.041666666666666664 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 620

   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. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 63.7%

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

   if 620 < im < 3.19999999999999976e76

   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. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in re around 0 58.3%

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

   7. Simplified 58.3%

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

   9. Applied egg-rr 25.9%

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

   if 3.19999999999999976e76 < 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. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 97.9%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + 0.08333333333333333 \cdot {im}^{2}\right)\right)} \]

   7. Simplified 97.9%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot 0.08333333333333333\right)\right)} \]

   8. Taylor expanded in im around inf 97.9%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 620:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 3.2 \cdot 10^{+76}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 21:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left({im}^{4} \cdot 0.041666666666666664\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 21.0) (sin re) (* re (* (pow im 4.0) 0.041666666666666664))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 21.0) {
		tmp = sin(re);
	} else {
		tmp = re * (pow(im, 4.0) * 0.041666666666666664);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 21.0d0) then
        tmp = sin(re)
    else
        tmp = re * ((im ** 4.0d0) * 0.041666666666666664d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 21.0) {
		tmp = Math.sin(re);
	} else {
		tmp = re * (Math.pow(im, 4.0) * 0.041666666666666664);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 21.0:
		tmp = math.sin(re)
	else:
		tmp = re * (math.pow(im, 4.0) * 0.041666666666666664)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 21.0)
		tmp = sin(re);
	else
		tmp = Float64(re * Float64((im ^ 4.0) * 0.041666666666666664));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 21.0)
		tmp = sin(re);
	else
		tmp = re * ((im ^ 4.0) * 0.041666666666666664);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 21.0], N[Sin[re], $MachinePrecision], N[(re * N[(N[Power[im, 4.0], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 21

   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. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 64.0%

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

   if 21 < 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. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 65.4%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + 0.08333333333333333 \cdot {im}^{2}\right)\right)} \]

   7. Simplified 65.4%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot 0.08333333333333333\right)\right)} \]

   8. Taylor expanded in im around inf 65.4%

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

   9. Taylor expanded in re around 0 44.5%

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

       1. *-commutative 44.5%

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

       2. *-commutative 44.5%

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

       3. associate-*r* 44.5%

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

   11. Simplified 44.5%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 21:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left({im}^{4} \cdot 0.041666666666666664\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 54.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 650:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;{re}^{-4}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 650.0) (sin re) (pow re -4.0)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 650.0) {
		tmp = sin(re);
	} else {
		tmp = pow(re, -4.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 650.0d0) then
        tmp = sin(re)
    else
        tmp = re ** (-4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 650.0) {
		tmp = Math.sin(re);
	} else {
		tmp = Math.pow(re, -4.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 650.0:
		tmp = math.sin(re)
	else:
		tmp = math.pow(re, -4.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 650.0)
		tmp = sin(re);
	else
		tmp = re ^ -4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 650.0)
		tmp = sin(re);
	else
		tmp = re ^ -4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 650.0], N[Sin[re], $MachinePrecision], N[Power[re, -4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 650

   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. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 63.7%

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

   if 650 < 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. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in re around 0 60.0%

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

   7. Simplified 60.0%

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

   9. Applied egg-rr 18.5%

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

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 650:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;{re}^{-4}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 49.8% accurate, 3.1× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 (sin re))

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return math.sin(re)

Julia

function code(re, im)
	return sin(re)
end

MATLAB

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

Wolfram

code[re_, im_] := N[Sin[re], $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. distribute-rgt-in 100.0%

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

   2. cancel-sign-sub 100.0%

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

   3. distribute-rgt-out-- 100.0%

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

   4. sub-neg 100.0%

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

   5. remove-double-neg 100.0%

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

   6. neg-sub0 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in im around 0 47.1%

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

6. Final simplification 47.1%

   \[\leadsto \sin re \]
7. Add Preprocessing

Alternative 9: 4.7% 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. distribute-rgt-in 100.0%

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

   2. cancel-sign-sub 100.0%

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

   3. distribute-rgt-out-- 100.0%

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

   4. sub-neg 100.0%

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

   5. remove-double-neg 100.0%

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

   6. neg-sub0 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in im around 0 84.9%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + 0.08333333333333333 \cdot {im}^{2}\right)\right)} \]

7. Simplified 84.9%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot 0.08333333333333333\right)\right)} \]

9. Applied egg-rr 4.1%

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

    1. +-inverses 4.1%

       \[\leadsto \frac{\sin re}{\sin re + \color{blue}{0}} \]

    2. +-rgt-identity 4.1%

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

    3. *-inverses 4.1%

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

11. Simplified 4.1%

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

12. Final simplification 4.1%

    \[\leadsto 1 \]
13. Add Preprocessing

Alternative 10: 26.1% 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. distribute-rgt-in 100.0%

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

   2. cancel-sign-sub 100.0%

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

   3. distribute-rgt-out-- 100.0%

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

   4. sub-neg 100.0%

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

   5. remove-double-neg 100.0%

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

   6. neg-sub0 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in re around 0 62.4%

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

7. Simplified 62.4%

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

8. Taylor expanded in im around 0 26.7%

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

9. Final simplification 26.7%

   \[\leadsto re \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024071 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))