math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 10.0s

Alternatives: 20

Speedup: 1.0×

• 49.0% 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. Add Preprocessing

Alternative 2: 63.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{log1p}\left(\mathsf{expm1}\left(re \cdot -0.16666666666666666\right)\right)\\ t_1 := 0.041666666666666664 \cdot \left(re \cdot {im}^{4}\right)\\ \mathbf{if}\;im \leq 2.6 \cdot 10^{-15}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 3.3 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(re\right)\right)\\ \mathbf{elif}\;im \leq 3.3 \cdot 10^{+61}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im \leq 2.7 \cdot 10^{+208}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;im \leq 1.58 \cdot 10^{+227}:\\ \;\;\;\;re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{2}\right)\\ \mathbf{elif}\;im \leq 6.5 \cdot 10^{+239}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;im \leq 6.6 \cdot 10^{+252} \lor \neg \left(im \leq 6.6 \cdot 10^{+273}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (log1p (expm1 (* re -0.16666666666666666))))
        (t_1 (* 0.041666666666666664 (* re (pow im 4.0)))))
   (if (<= im 2.6e-15)
     (sin re)
     (if (<= im 3.3e+19)
       (log1p (expm1 re))
       (if (<= im 3.3e+61)
         t_0
         (if (<= im 2.7e+208)
           t_1
           (if (<= im 1.58e+227)
             (* re (+ 1.0 (* -0.16666666666666666 (pow re 2.0))))
             (if (<= im 6.5e+239)
               t_1
               (if (or (<= im 6.6e+252) (not (<= im 6.6e+273)))
                 t_0
                 (* (fma im im 2.0) (* 0.5 re)))))))))))

C

double code(double re, double im) {
	double t_0 = log1p(expm1((re * -0.16666666666666666)));
	double t_1 = 0.041666666666666664 * (re * pow(im, 4.0));
	double tmp;
	if (im <= 2.6e-15) {
		tmp = sin(re);
	} else if (im <= 3.3e+19) {
		tmp = log1p(expm1(re));
	} else if (im <= 3.3e+61) {
		tmp = t_0;
	} else if (im <= 2.7e+208) {
		tmp = t_1;
	} else if (im <= 1.58e+227) {
		tmp = re * (1.0 + (-0.16666666666666666 * pow(re, 2.0)));
	} else if (im <= 6.5e+239) {
		tmp = t_1;
	} else if ((im <= 6.6e+252) || !(im <= 6.6e+273)) {
		tmp = t_0;
	} else {
		tmp = fma(im, im, 2.0) * (0.5 * re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = log1p(expm1(Float64(re * -0.16666666666666666)))
	t_1 = Float64(0.041666666666666664 * Float64(re * (im ^ 4.0)))
	tmp = 0.0
	if (im <= 2.6e-15)
		tmp = sin(re);
	elseif (im <= 3.3e+19)
		tmp = log1p(expm1(re));
	elseif (im <= 3.3e+61)
		tmp = t_0;
	elseif (im <= 2.7e+208)
		tmp = t_1;
	elseif (im <= 1.58e+227)
		tmp = Float64(re * Float64(1.0 + Float64(-0.16666666666666666 * (re ^ 2.0))));
	elseif (im <= 6.5e+239)
		tmp = t_1;
	elseif ((im <= 6.6e+252) || !(im <= 6.6e+273))
		tmp = t_0;
	else
		tmp = Float64(fma(im, im, 2.0) * Float64(0.5 * re));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[Log[1 + N[(Exp[N[(re * -0.16666666666666666), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(0.041666666666666664 * N[(re * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 2.6e-15], N[Sin[re], $MachinePrecision], If[LessEqual[im, 3.3e+19], N[Log[1 + N[(Exp[re] - 1), $MachinePrecision]], $MachinePrecision], If[LessEqual[im, 3.3e+61], t$95$0, If[LessEqual[im, 2.7e+208], t$95$1, If[LessEqual[im, 1.58e+227], N[(re * N[(1.0 + N[(-0.16666666666666666 * N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 6.5e+239], t$95$1, If[Or[LessEqual[im, 6.6e+252], N[Not[LessEqual[im, 6.6e+273]], $MachinePrecision]], t$95$0, N[(N[(im * im + 2.0), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if im < 2.60000000000000004e-15

   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 60.7%

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

   if 2.60000000000000004e-15 < im < 3.3e19

   1. Initial program 99.8%

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

      1. distribute-rgt-in 99.8%

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

      2. cancel-sign-sub 99.8%

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

      3. distribute-rgt-out-- 99.8%

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

      4. sub-neg 99.8%

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

      5. remove-double-neg 99.8%

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

      6. neg-sub0 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in re around 0 72.4%

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

   7. Applied egg-rr 44.3%

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

   if 3.3e19 < im < 3.2999999999999998e61 or 6.5e239 < im < 6.6000000000000002e252 or 6.59999999999999971e273 < 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 2.9%

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

   6. Taylor expanded in re around 0 54.7%

      \[\leadsto \color{blue}{re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative 54.7%

         \[\leadsto re \cdot \color{blue}{\left(-0.16666666666666666 \cdot {re}^{2} + 1\right)} \]

      2. distribute-rgt-in 54.7%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot {re}^{2}\right) \cdot re + 1 \cdot re} \]

      3. *-lft-identity 54.7%

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

      4. associate-*l* 54.7%

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

      5. fma-define 54.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.16666666666666666, {re}^{2} \cdot re, re\right)} \]

      6. pow-plus 54.7%

         \[\leadsto \mathsf{fma}\left(-0.16666666666666666, \color{blue}{{re}^{\left(2 + 1\right)}}, re\right) \]

      7. metadata-eval 54.7%

         \[\leadsto \mathsf{fma}\left(-0.16666666666666666, {re}^{\color{blue}{3}}, re\right) \]

   8. Simplified 54.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.16666666666666666, {re}^{3}, re\right)} \]

   9. Taylor expanded in re around inf 54.3%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {re}^{3}} \]

   11. Applied egg-rr 65.0%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(-0.16666666666666666 \cdot re\right)\right)} \]

   if 3.2999999999999998e61 < im < 2.7e208 or 1.57999999999999994e227 < im < 6.5e239

   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 88.3%

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

      1. +-commutative 88.3%

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

      2. fma-define 88.3%

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

      3. associate-*r* 88.3%

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

      4. distribute-rgt-out 88.3%

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

      5. +-commutative 88.3%

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

      6. *-commutative 88.3%

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

   7. Simplified 88.3%

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

   8. Taylor expanded in im around inf 97.2%

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

   9. Taylor expanded in re around 0 69.0%

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

   if 2.7e208 < im < 1.57999999999999994e227

   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 2.7%

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

   6. Taylor expanded in re around 0 60.9%

      \[\leadsto \color{blue}{re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{2}\right)} \]
   7. Step-by-step derivation

      1. *-commutative 60.9%

         \[\leadsto re \cdot \left(1 + \color{blue}{{re}^{2} \cdot -0.16666666666666666}\right) \]

   8. Simplified 60.9%

      \[\leadsto \color{blue}{re \cdot \left(1 + {re}^{2} \cdot -0.16666666666666666\right)} \]

   if 6.6000000000000002e252 < im < 6.59999999999999971e273

   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 0 100.0%

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

   6. Taylor expanded in im around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. unpow2 100.0%

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

      3. fma-define 100.0%

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

   8. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]
3. Recombined 6 regimes into one program.

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.6 \cdot 10^{-15}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 3.3 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(re\right)\right)\\ \mathbf{elif}\;im \leq 3.3 \cdot 10^{+61}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(re \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;im \leq 2.7 \cdot 10^{+208}:\\ \;\;\;\;0.041666666666666664 \cdot \left(re \cdot {im}^{4}\right)\\ \mathbf{elif}\;im \leq 1.58 \cdot 10^{+227}:\\ \;\;\;\;re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{2}\right)\\ \mathbf{elif}\;im \leq 6.5 \cdot 10^{+239}:\\ \;\;\;\;0.041666666666666664 \cdot \left(re \cdot {im}^{4}\right)\\ \mathbf{elif}\;im \leq 6.6 \cdot 10^{+252} \lor \neg \left(im \leq 6.6 \cdot 10^{+273}\right):\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(re \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.2:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;im \leq 2.75 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(re\right)\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+62}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(re \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.2)
   (* (* 0.5 (sin re)) (fma im im 2.0))
   (if (<= im 2.75e+26)
     (log1p (expm1 re))
     (if (<= im 1.15e+62)
       (log1p (expm1 (* re -0.16666666666666666)))
       (* 0.041666666666666664 (* (sin re) (pow im 4.0)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.2) {
		tmp = (0.5 * sin(re)) * fma(im, im, 2.0);
	} else if (im <= 2.75e+26) {
		tmp = log1p(expm1(re));
	} else if (im <= 1.15e+62) {
		tmp = log1p(expm1((re * -0.16666666666666666)));
	} else {
		tmp = 0.041666666666666664 * (sin(re) * pow(im, 4.0));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.2)
		tmp = Float64(Float64(0.5 * sin(re)) * fma(im, im, 2.0));
	elseif (im <= 2.75e+26)
		tmp = log1p(expm1(re));
	elseif (im <= 1.15e+62)
		tmp = log1p(expm1(Float64(re * -0.16666666666666666)));
	else
		tmp = Float64(0.041666666666666664 * Float64(sin(re) * (im ^ 4.0)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.2], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.75e+26], N[Log[1 + N[(Exp[re] - 1), $MachinePrecision]], $MachinePrecision], If[LessEqual[im, 1.15e+62], N[Log[1 + N[(Exp[N[(re * -0.16666666666666666), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision], 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 < 0.20000000000000001

   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 85.9%

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

      1. +-commutative 85.9%

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

      2. unpow2 85.9%

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

      3. fma-define 85.9%

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

   7. Simplified 85.9%

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

   if 0.20000000000000001 < im < 2.7499999999999998e26

   1. Initial program 99.7%

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

      1. distribute-rgt-in 99.7%

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

      2. cancel-sign-sub 99.7%

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

      3. distribute-rgt-out-- 99.7%

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

      4. sub-neg 99.7%

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

      5. remove-double-neg 99.7%

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

      6. neg-sub0 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in re around 0 81.1%

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

   7. Applied egg-rr 41.9%

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

   if 2.7499999999999998e26 < im < 1.14999999999999992e62

   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 2.8%

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

   6. Taylor expanded in re around 0 41.5%

      \[\leadsto \color{blue}{re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative 41.5%

         \[\leadsto re \cdot \color{blue}{\left(-0.16666666666666666 \cdot {re}^{2} + 1\right)} \]

      2. distribute-rgt-in 41.5%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot {re}^{2}\right) \cdot re + 1 \cdot re} \]

      3. *-lft-identity 41.5%

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

      4. associate-*l* 41.5%

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

      5. fma-define 41.5%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.16666666666666666, {re}^{2} \cdot re, re\right)} \]

      6. pow-plus 41.5%

         \[\leadsto \mathsf{fma}\left(-0.16666666666666666, \color{blue}{{re}^{\left(2 + 1\right)}}, re\right) \]

      7. metadata-eval 41.5%

         \[\leadsto \mathsf{fma}\left(-0.16666666666666666, {re}^{\color{blue}{3}}, re\right) \]

   8. Simplified 41.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.16666666666666666, {re}^{3}, re\right)} \]

   9. Taylor expanded in re around inf 41.0%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {re}^{3}} \]

   11. Applied egg-rr 50.5%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(-0.16666666666666666 \cdot re\right)\right)} \]

   if 1.14999999999999992e62 < 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 92.5%

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

      1. +-commutative 92.5%

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

      2. fma-define 92.5%

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

      3. associate-*r* 92.5%

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

      4. distribute-rgt-out 92.5%

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

      5. +-commutative 92.5%

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

      6. *-commutative 92.5%

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

   7. Simplified 92.5%

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

   8. Taylor expanded in im around inf 98.2%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.2:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;im \leq 2.75 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(re\right)\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+62}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(re \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.6 \cdot 10^{-15}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(re\right)\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+62}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(re \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.6e-15)
   (sin re)
   (if (<= im 1.1e+21)
     (log1p (expm1 re))
     (if (<= im 1.15e+62)
       (log1p (expm1 (* re -0.16666666666666666)))
       (* 0.041666666666666664 (* (sin re) (pow im 4.0)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.6e-15) {
		tmp = sin(re);
	} else if (im <= 1.1e+21) {
		tmp = log1p(expm1(re));
	} else if (im <= 1.15e+62) {
		tmp = log1p(expm1((re * -0.16666666666666666)));
	} 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 <= 2.6e-15) {
		tmp = Math.sin(re);
	} else if (im <= 1.1e+21) {
		tmp = Math.log1p(Math.expm1(re));
	} else if (im <= 1.15e+62) {
		tmp = Math.log1p(Math.expm1((re * -0.16666666666666666)));
	} else {
		tmp = 0.041666666666666664 * (Math.sin(re) * Math.pow(im, 4.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.6e-15:
		tmp = math.sin(re)
	elif im <= 1.1e+21:
		tmp = math.log1p(math.expm1(re))
	elif im <= 1.15e+62:
		tmp = math.log1p(math.expm1((re * -0.16666666666666666)))
	else:
		tmp = 0.041666666666666664 * (math.sin(re) * math.pow(im, 4.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.6e-15)
		tmp = sin(re);
	elseif (im <= 1.1e+21)
		tmp = log1p(expm1(re));
	elseif (im <= 1.15e+62)
		tmp = log1p(expm1(Float64(re * -0.16666666666666666)));
	else
		tmp = Float64(0.041666666666666664 * Float64(sin(re) * (im ^ 4.0)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.6e-15], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.1e+21], N[Log[1 + N[(Exp[re] - 1), $MachinePrecision]], $MachinePrecision], If[LessEqual[im, 1.15e+62], N[Log[1 + N[(Exp[N[(re * -0.16666666666666666), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision], 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 < 2.60000000000000004e-15

   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 60.7%

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

   if 2.60000000000000004e-15 < im < 1.1e21

   1. Initial program 99.8%

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

      1. distribute-rgt-in 99.8%

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

      2. cancel-sign-sub 99.8%

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

      3. distribute-rgt-out-- 99.8%

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

      4. sub-neg 99.8%

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

      5. remove-double-neg 99.8%

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

      6. neg-sub0 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in re around 0 72.4%

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

   7. Applied egg-rr 44.3%

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

   if 1.1e21 < im < 1.14999999999999992e62

   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 2.8%

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

   6. Taylor expanded in re around 0 41.5%

      \[\leadsto \color{blue}{re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative 41.5%

         \[\leadsto re \cdot \color{blue}{\left(-0.16666666666666666 \cdot {re}^{2} + 1\right)} \]

      2. distribute-rgt-in 41.5%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot {re}^{2}\right) \cdot re + 1 \cdot re} \]

      3. *-lft-identity 41.5%

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

      4. associate-*l* 41.5%

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

      5. fma-define 41.5%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.16666666666666666, {re}^{2} \cdot re, re\right)} \]

      6. pow-plus 41.5%

         \[\leadsto \mathsf{fma}\left(-0.16666666666666666, \color{blue}{{re}^{\left(2 + 1\right)}}, re\right) \]

      7. metadata-eval 41.5%

         \[\leadsto \mathsf{fma}\left(-0.16666666666666666, {re}^{\color{blue}{3}}, re\right) \]

   8. Simplified 41.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.16666666666666666, {re}^{3}, re\right)} \]

   9. Taylor expanded in re around inf 41.0%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {re}^{3}} \]

   11. Applied egg-rr 50.5%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(-0.16666666666666666 \cdot re\right)\right)} \]

   if 1.14999999999999992e62 < 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 92.5%

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

      1. +-commutative 92.5%

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

      2. fma-define 92.5%

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

      3. associate-*r* 92.5%

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

      4. distribute-rgt-out 92.5%

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

      5. +-commutative 92.5%

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

      6. *-commutative 92.5%

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

   7. Simplified 92.5%

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

   8. Taylor expanded in im around inf 98.2%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.6 \cdot 10^{-15}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(re\right)\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+62}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(re \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.102:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{+66}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.102)
   (* (* 0.5 (sin re)) (fma im im 2.0))
   (if (<= im 4.2e+66)
     (* (+ (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 <= 0.102) {
		tmp = (0.5 * sin(re)) * fma(im, im, 2.0);
	} else if (im <= 4.2e+66) {
		tmp = (exp(-im) + exp(im)) * (0.5 * re);
	} else {
		tmp = 0.041666666666666664 * (sin(re) * pow(im, 4.0));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.102)
		tmp = Float64(Float64(0.5 * sin(re)) * fma(im, im, 2.0));
	elseif (im <= 4.2e+66)
		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

Wolfram

code[re_, im_] := If[LessEqual[im, 0.102], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 4.2e+66], 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 < 0.101999999999999993

   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 85.9%

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

      1. +-commutative 85.9%

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

      2. unpow2 85.9%

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

      3. fma-define 85.9%

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

   7. Simplified 85.9%

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

   if 0.101999999999999993 < im < 4.20000000000000011e66

   1. Initial program 99.9%

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

      1. distribute-rgt-in 99.9%

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

      2. cancel-sign-sub 99.9%

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

      3. distribute-rgt-out-- 99.9%

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

      4. sub-neg 99.9%

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

      5. remove-double-neg 99.9%

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

      6. neg-sub0 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in re around 0 60.4%

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

   if 4.20000000000000011e66 < 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 92.5%

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

      1. +-commutative 92.5%

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

      2. fma-define 92.5%

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

      3. associate-*r* 92.5%

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

      4. distribute-rgt-out 92.5%

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

      5. +-commutative 92.5%

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

      6. *-commutative 92.5%

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

   7. Simplified 92.5%

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

   8. Taylor expanded in im around inf 98.2%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.102:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{+66}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.6 \cdot 10^{-15}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(re\right)\right)\\ \mathbf{elif}\;im \leq 2.45 \cdot 10^{+67}:\\ \;\;\;\;{re}^{-4}\\ \mathbf{elif}\;im \leq 2.3 \cdot 10^{+208}:\\ \;\;\;\;0.041666666666666664 \cdot \left(re \cdot {im}^{4}\right)\\ \mathbf{elif}\;im \leq 1.58 \cdot 10^{+227}:\\ \;\;\;\;re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot {im}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.6e-15)
   (sin re)
   (if (<= im 1.9e+19)
     (log1p (expm1 re))
     (if (<= im 2.45e+67)
       (pow re -4.0)
       (if (<= im 2.3e+208)
         (* 0.041666666666666664 (* re (pow im 4.0)))
         (if (<= im 1.58e+227)
           (* re (+ 1.0 (* -0.16666666666666666 (pow re 2.0))))
           (* 0.5 (* re (pow im 2.0)))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.6e-15) {
		tmp = sin(re);
	} else if (im <= 1.9e+19) {
		tmp = log1p(expm1(re));
	} else if (im <= 2.45e+67) {
		tmp = pow(re, -4.0);
	} else if (im <= 2.3e+208) {
		tmp = 0.041666666666666664 * (re * pow(im, 4.0));
	} else if (im <= 1.58e+227) {
		tmp = re * (1.0 + (-0.16666666666666666 * pow(re, 2.0)));
	} else {
		tmp = 0.5 * (re * pow(im, 2.0));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.6e-15) {
		tmp = Math.sin(re);
	} else if (im <= 1.9e+19) {
		tmp = Math.log1p(Math.expm1(re));
	} else if (im <= 2.45e+67) {
		tmp = Math.pow(re, -4.0);
	} else if (im <= 2.3e+208) {
		tmp = 0.041666666666666664 * (re * Math.pow(im, 4.0));
	} else if (im <= 1.58e+227) {
		tmp = re * (1.0 + (-0.16666666666666666 * Math.pow(re, 2.0)));
	} else {
		tmp = 0.5 * (re * Math.pow(im, 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.6e-15:
		tmp = math.sin(re)
	elif im <= 1.9e+19:
		tmp = math.log1p(math.expm1(re))
	elif im <= 2.45e+67:
		tmp = math.pow(re, -4.0)
	elif im <= 2.3e+208:
		tmp = 0.041666666666666664 * (re * math.pow(im, 4.0))
	elif im <= 1.58e+227:
		tmp = re * (1.0 + (-0.16666666666666666 * math.pow(re, 2.0)))
	else:
		tmp = 0.5 * (re * math.pow(im, 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.6e-15)
		tmp = sin(re);
	elseif (im <= 1.9e+19)
		tmp = log1p(expm1(re));
	elseif (im <= 2.45e+67)
		tmp = re ^ -4.0;
	elseif (im <= 2.3e+208)
		tmp = Float64(0.041666666666666664 * Float64(re * (im ^ 4.0)));
	elseif (im <= 1.58e+227)
		tmp = Float64(re * Float64(1.0 + Float64(-0.16666666666666666 * (re ^ 2.0))));
	else
		tmp = Float64(0.5 * Float64(re * (im ^ 2.0)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.6e-15], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.9e+19], N[Log[1 + N[(Exp[re] - 1), $MachinePrecision]], $MachinePrecision], If[LessEqual[im, 2.45e+67], N[Power[re, -4.0], $MachinePrecision], If[LessEqual[im, 2.3e+208], N[(0.041666666666666664 * N[(re * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.58e+227], N[(re * N[(1.0 + N[(-0.16666666666666666 * N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(re * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 6 regimes

2. if im < 2.60000000000000004e-15

   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 60.7%

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

   if 2.60000000000000004e-15 < im < 1.9e19

   1. Initial program 99.8%

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

      1. distribute-rgt-in 99.8%

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

      2. cancel-sign-sub 99.8%

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

      3. distribute-rgt-out-- 99.8%

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

      4. sub-neg 99.8%

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

      5. remove-double-neg 99.8%

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

      6. neg-sub0 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in re around 0 72.4%

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

   7. Applied egg-rr 44.3%

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

   if 1.9e19 < im < 2.44999999999999995e67

   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 50.0%

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

   7. Applied egg-rr 21.5%

      \[\leadsto \color{blue}{{re}^{-4}} \]

   if 2.44999999999999995e67 < im < 2.3e208

   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 86.6%

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

      1. +-commutative 86.6%

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

      2. fma-define 86.6%

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

      3. associate-*r* 86.6%

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

      4. distribute-rgt-out 86.6%

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

      5. +-commutative 86.6%

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

      6. *-commutative 86.6%

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

   7. Simplified 86.6%

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

   8. Taylor expanded in im around inf 96.7%

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

   9. Taylor expanded in re around 0 64.6%

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

   if 2.3e208 < im < 1.57999999999999994e227

   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 2.7%

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

   6. Taylor expanded in re around 0 60.9%

      \[\leadsto \color{blue}{re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{2}\right)} \]
   7. Step-by-step derivation

      1. *-commutative 60.9%

         \[\leadsto re \cdot \left(1 + \color{blue}{{re}^{2} \cdot -0.16666666666666666}\right) \]

   8. Simplified 60.9%

      \[\leadsto \color{blue}{re \cdot \left(1 + {re}^{2} \cdot -0.16666666666666666\right)} \]

   if 1.57999999999999994e227 < 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 64.7%

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

   6. Taylor expanded in im around 0 64.7%

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

      1. +-commutative 100.0%

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

      2. unpow2 100.0%

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

      3. fma-define 100.0%

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

   8. Simplified 64.7%

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

   9. Taylor expanded in im around inf 64.7%

      \[\leadsto \color{blue}{0.5 \cdot \left({im}^{2} \cdot re\right)} \]
3. Recombined 6 regimes into one program.

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.6 \cdot 10^{-15}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(re\right)\right)\\ \mathbf{elif}\;im \leq 2.45 \cdot 10^{+67}:\\ \;\;\;\;{re}^{-4}\\ \mathbf{elif}\;im \leq 2.3 \cdot 10^{+208}:\\ \;\;\;\;0.041666666666666664 \cdot \left(re \cdot {im}^{4}\right)\\ \mathbf{elif}\;im \leq 1.58 \cdot 10^{+227}:\\ \;\;\;\;re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot {im}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 55.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.056:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 3.1 \cdot 10^{+83}:\\ \;\;\;\;{re}^{-4}\\ \mathbf{elif}\;im \leq 9 \cdot 10^{+109}:\\ \;\;\;\;{re}^{6}\\ \mathbf{elif}\;im \leq 9.6 \cdot 10^{+144} \lor \neg \left(im \leq 4.5 \cdot 10^{+158}\right) \land \left(im \leq 6.8 \cdot 10^{+198} \lor \neg \left(im \leq 1.72 \cdot 10^{+227}\right) \land im \leq 1.85 \cdot 10^{+239}\right):\\ \;\;\;\;{re}^{-4}\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot {re}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.056)
   (sin re)
   (if (<= im 3.1e+83)
     (pow re -4.0)
     (if (<= im 9e+109)
       (pow re 6.0)
       (if (or (<= im 9.6e+144)
               (and (not (<= im 4.5e+158))
                    (or (<= im 6.8e+198)
                        (and (not (<= im 1.72e+227)) (<= im 1.85e+239)))))
         (pow re -4.0)
         (* -0.16666666666666666 (pow re 3.0)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.056) {
		tmp = sin(re);
	} else if (im <= 3.1e+83) {
		tmp = pow(re, -4.0);
	} else if (im <= 9e+109) {
		tmp = pow(re, 6.0);
	} else if ((im <= 9.6e+144) || (!(im <= 4.5e+158) && ((im <= 6.8e+198) || (!(im <= 1.72e+227) && (im <= 1.85e+239))))) {
		tmp = pow(re, -4.0);
	} else {
		tmp = -0.16666666666666666 * pow(re, 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.056d0) then
        tmp = sin(re)
    else if (im <= 3.1d+83) then
        tmp = re ** (-4.0d0)
    else if (im <= 9d+109) then
        tmp = re ** 6.0d0
    else if ((im <= 9.6d+144) .or. (.not. (im <= 4.5d+158)) .and. (im <= 6.8d+198) .or. (.not. (im <= 1.72d+227)) .and. (im <= 1.85d+239)) then
        tmp = re ** (-4.0d0)
    else
        tmp = (-0.16666666666666666d0) * (re ** 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.056) {
		tmp = Math.sin(re);
	} else if (im <= 3.1e+83) {
		tmp = Math.pow(re, -4.0);
	} else if (im <= 9e+109) {
		tmp = Math.pow(re, 6.0);
	} else if ((im <= 9.6e+144) || (!(im <= 4.5e+158) && ((im <= 6.8e+198) || (!(im <= 1.72e+227) && (im <= 1.85e+239))))) {
		tmp = Math.pow(re, -4.0);
	} else {
		tmp = -0.16666666666666666 * Math.pow(re, 3.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.056:
		tmp = math.sin(re)
	elif im <= 3.1e+83:
		tmp = math.pow(re, -4.0)
	elif im <= 9e+109:
		tmp = math.pow(re, 6.0)
	elif (im <= 9.6e+144) or (not (im <= 4.5e+158) and ((im <= 6.8e+198) or (not (im <= 1.72e+227) and (im <= 1.85e+239)))):
		tmp = math.pow(re, -4.0)
	else:
		tmp = -0.16666666666666666 * math.pow(re, 3.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.056)
		tmp = sin(re);
	elseif (im <= 3.1e+83)
		tmp = re ^ -4.0;
	elseif (im <= 9e+109)
		tmp = re ^ 6.0;
	elseif ((im <= 9.6e+144) || (!(im <= 4.5e+158) && ((im <= 6.8e+198) || (!(im <= 1.72e+227) && (im <= 1.85e+239)))))
		tmp = re ^ -4.0;
	else
		tmp = Float64(-0.16666666666666666 * (re ^ 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.056)
		tmp = sin(re);
	elseif (im <= 3.1e+83)
		tmp = re ^ -4.0;
	elseif (im <= 9e+109)
		tmp = re ^ 6.0;
	elseif ((im <= 9.6e+144) || (~((im <= 4.5e+158)) && ((im <= 6.8e+198) || (~((im <= 1.72e+227)) && (im <= 1.85e+239)))))
		tmp = re ^ -4.0;
	else
		tmp = -0.16666666666666666 * (re ^ 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.056], N[Sin[re], $MachinePrecision], If[LessEqual[im, 3.1e+83], N[Power[re, -4.0], $MachinePrecision], If[LessEqual[im, 9e+109], N[Power[re, 6.0], $MachinePrecision], If[Or[LessEqual[im, 9.6e+144], And[N[Not[LessEqual[im, 4.5e+158]], $MachinePrecision], Or[LessEqual[im, 6.8e+198], And[N[Not[LessEqual[im, 1.72e+227]], $MachinePrecision], LessEqual[im, 1.85e+239]]]]], N[Power[re, -4.0], $MachinePrecision], N[(-0.16666666666666666 * N[Power[re, 3.0], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if im < 0.0560000000000000012

   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 60.9%

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

   if 0.0560000000000000012 < im < 3.09999999999999992e83 or 8.9999999999999992e109 < im < 9.6000000000000002e144 or 4.50000000000000046e158 < im < 6.8000000000000001e198 or 1.71999999999999995e227 < im < 1.84999999999999999e239

   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 69.6%

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

   7. Applied egg-rr 31.6%

      \[\leadsto \color{blue}{{re}^{-4}} \]

   if 3.09999999999999992e83 < im < 8.9999999999999992e109

   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 0.0%

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

   7. Applied egg-rr 0.0%

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

   9. Applied egg-rr 50.0%

      \[\leadsto \color{blue}{{re}^{6}} \]

   if 9.6000000000000002e144 < im < 4.50000000000000046e158 or 6.8000000000000001e198 < im < 1.71999999999999995e227 or 1.84999999999999999e239 < 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 2.8%

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

   6. Taylor expanded in re around 0 40.6%

      \[\leadsto \color{blue}{re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative 40.6%

         \[\leadsto re \cdot \color{blue}{\left(-0.16666666666666666 \cdot {re}^{2} + 1\right)} \]

      2. distribute-rgt-in 40.6%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot {re}^{2}\right) \cdot re + 1 \cdot re} \]

      3. *-lft-identity 40.6%

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

      4. associate-*l* 40.6%

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

      5. fma-define 40.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.16666666666666666, {re}^{2} \cdot re, re\right)} \]

      6. pow-plus 40.6%

         \[\leadsto \mathsf{fma}\left(-0.16666666666666666, \color{blue}{{re}^{\left(2 + 1\right)}}, re\right) \]

      7. metadata-eval 40.6%

         \[\leadsto \mathsf{fma}\left(-0.16666666666666666, {re}^{\color{blue}{3}}, re\right) \]

   8. Simplified 40.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.16666666666666666, {re}^{3}, re\right)} \]

   9. Taylor expanded in re around inf 40.3%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {re}^{3}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.056:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 3.1 \cdot 10^{+83}:\\ \;\;\;\;{re}^{-4}\\ \mathbf{elif}\;im \leq 9 \cdot 10^{+109}:\\ \;\;\;\;{re}^{6}\\ \mathbf{elif}\;im \leq 9.6 \cdot 10^{+144} \lor \neg \left(im \leq 4.5 \cdot 10^{+158}\right) \land \left(im \leq 6.8 \cdot 10^{+198} \lor \neg \left(im \leq 1.72 \cdot 10^{+227}\right) \land im \leq 1.85 \cdot 10^{+239}\right):\\ \;\;\;\;{re}^{-4}\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot {re}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.056:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.7 \cdot 10^{+49}:\\ \;\;\;\;{re}^{-4}\\ \mathbf{elif}\;im \leq 2.7 \cdot 10^{+208}:\\ \;\;\;\;0.041666666666666664 \cdot \left(re \cdot {im}^{4}\right)\\ \mathbf{elif}\;im \leq 1.58 \cdot 10^{+227}:\\ \;\;\;\;re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot {im}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.056)
   (sin re)
   (if (<= im 1.7e+49)
     (pow re -4.0)
     (if (<= im 2.7e+208)
       (* 0.041666666666666664 (* re (pow im 4.0)))
       (if (<= im 1.58e+227)
         (* re (+ 1.0 (* -0.16666666666666666 (pow re 2.0))))
         (* 0.5 (* re (pow im 2.0))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.056) {
		tmp = sin(re);
	} else if (im <= 1.7e+49) {
		tmp = pow(re, -4.0);
	} else if (im <= 2.7e+208) {
		tmp = 0.041666666666666664 * (re * pow(im, 4.0));
	} else if (im <= 1.58e+227) {
		tmp = re * (1.0 + (-0.16666666666666666 * pow(re, 2.0)));
	} else {
		tmp = 0.5 * (re * pow(im, 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.056d0) then
        tmp = sin(re)
    else if (im <= 1.7d+49) then
        tmp = re ** (-4.0d0)
    else if (im <= 2.7d+208) then
        tmp = 0.041666666666666664d0 * (re * (im ** 4.0d0))
    else if (im <= 1.58d+227) then
        tmp = re * (1.0d0 + ((-0.16666666666666666d0) * (re ** 2.0d0)))
    else
        tmp = 0.5d0 * (re * (im ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.056) {
		tmp = Math.sin(re);
	} else if (im <= 1.7e+49) {
		tmp = Math.pow(re, -4.0);
	} else if (im <= 2.7e+208) {
		tmp = 0.041666666666666664 * (re * Math.pow(im, 4.0));
	} else if (im <= 1.58e+227) {
		tmp = re * (1.0 + (-0.16666666666666666 * Math.pow(re, 2.0)));
	} else {
		tmp = 0.5 * (re * Math.pow(im, 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.056:
		tmp = math.sin(re)
	elif im <= 1.7e+49:
		tmp = math.pow(re, -4.0)
	elif im <= 2.7e+208:
		tmp = 0.041666666666666664 * (re * math.pow(im, 4.0))
	elif im <= 1.58e+227:
		tmp = re * (1.0 + (-0.16666666666666666 * math.pow(re, 2.0)))
	else:
		tmp = 0.5 * (re * math.pow(im, 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.056)
		tmp = sin(re);
	elseif (im <= 1.7e+49)
		tmp = re ^ -4.0;
	elseif (im <= 2.7e+208)
		tmp = Float64(0.041666666666666664 * Float64(re * (im ^ 4.0)));
	elseif (im <= 1.58e+227)
		tmp = Float64(re * Float64(1.0 + Float64(-0.16666666666666666 * (re ^ 2.0))));
	else
		tmp = Float64(0.5 * Float64(re * (im ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.056)
		tmp = sin(re);
	elseif (im <= 1.7e+49)
		tmp = re ^ -4.0;
	elseif (im <= 2.7e+208)
		tmp = 0.041666666666666664 * (re * (im ^ 4.0));
	elseif (im <= 1.58e+227)
		tmp = re * (1.0 + (-0.16666666666666666 * (re ^ 2.0)));
	else
		tmp = 0.5 * (re * (im ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.056], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.7e+49], N[Power[re, -4.0], $MachinePrecision], If[LessEqual[im, 2.7e+208], N[(0.041666666666666664 * N[(re * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.58e+227], N[(re * N[(1.0 + N[(-0.16666666666666666 * N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(re * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if im < 0.0560000000000000012

   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 60.9%

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

   if 0.0560000000000000012 < im < 1.7e49

   1. Initial program 99.9%

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

      1. distribute-rgt-in 99.9%

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

      2. cancel-sign-sub 99.9%

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

      3. distribute-rgt-out-- 99.9%

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

      4. sub-neg 99.9%

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

      5. remove-double-neg 99.9%

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

      6. neg-sub0 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in re around 0 58.9%

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

   7. Applied egg-rr 26.7%

      \[\leadsto \color{blue}{{re}^{-4}} \]

   if 1.7e49 < im < 2.7e208

   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 76.4%

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

      1. +-commutative 76.4%

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

      2. fma-define 76.4%

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

      3. associate-*r* 76.4%

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

      4. distribute-rgt-out 76.4%

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

      5. +-commutative 76.4%

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

      6. *-commutative 76.4%

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

   7. Simplified 76.4%

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

   8. Taylor expanded in im around inf 85.3%

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

   9. Taylor expanded in re around 0 56.8%

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

   if 2.7e208 < im < 1.57999999999999994e227

   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 2.7%

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

   6. Taylor expanded in re around 0 60.9%

      \[\leadsto \color{blue}{re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{2}\right)} \]
   7. Step-by-step derivation

      1. *-commutative 60.9%

         \[\leadsto re \cdot \left(1 + \color{blue}{{re}^{2} \cdot -0.16666666666666666}\right) \]

   8. Simplified 60.9%

      \[\leadsto \color{blue}{re \cdot \left(1 + {re}^{2} \cdot -0.16666666666666666\right)} \]

   if 1.57999999999999994e227 < 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 64.7%

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

   6. Taylor expanded in im around 0 64.7%

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

      1. +-commutative 100.0%

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

      2. unpow2 100.0%

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

      3. fma-define 100.0%

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

   8. Simplified 64.7%

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

   9. Taylor expanded in im around inf 64.7%

      \[\leadsto \color{blue}{0.5 \cdot \left({im}^{2} \cdot re\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.056:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.7 \cdot 10^{+49}:\\ \;\;\;\;{re}^{-4}\\ \mathbf{elif}\;im \leq 2.7 \cdot 10^{+208}:\\ \;\;\;\;0.041666666666666664 \cdot \left(re \cdot {im}^{4}\right)\\ \mathbf{elif}\;im \leq 1.58 \cdot 10^{+227}:\\ \;\;\;\;re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot {im}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.056:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.7 \cdot 10^{+49}:\\ \;\;\;\;{re}^{-4}\\ \mathbf{elif}\;im \leq 2.7 \cdot 10^{+208} \lor \neg \left(im \leq 1.58 \cdot 10^{+227}\right):\\ \;\;\;\;0.041666666666666664 \cdot \left(re \cdot {im}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot {re}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.056)
   (sin re)
   (if (<= im 1.7e+49)
     (pow re -4.0)
     (if (or (<= im 2.7e+208) (not (<= im 1.58e+227)))
       (* 0.041666666666666664 (* re (pow im 4.0)))
       (* -0.16666666666666666 (pow re 3.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.056) {
		tmp = sin(re);
	} else if (im <= 1.7e+49) {
		tmp = pow(re, -4.0);
	} else if ((im <= 2.7e+208) || !(im <= 1.58e+227)) {
		tmp = 0.041666666666666664 * (re * pow(im, 4.0));
	} else {
		tmp = -0.16666666666666666 * pow(re, 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.056d0) then
        tmp = sin(re)
    else if (im <= 1.7d+49) then
        tmp = re ** (-4.0d0)
    else if ((im <= 2.7d+208) .or. (.not. (im <= 1.58d+227))) then
        tmp = 0.041666666666666664d0 * (re * (im ** 4.0d0))
    else
        tmp = (-0.16666666666666666d0) * (re ** 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.056) {
		tmp = Math.sin(re);
	} else if (im <= 1.7e+49) {
		tmp = Math.pow(re, -4.0);
	} else if ((im <= 2.7e+208) || !(im <= 1.58e+227)) {
		tmp = 0.041666666666666664 * (re * Math.pow(im, 4.0));
	} else {
		tmp = -0.16666666666666666 * Math.pow(re, 3.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.056:
		tmp = math.sin(re)
	elif im <= 1.7e+49:
		tmp = math.pow(re, -4.0)
	elif (im <= 2.7e+208) or not (im <= 1.58e+227):
		tmp = 0.041666666666666664 * (re * math.pow(im, 4.0))
	else:
		tmp = -0.16666666666666666 * math.pow(re, 3.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.056)
		tmp = sin(re);
	elseif (im <= 1.7e+49)
		tmp = re ^ -4.0;
	elseif ((im <= 2.7e+208) || !(im <= 1.58e+227))
		tmp = Float64(0.041666666666666664 * Float64(re * (im ^ 4.0)));
	else
		tmp = Float64(-0.16666666666666666 * (re ^ 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.056)
		tmp = sin(re);
	elseif (im <= 1.7e+49)
		tmp = re ^ -4.0;
	elseif ((im <= 2.7e+208) || ~((im <= 1.58e+227)))
		tmp = 0.041666666666666664 * (re * (im ^ 4.0));
	else
		tmp = -0.16666666666666666 * (re ^ 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.056], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.7e+49], N[Power[re, -4.0], $MachinePrecision], If[Or[LessEqual[im, 2.7e+208], N[Not[LessEqual[im, 1.58e+227]], $MachinePrecision]], N[(0.041666666666666664 * N[(re * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.16666666666666666 * N[Power[re, 3.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < 0.0560000000000000012

   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 60.9%

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

   if 0.0560000000000000012 < im < 1.7e49

   1. Initial program 99.9%

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

      1. distribute-rgt-in 99.9%

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

      2. cancel-sign-sub 99.9%

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

      3. distribute-rgt-out-- 99.9%

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

      4. sub-neg 99.9%

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

      5. remove-double-neg 99.9%

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

      6. neg-sub0 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in re around 0 58.9%

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

   7. Applied egg-rr 26.7%

      \[\leadsto \color{blue}{{re}^{-4}} \]

   if 1.7e49 < im < 2.7e208 or 1.57999999999999994e227 < 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 84.6%

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

      1. +-commutative 84.6%

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

      2. fma-define 84.6%

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

      3. associate-*r* 84.6%

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

      4. distribute-rgt-out 84.6%

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

      5. +-commutative 84.6%

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

      6. *-commutative 84.6%

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

   7. Simplified 84.6%

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

   8. Taylor expanded in im around inf 90.4%

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

   9. Taylor expanded in re around 0 59.5%

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

   if 2.7e208 < im < 1.57999999999999994e227

   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 2.7%

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

   6. Taylor expanded in re around 0 60.9%

      \[\leadsto \color{blue}{re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative 60.9%

         \[\leadsto re \cdot \color{blue}{\left(-0.16666666666666666 \cdot {re}^{2} + 1\right)} \]

      2. distribute-rgt-in 60.9%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot {re}^{2}\right) \cdot re + 1 \cdot re} \]

      3. *-lft-identity 60.9%

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

      4. associate-*l* 60.9%

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

      5. fma-define 60.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.16666666666666666, {re}^{2} \cdot re, re\right)} \]

      6. pow-plus 60.9%

         \[\leadsto \mathsf{fma}\left(-0.16666666666666666, \color{blue}{{re}^{\left(2 + 1\right)}}, re\right) \]

      7. metadata-eval 60.9%

         \[\leadsto \mathsf{fma}\left(-0.16666666666666666, {re}^{\color{blue}{3}}, re\right) \]

   8. Simplified 60.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.16666666666666666, {re}^{3}, re\right)} \]

   9. Taylor expanded in re around inf 60.6%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {re}^{3}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.056:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.7 \cdot 10^{+49}:\\ \;\;\;\;{re}^{-4}\\ \mathbf{elif}\;im \leq 2.7 \cdot 10^{+208} \lor \neg \left(im \leq 1.58 \cdot 10^{+227}\right):\\ \;\;\;\;0.041666666666666664 \cdot \left(re \cdot {im}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot {re}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 65.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.056:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.7 \cdot 10^{+49}:\\ \;\;\;\;{re}^{-4}\\ \mathbf{elif}\;im \leq 2.7 \cdot 10^{+208}:\\ \;\;\;\;0.041666666666666664 \cdot \left(re \cdot {im}^{4}\right)\\ \mathbf{elif}\;im \leq 1.58 \cdot 10^{+227}:\\ \;\;\;\;-0.16666666666666666 \cdot {re}^{3}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot {im}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.056)
   (sin re)
   (if (<= im 1.7e+49)
     (pow re -4.0)
     (if (<= im 2.7e+208)
       (* 0.041666666666666664 (* re (pow im 4.0)))
       (if (<= im 1.58e+227)
         (* -0.16666666666666666 (pow re 3.0))
         (* 0.5 (* re (pow im 2.0))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.056) {
		tmp = sin(re);
	} else if (im <= 1.7e+49) {
		tmp = pow(re, -4.0);
	} else if (im <= 2.7e+208) {
		tmp = 0.041666666666666664 * (re * pow(im, 4.0));
	} else if (im <= 1.58e+227) {
		tmp = -0.16666666666666666 * pow(re, 3.0);
	} else {
		tmp = 0.5 * (re * pow(im, 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.056d0) then
        tmp = sin(re)
    else if (im <= 1.7d+49) then
        tmp = re ** (-4.0d0)
    else if (im <= 2.7d+208) then
        tmp = 0.041666666666666664d0 * (re * (im ** 4.0d0))
    else if (im <= 1.58d+227) then
        tmp = (-0.16666666666666666d0) * (re ** 3.0d0)
    else
        tmp = 0.5d0 * (re * (im ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.056) {
		tmp = Math.sin(re);
	} else if (im <= 1.7e+49) {
		tmp = Math.pow(re, -4.0);
	} else if (im <= 2.7e+208) {
		tmp = 0.041666666666666664 * (re * Math.pow(im, 4.0));
	} else if (im <= 1.58e+227) {
		tmp = -0.16666666666666666 * Math.pow(re, 3.0);
	} else {
		tmp = 0.5 * (re * Math.pow(im, 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.056:
		tmp = math.sin(re)
	elif im <= 1.7e+49:
		tmp = math.pow(re, -4.0)
	elif im <= 2.7e+208:
		tmp = 0.041666666666666664 * (re * math.pow(im, 4.0))
	elif im <= 1.58e+227:
		tmp = -0.16666666666666666 * math.pow(re, 3.0)
	else:
		tmp = 0.5 * (re * math.pow(im, 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.056)
		tmp = sin(re);
	elseif (im <= 1.7e+49)
		tmp = re ^ -4.0;
	elseif (im <= 2.7e+208)
		tmp = Float64(0.041666666666666664 * Float64(re * (im ^ 4.0)));
	elseif (im <= 1.58e+227)
		tmp = Float64(-0.16666666666666666 * (re ^ 3.0));
	else
		tmp = Float64(0.5 * Float64(re * (im ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.056)
		tmp = sin(re);
	elseif (im <= 1.7e+49)
		tmp = re ^ -4.0;
	elseif (im <= 2.7e+208)
		tmp = 0.041666666666666664 * (re * (im ^ 4.0));
	elseif (im <= 1.58e+227)
		tmp = -0.16666666666666666 * (re ^ 3.0);
	else
		tmp = 0.5 * (re * (im ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.056], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.7e+49], N[Power[re, -4.0], $MachinePrecision], If[LessEqual[im, 2.7e+208], N[(0.041666666666666664 * N[(re * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.58e+227], N[(-0.16666666666666666 * N[Power[re, 3.0], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(re * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if im < 0.0560000000000000012

   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 60.9%

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

   if 0.0560000000000000012 < im < 1.7e49

   1. Initial program 99.9%

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

      1. distribute-rgt-in 99.9%

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

      2. cancel-sign-sub 99.9%

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

      3. distribute-rgt-out-- 99.9%

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

      4. sub-neg 99.9%

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

      5. remove-double-neg 99.9%

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

      6. neg-sub0 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in re around 0 58.9%

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

   7. Applied egg-rr 26.7%

      \[\leadsto \color{blue}{{re}^{-4}} \]

   if 1.7e49 < im < 2.7e208

   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 76.4%

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

      1. +-commutative 76.4%

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

      2. fma-define 76.4%

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

      3. associate-*r* 76.4%

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

      4. distribute-rgt-out 76.4%

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

      5. +-commutative 76.4%

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

      6. *-commutative 76.4%

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

   7. Simplified 76.4%

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

   8. Taylor expanded in im around inf 85.3%

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

   9. Taylor expanded in re around 0 56.8%

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

   if 2.7e208 < im < 1.57999999999999994e227

   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 2.7%

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

   6. Taylor expanded in re around 0 60.9%

      \[\leadsto \color{blue}{re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative 60.9%

         \[\leadsto re \cdot \color{blue}{\left(-0.16666666666666666 \cdot {re}^{2} + 1\right)} \]

      2. distribute-rgt-in 60.9%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot {re}^{2}\right) \cdot re + 1 \cdot re} \]

      3. *-lft-identity 60.9%

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

      4. associate-*l* 60.9%

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

      5. fma-define 60.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.16666666666666666, {re}^{2} \cdot re, re\right)} \]

      6. pow-plus 60.9%

         \[\leadsto \mathsf{fma}\left(-0.16666666666666666, \color{blue}{{re}^{\left(2 + 1\right)}}, re\right) \]

      7. metadata-eval 60.9%

         \[\leadsto \mathsf{fma}\left(-0.16666666666666666, {re}^{\color{blue}{3}}, re\right) \]

   8. Simplified 60.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.16666666666666666, {re}^{3}, re\right)} \]

   9. Taylor expanded in re around inf 60.6%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {re}^{3}} \]

   if 1.57999999999999994e227 < 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 64.7%

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

   6. Taylor expanded in im around 0 64.7%

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

      1. +-commutative 100.0%

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

      2. unpow2 100.0%

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

      3. fma-define 100.0%

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

   8. Simplified 64.7%

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

   9. Taylor expanded in im around inf 64.7%

      \[\leadsto \color{blue}{0.5 \cdot \left({im}^{2} \cdot re\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.056:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.7 \cdot 10^{+49}:\\ \;\;\;\;{re}^{-4}\\ \mathbf{elif}\;im \leq 2.7 \cdot 10^{+208}:\\ \;\;\;\;0.041666666666666664 \cdot \left(re \cdot {im}^{4}\right)\\ \mathbf{elif}\;im \leq 1.58 \cdot 10^{+227}:\\ \;\;\;\;-0.16666666666666666 \cdot {re}^{3}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot {im}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 55.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.056:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 3.6 \cdot 10^{+83} \lor \neg \left(im \leq 5 \cdot 10^{+108}\right):\\ \;\;\;\;{re}^{-4}\\ \mathbf{else}:\\ \;\;\;\;{re}^{6}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.056)
   (sin re)
   (if (or (<= im 3.6e+83) (not (<= im 5e+108))) (pow re -4.0) (pow re 6.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.056) {
		tmp = sin(re);
	} else if ((im <= 3.6e+83) || !(im <= 5e+108)) {
		tmp = pow(re, -4.0);
	} else {
		tmp = pow(re, 6.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.056d0) then
        tmp = sin(re)
    else if ((im <= 3.6d+83) .or. (.not. (im <= 5d+108))) then
        tmp = re ** (-4.0d0)
    else
        tmp = re ** 6.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.056) {
		tmp = Math.sin(re);
	} else if ((im <= 3.6e+83) || !(im <= 5e+108)) {
		tmp = Math.pow(re, -4.0);
	} else {
		tmp = Math.pow(re, 6.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.056:
		tmp = math.sin(re)
	elif (im <= 3.6e+83) or not (im <= 5e+108):
		tmp = math.pow(re, -4.0)
	else:
		tmp = math.pow(re, 6.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.056)
		tmp = sin(re);
	elseif ((im <= 3.6e+83) || !(im <= 5e+108))
		tmp = re ^ -4.0;
	else
		tmp = re ^ 6.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.056)
		tmp = sin(re);
	elseif ((im <= 3.6e+83) || ~((im <= 5e+108)))
		tmp = re ^ -4.0;
	else
		tmp = re ^ 6.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.056], N[Sin[re], $MachinePrecision], If[Or[LessEqual[im, 3.6e+83], N[Not[LessEqual[im, 5e+108]], $MachinePrecision]], N[Power[re, -4.0], $MachinePrecision], N[Power[re, 6.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.0560000000000000012

   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 60.9%

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

   if 0.0560000000000000012 < im < 3.5999999999999997e83 or 4.99999999999999991e108 < 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 64.2%

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

   7. Applied egg-rr 26.3%

      \[\leadsto \color{blue}{{re}^{-4}} \]

   if 3.5999999999999997e83 < im < 4.99999999999999991e108

   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 0.0%

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

   7. Applied egg-rr 0.0%

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

   9. Applied egg-rr 50.0%

      \[\leadsto \color{blue}{{re}^{6}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.056:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 3.6 \cdot 10^{+83} \lor \neg \left(im \leq 5 \cdot 10^{+108}\right):\\ \;\;\;\;{re}^{-4}\\ \mathbf{else}:\\ \;\;\;\;{re}^{6}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 55.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.056:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 4.1 \cdot 10^{+83} \lor \neg \left(im \leq 3.5 \cdot 10^{+112}\right):\\ \;\;\;\;{re}^{-4}\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.056)
   (sin re)
   (if (or (<= im 4.1e+83) (not (<= im 3.5e+112))) (pow re -4.0) (* re re))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.056) {
		tmp = sin(re);
	} else if ((im <= 4.1e+83) || !(im <= 3.5e+112)) {
		tmp = pow(re, -4.0);
	} else {
		tmp = re * re;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.056d0) then
        tmp = sin(re)
    else if ((im <= 4.1d+83) .or. (.not. (im <= 3.5d+112))) then
        tmp = re ** (-4.0d0)
    else
        tmp = re * re
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.056) {
		tmp = Math.sin(re);
	} else if ((im <= 4.1e+83) || !(im <= 3.5e+112)) {
		tmp = Math.pow(re, -4.0);
	} else {
		tmp = re * re;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.056:
		tmp = math.sin(re)
	elif (im <= 4.1e+83) or not (im <= 3.5e+112):
		tmp = math.pow(re, -4.0)
	else:
		tmp = re * re
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.056)
		tmp = sin(re);
	elseif ((im <= 4.1e+83) || !(im <= 3.5e+112))
		tmp = re ^ -4.0;
	else
		tmp = Float64(re * re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.056)
		tmp = sin(re);
	elseif ((im <= 4.1e+83) || ~((im <= 3.5e+112)))
		tmp = re ^ -4.0;
	else
		tmp = re * re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.056], N[Sin[re], $MachinePrecision], If[Or[LessEqual[im, 4.1e+83], N[Not[LessEqual[im, 3.5e+112]], $MachinePrecision]], N[Power[re, -4.0], $MachinePrecision], N[(re * re), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.0560000000000000012

   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 60.9%

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

   if 0.0560000000000000012 < im < 4.1000000000000001e83 or 3.49999999999999997e112 < 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 63.0%

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

   7. Applied egg-rr 25.5%

      \[\leadsto \color{blue}{{re}^{-4}} \]

   if 4.1000000000000001e83 < im < 3.49999999999999997e112

   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 50.0%

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

   7. Applied egg-rr 25.5%

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

   9. Applied egg-rr 1.5%

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

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.056:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 4.1 \cdot 10^{+83} \lor \neg \left(im \leq 3.5 \cdot 10^{+112}\right):\\ \;\;\;\;{re}^{-4}\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 53.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.6 \cdot 10^{-15}:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 2.6e-15) (sin re) (* re re)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.6e-15) {
		tmp = sin(re);
	} else {
		tmp = re * re;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 2.6d-15) then
        tmp = sin(re)
    else
        tmp = re * re
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.6e-15) {
		tmp = Math.sin(re);
	} else {
		tmp = re * re;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.6e-15:
		tmp = math.sin(re)
	else:
		tmp = re * re
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.6e-15)
		tmp = sin(re);
	else
		tmp = Float64(re * re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.6e-15)
		tmp = sin(re);
	else
		tmp = re * re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.6e-15], N[Sin[re], $MachinePrecision], N[(re * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2.60000000000000004e-15

   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 60.7%

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

   if 2.60000000000000004e-15 < 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 62.8%

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

   7. Applied egg-rr 17.5%

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

   9. Applied egg-rr 10.8%

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

Alternative 14: 29.3% accurate, 38.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.056:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 0.056) re (* re re)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.056) {
		tmp = re;
	} else {
		tmp = re * re;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.056d0) then
        tmp = re
    else
        tmp = re * re
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.056) {
		tmp = re;
	} else {
		tmp = re * re;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.056:
		tmp = re
	else:
		tmp = re * re
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.056)
		tmp = re;
	else
		tmp = Float64(re * re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.056)
		tmp = re;
	else
		tmp = re * re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.056], re, N[(re * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 0.0560000000000000012

   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 57.8%

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

   6. Taylor expanded in im around 0 30.6%

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

   if 0.0560000000000000012 < 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 62.2%

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

   7. Applied egg-rr 16.3%

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

   9. Applied egg-rr 10.9%

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

Alternative 15: 26.6% accurate, 51.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.75 \cdot 10^{-21}:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= re 1.75e-21) re -1.0))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.75e-21) {
		tmp = re;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 1.75d-21) then
        tmp = re
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.75e-21) {
		tmp = re;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.75e-21:
		tmp = re
	else:
		tmp = -1.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.75e-21)
		tmp = re;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.75e-21)
		tmp = re;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.75e-21], re, -1.0]

Derivation

1. Split input into 2 regimes

2. if re < 1.7500000000000002e-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 re around 0 73.5%

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

   6. Taylor expanded in im around 0 31.2%

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

   if 1.7500000000000002e-21 < 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. 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 21.1%

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

   7. Applied egg-rr 2.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(re\right)} - -4} \]
   8. Step-by-step derivation

      1. sub-neg 2.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(re\right)} + \left(--4\right)} \]

      2. metadata-eval 2.5%

         \[\leadsto e^{\mathsf{log1p}\left(re\right)} + \color{blue}{4} \]

      3. +-commutative 2.5%

         \[\leadsto \color{blue}{4 + e^{\mathsf{log1p}\left(re\right)}} \]

      4. log1p-undefine 2.5%

         \[\leadsto 4 + e^{\color{blue}{\log \left(1 + re\right)}} \]

      5. rem-exp-log 2.5%

         \[\leadsto 4 + \color{blue}{\left(1 + re\right)} \]

      6. +-commutative 2.5%

         \[\leadsto 4 + \color{blue}{\left(re + 1\right)} \]

   9. Simplified 2.5%

      \[\leadsto \color{blue}{4 + \left(re + 1\right)} \]

   11. Applied egg-rr 7.9%

       \[\leadsto \color{blue}{-1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 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. 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

6. Applied egg-rr 4.0%

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

   1. +-inverses 4.0%

      \[\leadsto \frac{\sin re \cdot -2}{\sin re \cdot -2 + \color{blue}{0}} \]

   2. +-rgt-identity 4.0%

      \[\leadsto \frac{\sin re \cdot -2}{\color{blue}{\sin re \cdot -2}} \]

   3. *-inverses 4.0%

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

8. Simplified 4.0%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Alternative 17: 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. 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 58.9%

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

7. Applied egg-rr 1.9%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(re\right)} - -4} \]
8. Step-by-step derivation

   1. sub-neg 1.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(re\right)} + \left(--4\right)} \]

   2. metadata-eval 1.9%

      \[\leadsto e^{\mathsf{log1p}\left(re\right)} + \color{blue}{4} \]

   3. +-commutative 1.9%

      \[\leadsto \color{blue}{4 + e^{\mathsf{log1p}\left(re\right)}} \]

   4. log1p-undefine 1.9%

      \[\leadsto 4 + e^{\color{blue}{\log \left(1 + re\right)}} \]

   5. rem-exp-log 2.7%

      \[\leadsto 4 + \color{blue}{\left(1 + re\right)} \]

   6. +-commutative 2.7%

      \[\leadsto 4 + \color{blue}{\left(re + 1\right)} \]

9. Simplified 2.7%

   \[\leadsto \color{blue}{4 + \left(re + 1\right)} \]

11. Applied egg-rr 4.0%

    \[\leadsto \color{blue}{0.5} \]
12. Add Preprocessing

Alternative 18: 4.4% 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. 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 58.9%

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

7. Applied egg-rr 1.9%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(re\right)} - -4} \]
8. Step-by-step derivation

   1. sub-neg 1.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(re\right)} + \left(--4\right)} \]

   2. metadata-eval 1.9%

      \[\leadsto e^{\mathsf{log1p}\left(re\right)} + \color{blue}{4} \]

   3. +-commutative 1.9%

      \[\leadsto \color{blue}{4 + e^{\mathsf{log1p}\left(re\right)}} \]

   4. log1p-undefine 1.9%

      \[\leadsto 4 + e^{\color{blue}{\log \left(1 + re\right)}} \]

   5. rem-exp-log 2.7%

      \[\leadsto 4 + \color{blue}{\left(1 + re\right)} \]

   6. +-commutative 2.7%

      \[\leadsto 4 + \color{blue}{\left(re + 1\right)} \]

9. Simplified 2.7%

   \[\leadsto \color{blue}{4 + \left(re + 1\right)} \]

11. Applied egg-rr 3.7%

    \[\leadsto \color{blue}{0.125} \]
12. Add Preprocessing

Alternative 19: 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 re around 0 58.9%

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

7. Applied egg-rr 1.9%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(re\right)} - -4} \]
8. Step-by-step derivation

   1. sub-neg 1.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(re\right)} + \left(--4\right)} \]

   2. metadata-eval 1.9%

      \[\leadsto e^{\mathsf{log1p}\left(re\right)} + \color{blue}{4} \]

   3. +-commutative 1.9%

      \[\leadsto \color{blue}{4 + e^{\mathsf{log1p}\left(re\right)}} \]

   4. log1p-undefine 1.9%

      \[\leadsto 4 + e^{\color{blue}{\log \left(1 + re\right)}} \]

   5. rem-exp-log 2.7%

      \[\leadsto 4 + \color{blue}{\left(1 + re\right)} \]

   6. +-commutative 2.7%

      \[\leadsto 4 + \color{blue}{\left(re + 1\right)} \]

9. Simplified 2.7%

   \[\leadsto \color{blue}{4 + \left(re + 1\right)} \]

11. Applied egg-rr 5.1%

    \[\leadsto \color{blue}{-1} \]
12. Add Preprocessing

Alternative 20: 4.1% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ -4 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 -4.0)

C

double code(double re, double im) {
	return -4.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = -4.0d0
end function

Java

public static double code(double re, double im) {
	return -4.0;
}

Python

def code(re, im):
	return -4.0

Julia

function code(re, im)
	return -4.0
end

MATLAB

function tmp = code(re, im)
	tmp = -4.0;
end

Wolfram

code[re_, im_] := -4.0

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 re around 0 58.9%

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

7. Applied egg-rr 1.9%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(re\right)} - -4} \]
8. Step-by-step derivation

   1. sub-neg 1.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(re\right)} + \left(--4\right)} \]

   2. metadata-eval 1.9%

      \[\leadsto e^{\mathsf{log1p}\left(re\right)} + \color{blue}{4} \]

   3. +-commutative 1.9%

      \[\leadsto \color{blue}{4 + e^{\mathsf{log1p}\left(re\right)}} \]

   4. log1p-undefine 1.9%

      \[\leadsto 4 + e^{\color{blue}{\log \left(1 + re\right)}} \]

   5. rem-exp-log 2.7%

      \[\leadsto 4 + \color{blue}{\left(1 + re\right)} \]

   6. +-commutative 2.7%

      \[\leadsto 4 + \color{blue}{\left(re + 1\right)} \]

9. Simplified 2.7%

   \[\leadsto \color{blue}{4 + \left(re + 1\right)} \]

11. Applied egg-rr 4.3%

    \[\leadsto \color{blue}{-4} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024107 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))