math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 7.2s

Alternatives: 22

Speedup: 1.0×

• 48.6% 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: 82.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.22:\\ \;\;\;\;\sin re \cdot \left(1 + 0.5 \cdot {im}^{2}\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+98}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;im \leq 10^{+152}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(-0.16666666666666666 \cdot {re}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sin re \cdot {im}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.22)
   (* (sin re) (+ 1.0 (* 0.5 (pow im 2.0))))
   (if (<= im 1.1e+98)
     (* (+ (exp (- im)) (exp im)) (* 0.5 re))
     (if (<= im 1e+152)
       (log1p (expm1 (* -0.16666666666666666 (pow re 3.0))))
       (* 0.5 (* (sin re) (pow im 2.0)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.22) {
		tmp = sin(re) * (1.0 + (0.5 * pow(im, 2.0)));
	} else if (im <= 1.1e+98) {
		tmp = (exp(-im) + exp(im)) * (0.5 * re);
	} else if (im <= 1e+152) {
		tmp = log1p(expm1((-0.16666666666666666 * pow(re, 3.0))));
	} else {
		tmp = 0.5 * (sin(re) * pow(im, 2.0));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.22) {
		tmp = Math.sin(re) * (1.0 + (0.5 * Math.pow(im, 2.0)));
	} else if (im <= 1.1e+98) {
		tmp = (Math.exp(-im) + Math.exp(im)) * (0.5 * re);
	} else if (im <= 1e+152) {
		tmp = Math.log1p(Math.expm1((-0.16666666666666666 * Math.pow(re, 3.0))));
	} else {
		tmp = 0.5 * (Math.sin(re) * Math.pow(im, 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.22:
		tmp = math.sin(re) * (1.0 + (0.5 * math.pow(im, 2.0)))
	elif im <= 1.1e+98:
		tmp = (math.exp(-im) + math.exp(im)) * (0.5 * re)
	elif im <= 1e+152:
		tmp = math.log1p(math.expm1((-0.16666666666666666 * math.pow(re, 3.0))))
	else:
		tmp = 0.5 * (math.sin(re) * math.pow(im, 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.22)
		tmp = Float64(sin(re) * Float64(1.0 + Float64(0.5 * (im ^ 2.0))));
	elseif (im <= 1.1e+98)
		tmp = Float64(Float64(exp(Float64(-im)) + exp(im)) * Float64(0.5 * re));
	elseif (im <= 1e+152)
		tmp = log1p(expm1(Float64(-0.16666666666666666 * (re ^ 3.0))));
	else
		tmp = Float64(0.5 * Float64(sin(re) * (im ^ 2.0)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.22], N[(N[Sin[re], $MachinePrecision] * N[(1.0 + N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.1e+98], N[(N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1e+152], N[Log[1 + N[(Exp[N[(-0.16666666666666666 * N[Power[re, 3.0], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision], N[(0.5 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < 0.220000000000000001

   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 82.0%

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

      1. *-lft-identity 82.0%

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

      2. associate-*r* 82.0%

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

      3. distribute-rgt-out 82.0%

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

   7. Simplified 82.0%

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

   if 0.220000000000000001 < im < 1.10000000000000004e98

   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 87.5%

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

   if 1.10000000000000004e98 < im < 1e152

   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 3.0%

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

   6. Taylor expanded in re around 0 32.7%

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

   7. Taylor expanded in re around inf 32.3%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {re}^{3}} \]
   8. Step-by-step derivation

      1. log1p-expm1-u 54.2%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(-0.16666666666666666 \cdot {re}^{3}\right)\right)} \]

   9. Applied egg-rr 54.2%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(-0.16666666666666666 \cdot {re}^{3}\right)\right)} \]

   if 1e152 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 97.3%

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

      1. *-lft-identity 97.3%

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

      2. associate-*r* 97.3%

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

      3. distribute-rgt-out 97.3%

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

   7. Simplified 97.3%

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

   8. Taylor expanded in im around inf 97.3%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.22:\\ \;\;\;\;\sin re \cdot \left(1 + 0.5 \cdot {im}^{2}\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+98}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;im \leq 10^{+152}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(-0.16666666666666666 \cdot {re}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sin re \cdot {im}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ t_1 := 1 + 0.5 \cdot {im}^{2}\\ \mathbf{if}\;im \leq 0.15:\\ \;\;\;\;\sin re \cdot t\_1\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+98}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im \leq 1.7 \cdot 10^{+142}:\\ \;\;\;\;t\_1 \cdot \left(re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{2}\right)\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sin re \cdot {im}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (+ (exp (- im)) (exp im)) (* 0.5 re)))
        (t_1 (+ 1.0 (* 0.5 (pow im 2.0)))))
   (if (<= im 0.15)
     (* (sin re) t_1)
     (if (<= im 1.1e+98)
       t_0
       (if (<= im 1.7e+142)
         (* t_1 (* re (+ 1.0 (* -0.16666666666666666 (pow re 2.0)))))
         (if (<= im 1.35e+154) t_0 (* 0.5 (* (sin re) (pow im 2.0)))))))))

C

double code(double re, double im) {
	double t_0 = (exp(-im) + exp(im)) * (0.5 * re);
	double t_1 = 1.0 + (0.5 * pow(im, 2.0));
	double tmp;
	if (im <= 0.15) {
		tmp = sin(re) * t_1;
	} else if (im <= 1.1e+98) {
		tmp = t_0;
	} else if (im <= 1.7e+142) {
		tmp = t_1 * (re * (1.0 + (-0.16666666666666666 * pow(re, 2.0))));
	} else if (im <= 1.35e+154) {
		tmp = t_0;
	} else {
		tmp = 0.5 * (sin(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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (exp(-im) + exp(im)) * (0.5d0 * re)
    t_1 = 1.0d0 + (0.5d0 * (im ** 2.0d0))
    if (im <= 0.15d0) then
        tmp = sin(re) * t_1
    else if (im <= 1.1d+98) then
        tmp = t_0
    else if (im <= 1.7d+142) then
        tmp = t_1 * (re * (1.0d0 + ((-0.16666666666666666d0) * (re ** 2.0d0))))
    else if (im <= 1.35d+154) then
        tmp = t_0
    else
        tmp = 0.5d0 * (sin(re) * (im ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (Math.exp(-im) + Math.exp(im)) * (0.5 * re);
	double t_1 = 1.0 + (0.5 * Math.pow(im, 2.0));
	double tmp;
	if (im <= 0.15) {
		tmp = Math.sin(re) * t_1;
	} else if (im <= 1.1e+98) {
		tmp = t_0;
	} else if (im <= 1.7e+142) {
		tmp = t_1 * (re * (1.0 + (-0.16666666666666666 * Math.pow(re, 2.0))));
	} else if (im <= 1.35e+154) {
		tmp = t_0;
	} else {
		tmp = 0.5 * (Math.sin(re) * Math.pow(im, 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (math.exp(-im) + math.exp(im)) * (0.5 * re)
	t_1 = 1.0 + (0.5 * math.pow(im, 2.0))
	tmp = 0
	if im <= 0.15:
		tmp = math.sin(re) * t_1
	elif im <= 1.1e+98:
		tmp = t_0
	elif im <= 1.7e+142:
		tmp = t_1 * (re * (1.0 + (-0.16666666666666666 * math.pow(re, 2.0))))
	elif im <= 1.35e+154:
		tmp = t_0
	else:
		tmp = 0.5 * (math.sin(re) * math.pow(im, 2.0))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(exp(Float64(-im)) + exp(im)) * Float64(0.5 * re))
	t_1 = Float64(1.0 + Float64(0.5 * (im ^ 2.0)))
	tmp = 0.0
	if (im <= 0.15)
		tmp = Float64(sin(re) * t_1);
	elseif (im <= 1.1e+98)
		tmp = t_0;
	elseif (im <= 1.7e+142)
		tmp = Float64(t_1 * Float64(re * Float64(1.0 + Float64(-0.16666666666666666 * (re ^ 2.0)))));
	elseif (im <= 1.35e+154)
		tmp = t_0;
	else
		tmp = Float64(0.5 * Float64(sin(re) * (im ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (exp(-im) + exp(im)) * (0.5 * re);
	t_1 = 1.0 + (0.5 * (im ^ 2.0));
	tmp = 0.0;
	if (im <= 0.15)
		tmp = sin(re) * t_1;
	elseif (im <= 1.1e+98)
		tmp = t_0;
	elseif (im <= 1.7e+142)
		tmp = t_1 * (re * (1.0 + (-0.16666666666666666 * (re ^ 2.0))));
	elseif (im <= 1.35e+154)
		tmp = t_0;
	else
		tmp = 0.5 * (sin(re) * (im ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 0.15], N[(N[Sin[re], $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[im, 1.1e+98], t$95$0, If[LessEqual[im, 1.7e+142], N[(t$95$1 * N[(re * N[(1.0 + N[(-0.16666666666666666 * N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.35e+154], t$95$0, N[(0.5 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < 0.149999999999999994

   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 82.0%

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

      1. *-lft-identity 82.0%

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

      2. associate-*r* 82.0%

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

      3. distribute-rgt-out 82.0%

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

   7. Simplified 82.0%

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

   if 0.149999999999999994 < im < 1.10000000000000004e98 or 1.6999999999999999e142 < im < 1.35000000000000003e154

   1. Initial program 100.0%

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

      1. 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 85.0%

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

   if 1.10000000000000004e98 < im < 1.6999999999999999e142

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 6.1%

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

      1. *-lft-identity 6.1%

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

      2. associate-*r* 6.1%

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

      3. distribute-rgt-out 6.1%

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

   7. Simplified 6.1%

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

   8. Taylor expanded in re around 0 43.4%

      \[\leadsto \color{blue}{\left(re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{2}\right)\right)} \cdot \left(1 + 0.5 \cdot {im}^{2}\right) \]

   if 1.35000000000000003e154 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

      1. *-lft-identity 100.0%

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

      2. associate-*r* 100.0%

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

      3. distribute-rgt-out 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around inf 100.0%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.15:\\ \;\;\;\;\sin re \cdot \left(1 + 0.5 \cdot {im}^{2}\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+98}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;im \leq 1.7 \cdot 10^{+142}:\\ \;\;\;\;\left(1 + 0.5 \cdot {im}^{2}\right) \cdot \left(re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{2}\right)\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sin re \cdot {im}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{if}\;im \leq 0.056:\\ \;\;\;\;\sin re \cdot \left(1 + 0.5 \cdot {im}^{2}\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+98}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im \leq 2.2 \cdot 10^{+142}:\\ \;\;\;\;0.5 \cdot \left({im}^{2} \cdot \left(re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{2}\right)\right)\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sin re \cdot {im}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (+ (exp (- im)) (exp im)) (* 0.5 re))))
   (if (<= im 0.056)
     (* (sin re) (+ 1.0 (* 0.5 (pow im 2.0))))
     (if (<= im 1.1e+98)
       t_0
       (if (<= im 2.2e+142)
         (*
          0.5
          (*
           (pow im 2.0)
           (* re (+ 1.0 (* -0.16666666666666666 (pow re 2.0))))))
         (if (<= im 1.35e+154) t_0 (* 0.5 (* (sin re) (pow im 2.0)))))))))

C

double code(double re, double im) {
	double t_0 = (exp(-im) + exp(im)) * (0.5 * re);
	double tmp;
	if (im <= 0.056) {
		tmp = sin(re) * (1.0 + (0.5 * pow(im, 2.0)));
	} else if (im <= 1.1e+98) {
		tmp = t_0;
	} else if (im <= 2.2e+142) {
		tmp = 0.5 * (pow(im, 2.0) * (re * (1.0 + (-0.16666666666666666 * pow(re, 2.0)))));
	} else if (im <= 1.35e+154) {
		tmp = t_0;
	} else {
		tmp = 0.5 * (sin(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) :: t_0
    real(8) :: tmp
    t_0 = (exp(-im) + exp(im)) * (0.5d0 * re)
    if (im <= 0.056d0) then
        tmp = sin(re) * (1.0d0 + (0.5d0 * (im ** 2.0d0)))
    else if (im <= 1.1d+98) then
        tmp = t_0
    else if (im <= 2.2d+142) then
        tmp = 0.5d0 * ((im ** 2.0d0) * (re * (1.0d0 + ((-0.16666666666666666d0) * (re ** 2.0d0)))))
    else if (im <= 1.35d+154) then
        tmp = t_0
    else
        tmp = 0.5d0 * (sin(re) * (im ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (Math.exp(-im) + Math.exp(im)) * (0.5 * re);
	double tmp;
	if (im <= 0.056) {
		tmp = Math.sin(re) * (1.0 + (0.5 * Math.pow(im, 2.0)));
	} else if (im <= 1.1e+98) {
		tmp = t_0;
	} else if (im <= 2.2e+142) {
		tmp = 0.5 * (Math.pow(im, 2.0) * (re * (1.0 + (-0.16666666666666666 * Math.pow(re, 2.0)))));
	} else if (im <= 1.35e+154) {
		tmp = t_0;
	} else {
		tmp = 0.5 * (Math.sin(re) * Math.pow(im, 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (math.exp(-im) + math.exp(im)) * (0.5 * re)
	tmp = 0
	if im <= 0.056:
		tmp = math.sin(re) * (1.0 + (0.5 * math.pow(im, 2.0)))
	elif im <= 1.1e+98:
		tmp = t_0
	elif im <= 2.2e+142:
		tmp = 0.5 * (math.pow(im, 2.0) * (re * (1.0 + (-0.16666666666666666 * math.pow(re, 2.0)))))
	elif im <= 1.35e+154:
		tmp = t_0
	else:
		tmp = 0.5 * (math.sin(re) * math.pow(im, 2.0))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(exp(Float64(-im)) + exp(im)) * Float64(0.5 * re))
	tmp = 0.0
	if (im <= 0.056)
		tmp = Float64(sin(re) * Float64(1.0 + Float64(0.5 * (im ^ 2.0))));
	elseif (im <= 1.1e+98)
		tmp = t_0;
	elseif (im <= 2.2e+142)
		tmp = Float64(0.5 * Float64((im ^ 2.0) * Float64(re * Float64(1.0 + Float64(-0.16666666666666666 * (re ^ 2.0))))));
	elseif (im <= 1.35e+154)
		tmp = t_0;
	else
		tmp = Float64(0.5 * Float64(sin(re) * (im ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (exp(-im) + exp(im)) * (0.5 * re);
	tmp = 0.0;
	if (im <= 0.056)
		tmp = sin(re) * (1.0 + (0.5 * (im ^ 2.0)));
	elseif (im <= 1.1e+98)
		tmp = t_0;
	elseif (im <= 2.2e+142)
		tmp = 0.5 * ((im ^ 2.0) * (re * (1.0 + (-0.16666666666666666 * (re ^ 2.0)))));
	elseif (im <= 1.35e+154)
		tmp = t_0;
	else
		tmp = 0.5 * (sin(re) * (im ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 0.056], N[(N[Sin[re], $MachinePrecision] * N[(1.0 + N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.1e+98], t$95$0, If[LessEqual[im, 2.2e+142], N[(0.5 * N[(N[Power[im, 2.0], $MachinePrecision] * N[(re * N[(1.0 + N[(-0.16666666666666666 * N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.35e+154], t$95$0, N[(0.5 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 2.0], $MachinePrecision]), $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 82.0%

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

      1. *-lft-identity 82.0%

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

      2. associate-*r* 82.0%

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

      3. distribute-rgt-out 82.0%

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

   7. Simplified 82.0%

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

   if 0.0560000000000000012 < im < 1.10000000000000004e98 or 2.19999999999999987e142 < im < 1.35000000000000003e154

   1. Initial program 100.0%

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

      1. 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 85.0%

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

   if 1.10000000000000004e98 < im < 2.19999999999999987e142

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 6.1%

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

      1. *-lft-identity 6.1%

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

      2. associate-*r* 6.1%

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

      3. distribute-rgt-out 6.1%

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

   7. Simplified 6.1%

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

   8. Taylor expanded in re around 0 43.4%

      \[\leadsto \color{blue}{\left(re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{2}\right)\right)} \cdot \left(1 + 0.5 \cdot {im}^{2}\right) \]

   9. Taylor expanded in im around inf 43.4%

      \[\leadsto \color{blue}{0.5 \cdot \left({im}^{2} \cdot \left(re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{2}\right)\right)\right)} \]

   if 1.35000000000000003e154 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

      1. *-lft-identity 100.0%

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

      2. associate-*r* 100.0%

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

      3. distribute-rgt-out 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around inf 100.0%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.056:\\ \;\;\;\;\sin re \cdot \left(1 + 0.5 \cdot {im}^{2}\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+98}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;im \leq 2.2 \cdot 10^{+142}:\\ \;\;\;\;0.5 \cdot \left({im}^{2} \cdot \left(re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{2}\right)\right)\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sin re \cdot {im}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{if}\;im \leq 0.019:\\ \;\;\;\;\sin re \cdot \left(1 + 0.5 \cdot {im}^{2}\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+98}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im \leq 1.7 \cdot 10^{+142}:\\ \;\;\;\;{re}^{3} \cdot \left(-0.16666666666666666 + {im}^{2} \cdot -0.08333333333333333\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sin re \cdot {im}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (+ (exp (- im)) (exp im)) (* 0.5 re))))
   (if (<= im 0.019)
     (* (sin re) (+ 1.0 (* 0.5 (pow im 2.0))))
     (if (<= im 1.1e+98)
       t_0
       (if (<= im 1.7e+142)
         (*
          (pow re 3.0)
          (+ -0.16666666666666666 (* (pow im 2.0) -0.08333333333333333)))
         (if (<= im 1.35e+154) t_0 (* 0.5 (* (sin re) (pow im 2.0)))))))))

C

double code(double re, double im) {
	double t_0 = (exp(-im) + exp(im)) * (0.5 * re);
	double tmp;
	if (im <= 0.019) {
		tmp = sin(re) * (1.0 + (0.5 * pow(im, 2.0)));
	} else if (im <= 1.1e+98) {
		tmp = t_0;
	} else if (im <= 1.7e+142) {
		tmp = pow(re, 3.0) * (-0.16666666666666666 + (pow(im, 2.0) * -0.08333333333333333));
	} else if (im <= 1.35e+154) {
		tmp = t_0;
	} else {
		tmp = 0.5 * (sin(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) :: t_0
    real(8) :: tmp
    t_0 = (exp(-im) + exp(im)) * (0.5d0 * re)
    if (im <= 0.019d0) then
        tmp = sin(re) * (1.0d0 + (0.5d0 * (im ** 2.0d0)))
    else if (im <= 1.1d+98) then
        tmp = t_0
    else if (im <= 1.7d+142) then
        tmp = (re ** 3.0d0) * ((-0.16666666666666666d0) + ((im ** 2.0d0) * (-0.08333333333333333d0)))
    else if (im <= 1.35d+154) then
        tmp = t_0
    else
        tmp = 0.5d0 * (sin(re) * (im ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (Math.exp(-im) + Math.exp(im)) * (0.5 * re);
	double tmp;
	if (im <= 0.019) {
		tmp = Math.sin(re) * (1.0 + (0.5 * Math.pow(im, 2.0)));
	} else if (im <= 1.1e+98) {
		tmp = t_0;
	} else if (im <= 1.7e+142) {
		tmp = Math.pow(re, 3.0) * (-0.16666666666666666 + (Math.pow(im, 2.0) * -0.08333333333333333));
	} else if (im <= 1.35e+154) {
		tmp = t_0;
	} else {
		tmp = 0.5 * (Math.sin(re) * Math.pow(im, 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (math.exp(-im) + math.exp(im)) * (0.5 * re)
	tmp = 0
	if im <= 0.019:
		tmp = math.sin(re) * (1.0 + (0.5 * math.pow(im, 2.0)))
	elif im <= 1.1e+98:
		tmp = t_0
	elif im <= 1.7e+142:
		tmp = math.pow(re, 3.0) * (-0.16666666666666666 + (math.pow(im, 2.0) * -0.08333333333333333))
	elif im <= 1.35e+154:
		tmp = t_0
	else:
		tmp = 0.5 * (math.sin(re) * math.pow(im, 2.0))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(exp(Float64(-im)) + exp(im)) * Float64(0.5 * re))
	tmp = 0.0
	if (im <= 0.019)
		tmp = Float64(sin(re) * Float64(1.0 + Float64(0.5 * (im ^ 2.0))));
	elseif (im <= 1.1e+98)
		tmp = t_0;
	elseif (im <= 1.7e+142)
		tmp = Float64((re ^ 3.0) * Float64(-0.16666666666666666 + Float64((im ^ 2.0) * -0.08333333333333333)));
	elseif (im <= 1.35e+154)
		tmp = t_0;
	else
		tmp = Float64(0.5 * Float64(sin(re) * (im ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (exp(-im) + exp(im)) * (0.5 * re);
	tmp = 0.0;
	if (im <= 0.019)
		tmp = sin(re) * (1.0 + (0.5 * (im ^ 2.0)));
	elseif (im <= 1.1e+98)
		tmp = t_0;
	elseif (im <= 1.7e+142)
		tmp = (re ^ 3.0) * (-0.16666666666666666 + ((im ^ 2.0) * -0.08333333333333333));
	elseif (im <= 1.35e+154)
		tmp = t_0;
	else
		tmp = 0.5 * (sin(re) * (im ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 0.019], N[(N[Sin[re], $MachinePrecision] * N[(1.0 + N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.1e+98], t$95$0, If[LessEqual[im, 1.7e+142], N[(N[Power[re, 3.0], $MachinePrecision] * N[(-0.16666666666666666 + N[(N[Power[im, 2.0], $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.35e+154], t$95$0, N[(0.5 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < 0.0189999999999999995

   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 82.0%

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

      1. *-lft-identity 82.0%

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

      2. associate-*r* 82.0%

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

      3. distribute-rgt-out 82.0%

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

   7. Simplified 82.0%

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

   if 0.0189999999999999995 < im < 1.10000000000000004e98 or 1.6999999999999999e142 < im < 1.35000000000000003e154

   1. Initial program 100.0%

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

      1. 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 85.0%

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

   if 1.10000000000000004e98 < im < 1.6999999999999999e142

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 6.1%

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

      1. *-lft-identity 6.1%

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

      2. associate-*r* 6.1%

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

      3. distribute-rgt-out 6.1%

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

   7. Simplified 6.1%

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

   8. Taylor expanded in re around 0 43.4%

      \[\leadsto \color{blue}{\left(re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{2}\right)\right)} \cdot \left(1 + 0.5 \cdot {im}^{2}\right) \]

   9. Taylor expanded in re around inf 42.2%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({re}^{3} \cdot \left(1 + 0.5 \cdot {im}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* 42.2%

          \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot {re}^{3}\right) \cdot \left(1 + 0.5 \cdot {im}^{2}\right)} \]

       2. *-commutative 42.2%

          \[\leadsto \color{blue}{\left({re}^{3} \cdot -0.16666666666666666\right)} \cdot \left(1 + 0.5 \cdot {im}^{2}\right) \]

       3. +-commutative 42.2%

          \[\leadsto \left({re}^{3} \cdot -0.16666666666666666\right) \cdot \color{blue}{\left(0.5 \cdot {im}^{2} + 1\right)} \]

       4. fma-undefine 42.2%

          \[\leadsto \left({re}^{3} \cdot -0.16666666666666666\right) \cdot \color{blue}{\mathsf{fma}\left(0.5, {im}^{2}, 1\right)} \]

       5. associate-*r* 42.2%

          \[\leadsto \color{blue}{{re}^{3} \cdot \left(-0.16666666666666666 \cdot \mathsf{fma}\left(0.5, {im}^{2}, 1\right)\right)} \]

       6. fma-undefine 42.2%

          \[\leadsto {re}^{3} \cdot \left(-0.16666666666666666 \cdot \color{blue}{\left(0.5 \cdot {im}^{2} + 1\right)}\right) \]

       7. +-commutative 42.2%

          \[\leadsto {re}^{3} \cdot \left(-0.16666666666666666 \cdot \color{blue}{\left(1 + 0.5 \cdot {im}^{2}\right)}\right) \]

       8. distribute-rgt-in 42.2%

          \[\leadsto {re}^{3} \cdot \color{blue}{\left(1 \cdot -0.16666666666666666 + \left(0.5 \cdot {im}^{2}\right) \cdot -0.16666666666666666\right)} \]

       9. metadata-eval 42.2%

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

       10. *-commutative 42.2%

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

       11. associate-*l* 42.2%

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

       12. metadata-eval 42.2%

           \[\leadsto {re}^{3} \cdot \left(-0.16666666666666666 + {im}^{2} \cdot \color{blue}{-0.08333333333333333}\right) \]

   11. Simplified 42.2%

       \[\leadsto \color{blue}{{re}^{3} \cdot \left(-0.16666666666666666 + {im}^{2} \cdot -0.08333333333333333\right)} \]

   if 1.35000000000000003e154 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

      1. *-lft-identity 100.0%

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

      2. associate-*r* 100.0%

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

      3. distribute-rgt-out 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around inf 100.0%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.019:\\ \;\;\;\;\sin re \cdot \left(1 + 0.5 \cdot {im}^{2}\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+98}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;im \leq 1.7 \cdot 10^{+142}:\\ \;\;\;\;{re}^{3} \cdot \left(-0.16666666666666666 + {im}^{2} \cdot -0.08333333333333333\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sin re \cdot {im}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{if}\;im \leq 0.45:\\ \;\;\;\;\sin re \cdot \left(1 + 0.5 \cdot {im}^{2}\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+98}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im \leq 5 \cdot 10^{+142}:\\ \;\;\;\;re \cdot \left(1 + \sqrt[3]{{re}^{6} \cdot -0.004629629629629629}\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sin re \cdot {im}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (+ (exp (- im)) (exp im)) (* 0.5 re))))
   (if (<= im 0.45)
     (* (sin re) (+ 1.0 (* 0.5 (pow im 2.0))))
     (if (<= im 1.1e+98)
       t_0
       (if (<= im 5e+142)
         (* re (+ 1.0 (cbrt (* (pow re 6.0) -0.004629629629629629))))
         (if (<= im 1.35e+154) t_0 (* 0.5 (* (sin re) (pow im 2.0)))))))))

C

double code(double re, double im) {
	double t_0 = (exp(-im) + exp(im)) * (0.5 * re);
	double tmp;
	if (im <= 0.45) {
		tmp = sin(re) * (1.0 + (0.5 * pow(im, 2.0)));
	} else if (im <= 1.1e+98) {
		tmp = t_0;
	} else if (im <= 5e+142) {
		tmp = re * (1.0 + cbrt((pow(re, 6.0) * -0.004629629629629629)));
	} else if (im <= 1.35e+154) {
		tmp = t_0;
	} else {
		tmp = 0.5 * (sin(re) * pow(im, 2.0));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = (Math.exp(-im) + Math.exp(im)) * (0.5 * re);
	double tmp;
	if (im <= 0.45) {
		tmp = Math.sin(re) * (1.0 + (0.5 * Math.pow(im, 2.0)));
	} else if (im <= 1.1e+98) {
		tmp = t_0;
	} else if (im <= 5e+142) {
		tmp = re * (1.0 + Math.cbrt((Math.pow(re, 6.0) * -0.004629629629629629)));
	} else if (im <= 1.35e+154) {
		tmp = t_0;
	} else {
		tmp = 0.5 * (Math.sin(re) * Math.pow(im, 2.0));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(exp(Float64(-im)) + exp(im)) * Float64(0.5 * re))
	tmp = 0.0
	if (im <= 0.45)
		tmp = Float64(sin(re) * Float64(1.0 + Float64(0.5 * (im ^ 2.0))));
	elseif (im <= 1.1e+98)
		tmp = t_0;
	elseif (im <= 5e+142)
		tmp = Float64(re * Float64(1.0 + cbrt(Float64((re ^ 6.0) * -0.004629629629629629))));
	elseif (im <= 1.35e+154)
		tmp = t_0;
	else
		tmp = Float64(0.5 * Float64(sin(re) * (im ^ 2.0)));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 0.45], N[(N[Sin[re], $MachinePrecision] * N[(1.0 + N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.1e+98], t$95$0, If[LessEqual[im, 5e+142], N[(re * N[(1.0 + N[Power[N[(N[Power[re, 6.0], $MachinePrecision] * -0.004629629629629629), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.35e+154], t$95$0, N[(0.5 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < 0.450000000000000011

   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 82.0%

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

      1. *-lft-identity 82.0%

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

      2. associate-*r* 82.0%

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

      3. distribute-rgt-out 82.0%

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

   7. Simplified 82.0%

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

   if 0.450000000000000011 < im < 1.10000000000000004e98 or 5.0000000000000001e142 < im < 1.35000000000000003e154

   1. Initial program 100.0%

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

      1. 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 85.0%

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

   if 1.10000000000000004e98 < im < 5.0000000000000001e142

   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 32.2%

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

      1. add-cbrt-cube 41.4%

         \[\leadsto re \cdot \left(1 + \color{blue}{\sqrt[3]{\left(\left(-0.16666666666666666 \cdot {re}^{2}\right) \cdot \left(-0.16666666666666666 \cdot {re}^{2}\right)\right) \cdot \left(-0.16666666666666666 \cdot {re}^{2}\right)}}\right) \]

      2. pow1/3 10.2%

         \[\leadsto re \cdot \left(1 + \color{blue}{{\left(\left(\left(-0.16666666666666666 \cdot {re}^{2}\right) \cdot \left(-0.16666666666666666 \cdot {re}^{2}\right)\right) \cdot \left(-0.16666666666666666 \cdot {re}^{2}\right)\right)}^{0.3333333333333333}}\right) \]

      3. pow3 10.2%

         \[\leadsto re \cdot \left(1 + {\color{blue}{\left({\left(-0.16666666666666666 \cdot {re}^{2}\right)}^{3}\right)}}^{0.3333333333333333}\right) \]

      4. *-commutative 10.2%

         \[\leadsto re \cdot \left(1 + {\left({\color{blue}{\left({re}^{2} \cdot -0.16666666666666666\right)}}^{3}\right)}^{0.3333333333333333}\right) \]

      5. unpow-prod-down 10.2%

         \[\leadsto re \cdot \left(1 + {\color{blue}{\left({\left({re}^{2}\right)}^{3} \cdot {-0.16666666666666666}^{3}\right)}}^{0.3333333333333333}\right) \]

      6. unpow2 10.2%

         \[\leadsto re \cdot \left(1 + {\left({\color{blue}{\left(re \cdot re\right)}}^{3} \cdot {-0.16666666666666666}^{3}\right)}^{0.3333333333333333}\right) \]

      7. pow-prod-down 10.2%

         \[\leadsto re \cdot \left(1 + {\left(\color{blue}{\left({re}^{3} \cdot {re}^{3}\right)} \cdot {-0.16666666666666666}^{3}\right)}^{0.3333333333333333}\right) \]

      8. pow-prod-up 10.2%

         \[\leadsto re \cdot \left(1 + {\left(\color{blue}{{re}^{\left(3 + 3\right)}} \cdot {-0.16666666666666666}^{3}\right)}^{0.3333333333333333}\right) \]

      9. metadata-eval 10.2%

         \[\leadsto re \cdot \left(1 + {\left({re}^{\color{blue}{6}} \cdot {-0.16666666666666666}^{3}\right)}^{0.3333333333333333}\right) \]

      10. metadata-eval 10.2%

          \[\leadsto re \cdot \left(1 + {\left({re}^{6} \cdot \color{blue}{-0.004629629629629629}\right)}^{0.3333333333333333}\right) \]

   8. Applied egg-rr 10.2%

      \[\leadsto re \cdot \left(1 + \color{blue}{{\left({re}^{6} \cdot -0.004629629629629629\right)}^{0.3333333333333333}}\right) \]
   9. Step-by-step derivation

      1. unpow1/3 41.4%

         \[\leadsto re \cdot \left(1 + \color{blue}{\sqrt[3]{{re}^{6} \cdot -0.004629629629629629}}\right) \]

   10. Simplified 41.4%

       \[\leadsto re \cdot \left(1 + \color{blue}{\sqrt[3]{{re}^{6} \cdot -0.004629629629629629}}\right) \]

   if 1.35000000000000003e154 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

      1. *-lft-identity 100.0%

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

      2. associate-*r* 100.0%

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

      3. distribute-rgt-out 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around inf 100.0%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.45:\\ \;\;\;\;\sin re \cdot \left(1 + 0.5 \cdot {im}^{2}\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+98}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;im \leq 5 \cdot 10^{+142}:\\ \;\;\;\;re \cdot \left(1 + \sqrt[3]{{re}^{6} \cdot -0.004629629629629629}\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sin re \cdot {im}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.1 \cdot 10^{+15}:\\ \;\;\;\;\sin re \cdot \left(1 + 0.5 \cdot {im}^{2}\right)\\ \mathbf{elif}\;im \leq 4.3 \cdot 10^{+97}:\\ \;\;\;\;2 \cdot {\sin re}^{-2}\\ \mathbf{elif}\;im \leq 10^{+152}:\\ \;\;\;\;re \cdot \left(1 + \sqrt[3]{{re}^{6} \cdot -0.004629629629629629}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sin re \cdot {im}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.1e+15)
   (* (sin re) (+ 1.0 (* 0.5 (pow im 2.0))))
   (if (<= im 4.3e+97)
     (* 2.0 (pow (sin re) -2.0))
     (if (<= im 1e+152)
       (* re (+ 1.0 (cbrt (* (pow re 6.0) -0.004629629629629629))))
       (* 0.5 (* (sin re) (pow im 2.0)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.1e+15) {
		tmp = sin(re) * (1.0 + (0.5 * pow(im, 2.0)));
	} else if (im <= 4.3e+97) {
		tmp = 2.0 * pow(sin(re), -2.0);
	} else if (im <= 1e+152) {
		tmp = re * (1.0 + cbrt((pow(re, 6.0) * -0.004629629629629629)));
	} else {
		tmp = 0.5 * (sin(re) * pow(im, 2.0));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.1e+15) {
		tmp = Math.sin(re) * (1.0 + (0.5 * Math.pow(im, 2.0)));
	} else if (im <= 4.3e+97) {
		tmp = 2.0 * Math.pow(Math.sin(re), -2.0);
	} else if (im <= 1e+152) {
		tmp = re * (1.0 + Math.cbrt((Math.pow(re, 6.0) * -0.004629629629629629)));
	} else {
		tmp = 0.5 * (Math.sin(re) * Math.pow(im, 2.0));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.1e+15)
		tmp = Float64(sin(re) * Float64(1.0 + Float64(0.5 * (im ^ 2.0))));
	elseif (im <= 4.3e+97)
		tmp = Float64(2.0 * (sin(re) ^ -2.0));
	elseif (im <= 1e+152)
		tmp = Float64(re * Float64(1.0 + cbrt(Float64((re ^ 6.0) * -0.004629629629629629))));
	else
		tmp = Float64(0.5 * Float64(sin(re) * (im ^ 2.0)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.1e+15], N[(N[Sin[re], $MachinePrecision] * N[(1.0 + N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 4.3e+97], N[(2.0 * N[Power[N[Sin[re], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1e+152], N[(re * N[(1.0 + N[Power[N[(N[Power[re, 6.0], $MachinePrecision] * -0.004629629629629629), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < 2.1e15

   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 82.0%

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

      1. *-lft-identity 82.0%

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

      2. associate-*r* 82.0%

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

      3. distribute-rgt-out 82.0%

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

   7. Simplified 82.0%

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

   if 2.1e15 < im < 4.2999999999999998e97

   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 2.4%

      \[\leadsto \color{blue}{\sin re + \sin re} \]
   7. Step-by-step derivation

      1. count-2 2.4%

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

   8. Simplified 2.4%

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

   10. Applied egg-rr 8.2%

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

   if 4.2999999999999998e97 < im < 1e152

   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 3.0%

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

   6. Taylor expanded in re around 0 32.7%

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

      1. add-cbrt-cube 39.8%

         \[\leadsto re \cdot \left(1 + \color{blue}{\sqrt[3]{\left(\left(-0.16666666666666666 \cdot {re}^{2}\right) \cdot \left(-0.16666666666666666 \cdot {re}^{2}\right)\right) \cdot \left(-0.16666666666666666 \cdot {re}^{2}\right)}}\right) \]

      2. pow1/3 7.8%

         \[\leadsto re \cdot \left(1 + \color{blue}{{\left(\left(\left(-0.16666666666666666 \cdot {re}^{2}\right) \cdot \left(-0.16666666666666666 \cdot {re}^{2}\right)\right) \cdot \left(-0.16666666666666666 \cdot {re}^{2}\right)\right)}^{0.3333333333333333}}\right) \]

      3. pow3 7.8%

         \[\leadsto re \cdot \left(1 + {\color{blue}{\left({\left(-0.16666666666666666 \cdot {re}^{2}\right)}^{3}\right)}}^{0.3333333333333333}\right) \]

      4. *-commutative 7.8%

         \[\leadsto re \cdot \left(1 + {\left({\color{blue}{\left({re}^{2} \cdot -0.16666666666666666\right)}}^{3}\right)}^{0.3333333333333333}\right) \]

      5. unpow-prod-down 7.8%

         \[\leadsto re \cdot \left(1 + {\color{blue}{\left({\left({re}^{2}\right)}^{3} \cdot {-0.16666666666666666}^{3}\right)}}^{0.3333333333333333}\right) \]

      6. unpow2 7.8%

         \[\leadsto re \cdot \left(1 + {\left({\color{blue}{\left(re \cdot re\right)}}^{3} \cdot {-0.16666666666666666}^{3}\right)}^{0.3333333333333333}\right) \]

      7. pow-prod-down 7.8%

         \[\leadsto re \cdot \left(1 + {\left(\color{blue}{\left({re}^{3} \cdot {re}^{3}\right)} \cdot {-0.16666666666666666}^{3}\right)}^{0.3333333333333333}\right) \]

      8. pow-prod-up 7.8%

         \[\leadsto re \cdot \left(1 + {\left(\color{blue}{{re}^{\left(3 + 3\right)}} \cdot {-0.16666666666666666}^{3}\right)}^{0.3333333333333333}\right) \]

      9. metadata-eval 7.8%

         \[\leadsto re \cdot \left(1 + {\left({re}^{\color{blue}{6}} \cdot {-0.16666666666666666}^{3}\right)}^{0.3333333333333333}\right) \]

      10. metadata-eval 7.8%

          \[\leadsto re \cdot \left(1 + {\left({re}^{6} \cdot \color{blue}{-0.004629629629629629}\right)}^{0.3333333333333333}\right) \]

   8. Applied egg-rr 7.8%

      \[\leadsto re \cdot \left(1 + \color{blue}{{\left({re}^{6} \cdot -0.004629629629629629\right)}^{0.3333333333333333}}\right) \]
   9. Step-by-step derivation

      1. unpow1/3 39.8%

         \[\leadsto re \cdot \left(1 + \color{blue}{\sqrt[3]{{re}^{6} \cdot -0.004629629629629629}}\right) \]

   10. Simplified 39.8%

       \[\leadsto re \cdot \left(1 + \color{blue}{\sqrt[3]{{re}^{6} \cdot -0.004629629629629629}}\right) \]

   if 1e152 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 97.3%

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

      1. *-lft-identity 97.3%

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

      2. associate-*r* 97.3%

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

      3. distribute-rgt-out 97.3%

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

   7. Simplified 97.3%

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

   8. Taylor expanded in im around inf 97.3%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.1 \cdot 10^{+15}:\\ \;\;\;\;\sin re \cdot \left(1 + 0.5 \cdot {im}^{2}\right)\\ \mathbf{elif}\;im \leq 4.3 \cdot 10^{+97}:\\ \;\;\;\;2 \cdot {\sin re}^{-2}\\ \mathbf{elif}\;im \leq 10^{+152}:\\ \;\;\;\;re \cdot \left(1 + \sqrt[3]{{re}^{6} \cdot -0.004629629629629629}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sin re \cdot {im}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 60000000000:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 4.6 \cdot 10^{+97}:\\ \;\;\;\;2 \cdot {\sin re}^{-2}\\ \mathbf{elif}\;im \leq 10^{+152}:\\ \;\;\;\;re \cdot \left(1 + \sqrt[3]{{re}^{6} \cdot -0.004629629629629629}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sin re \cdot {im}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 60000000000.0)
   (sin re)
   (if (<= im 4.6e+97)
     (* 2.0 (pow (sin re) -2.0))
     (if (<= im 1e+152)
       (* re (+ 1.0 (cbrt (* (pow re 6.0) -0.004629629629629629))))
       (* 0.5 (* (sin re) (pow im 2.0)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 60000000000.0) {
		tmp = sin(re);
	} else if (im <= 4.6e+97) {
		tmp = 2.0 * pow(sin(re), -2.0);
	} else if (im <= 1e+152) {
		tmp = re * (1.0 + cbrt((pow(re, 6.0) * -0.004629629629629629)));
	} else {
		tmp = 0.5 * (sin(re) * pow(im, 2.0));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 60000000000.0) {
		tmp = Math.sin(re);
	} else if (im <= 4.6e+97) {
		tmp = 2.0 * Math.pow(Math.sin(re), -2.0);
	} else if (im <= 1e+152) {
		tmp = re * (1.0 + Math.cbrt((Math.pow(re, 6.0) * -0.004629629629629629)));
	} else {
		tmp = 0.5 * (Math.sin(re) * Math.pow(im, 2.0));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 60000000000.0)
		tmp = sin(re);
	elseif (im <= 4.6e+97)
		tmp = Float64(2.0 * (sin(re) ^ -2.0));
	elseif (im <= 1e+152)
		tmp = Float64(re * Float64(1.0 + cbrt(Float64((re ^ 6.0) * -0.004629629629629629))));
	else
		tmp = Float64(0.5 * Float64(sin(re) * (im ^ 2.0)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 60000000000.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 4.6e+97], N[(2.0 * N[Power[N[Sin[re], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1e+152], N[(re * N[(1.0 + N[Power[N[(N[Power[re, 6.0], $MachinePrecision] * -0.004629629629629629), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < 6e10

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 64.1%

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

   if 6e10 < im < 4.60000000000000011e97

   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 2.4%

      \[\leadsto \color{blue}{\sin re + \sin re} \]
   7. Step-by-step derivation

      1. count-2 2.4%

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

   8. Simplified 2.4%

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

   10. Applied egg-rr 8.2%

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

   if 4.60000000000000011e97 < im < 1e152

   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 3.0%

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

   6. Taylor expanded in re around 0 32.7%

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

      1. add-cbrt-cube 39.8%

         \[\leadsto re \cdot \left(1 + \color{blue}{\sqrt[3]{\left(\left(-0.16666666666666666 \cdot {re}^{2}\right) \cdot \left(-0.16666666666666666 \cdot {re}^{2}\right)\right) \cdot \left(-0.16666666666666666 \cdot {re}^{2}\right)}}\right) \]

      2. pow1/3 7.8%

         \[\leadsto re \cdot \left(1 + \color{blue}{{\left(\left(\left(-0.16666666666666666 \cdot {re}^{2}\right) \cdot \left(-0.16666666666666666 \cdot {re}^{2}\right)\right) \cdot \left(-0.16666666666666666 \cdot {re}^{2}\right)\right)}^{0.3333333333333333}}\right) \]

      3. pow3 7.8%

         \[\leadsto re \cdot \left(1 + {\color{blue}{\left({\left(-0.16666666666666666 \cdot {re}^{2}\right)}^{3}\right)}}^{0.3333333333333333}\right) \]

      4. *-commutative 7.8%

         \[\leadsto re \cdot \left(1 + {\left({\color{blue}{\left({re}^{2} \cdot -0.16666666666666666\right)}}^{3}\right)}^{0.3333333333333333}\right) \]

      5. unpow-prod-down 7.8%

         \[\leadsto re \cdot \left(1 + {\color{blue}{\left({\left({re}^{2}\right)}^{3} \cdot {-0.16666666666666666}^{3}\right)}}^{0.3333333333333333}\right) \]

      6. unpow2 7.8%

         \[\leadsto re \cdot \left(1 + {\left({\color{blue}{\left(re \cdot re\right)}}^{3} \cdot {-0.16666666666666666}^{3}\right)}^{0.3333333333333333}\right) \]

      7. pow-prod-down 7.8%

         \[\leadsto re \cdot \left(1 + {\left(\color{blue}{\left({re}^{3} \cdot {re}^{3}\right)} \cdot {-0.16666666666666666}^{3}\right)}^{0.3333333333333333}\right) \]

      8. pow-prod-up 7.8%

         \[\leadsto re \cdot \left(1 + {\left(\color{blue}{{re}^{\left(3 + 3\right)}} \cdot {-0.16666666666666666}^{3}\right)}^{0.3333333333333333}\right) \]

      9. metadata-eval 7.8%

         \[\leadsto re \cdot \left(1 + {\left({re}^{\color{blue}{6}} \cdot {-0.16666666666666666}^{3}\right)}^{0.3333333333333333}\right) \]

      10. metadata-eval 7.8%

          \[\leadsto re \cdot \left(1 + {\left({re}^{6} \cdot \color{blue}{-0.004629629629629629}\right)}^{0.3333333333333333}\right) \]

   8. Applied egg-rr 7.8%

      \[\leadsto re \cdot \left(1 + \color{blue}{{\left({re}^{6} \cdot -0.004629629629629629\right)}^{0.3333333333333333}}\right) \]
   9. Step-by-step derivation

      1. unpow1/3 39.8%

         \[\leadsto re \cdot \left(1 + \color{blue}{\sqrt[3]{{re}^{6} \cdot -0.004629629629629629}}\right) \]

   10. Simplified 39.8%

       \[\leadsto re \cdot \left(1 + \color{blue}{\sqrt[3]{{re}^{6} \cdot -0.004629629629629629}}\right) \]

   if 1e152 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 97.3%

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

      1. *-lft-identity 97.3%

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

      2. associate-*r* 97.3%

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

      3. distribute-rgt-out 97.3%

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

   7. Simplified 97.3%

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

   8. Taylor expanded in im around inf 97.3%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 60000000000:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 4.6 \cdot 10^{+97}:\\ \;\;\;\;2 \cdot {\sin re}^{-2}\\ \mathbf{elif}\;im \leq 10^{+152}:\\ \;\;\;\;re \cdot \left(1 + \sqrt[3]{{re}^{6} \cdot -0.004629629629629629}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sin re \cdot {im}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 60000000000:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.55 \cdot 10^{+101}:\\ \;\;\;\;2 \cdot {\sin re}^{-2}\\ \mathbf{elif}\;im \leq 10^{+152}:\\ \;\;\;\;2 \cdot \left(re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sin re \cdot {im}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 60000000000.0)
   (sin re)
   (if (<= im 1.55e+101)
     (* 2.0 (pow (sin re) -2.0))
     (if (<= im 1e+152)
       (* 2.0 (* re (+ 1.0 (* -0.16666666666666666 (pow re 2.0)))))
       (* 0.5 (* (sin re) (pow im 2.0)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 60000000000.0) {
		tmp = sin(re);
	} else if (im <= 1.55e+101) {
		tmp = 2.0 * pow(sin(re), -2.0);
	} else if (im <= 1e+152) {
		tmp = 2.0 * (re * (1.0 + (-0.16666666666666666 * pow(re, 2.0))));
	} else {
		tmp = 0.5 * (sin(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 <= 60000000000.0d0) then
        tmp = sin(re)
    else if (im <= 1.55d+101) then
        tmp = 2.0d0 * (sin(re) ** (-2.0d0))
    else if (im <= 1d+152) then
        tmp = 2.0d0 * (re * (1.0d0 + ((-0.16666666666666666d0) * (re ** 2.0d0))))
    else
        tmp = 0.5d0 * (sin(re) * (im ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 60000000000.0) {
		tmp = Math.sin(re);
	} else if (im <= 1.55e+101) {
		tmp = 2.0 * Math.pow(Math.sin(re), -2.0);
	} else if (im <= 1e+152) {
		tmp = 2.0 * (re * (1.0 + (-0.16666666666666666 * Math.pow(re, 2.0))));
	} else {
		tmp = 0.5 * (Math.sin(re) * Math.pow(im, 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 60000000000.0:
		tmp = math.sin(re)
	elif im <= 1.55e+101:
		tmp = 2.0 * math.pow(math.sin(re), -2.0)
	elif im <= 1e+152:
		tmp = 2.0 * (re * (1.0 + (-0.16666666666666666 * math.pow(re, 2.0))))
	else:
		tmp = 0.5 * (math.sin(re) * math.pow(im, 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 60000000000.0)
		tmp = sin(re);
	elseif (im <= 1.55e+101)
		tmp = Float64(2.0 * (sin(re) ^ -2.0));
	elseif (im <= 1e+152)
		tmp = Float64(2.0 * Float64(re * Float64(1.0 + Float64(-0.16666666666666666 * (re ^ 2.0)))));
	else
		tmp = Float64(0.5 * Float64(sin(re) * (im ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 60000000000.0)
		tmp = sin(re);
	elseif (im <= 1.55e+101)
		tmp = 2.0 * (sin(re) ^ -2.0);
	elseif (im <= 1e+152)
		tmp = 2.0 * (re * (1.0 + (-0.16666666666666666 * (re ^ 2.0))));
	else
		tmp = 0.5 * (sin(re) * (im ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 60000000000.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.55e+101], N[(2.0 * N[Power[N[Sin[re], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1e+152], N[(2.0 * N[(re * N[(1.0 + N[(-0.16666666666666666 * N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < 6e10

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 64.1%

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

   if 6e10 < im < 1.55e101

   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 2.5%

      \[\leadsto \color{blue}{\sin re + \sin re} \]
   7. Step-by-step derivation

      1. count-2 2.5%

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

   8. Simplified 2.5%

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

   10. Applied egg-rr 12.2%

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

   if 1.55e101 < im < 1e152

   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 3.1%

      \[\leadsto \color{blue}{\sin re + \sin re} \]
   7. Step-by-step derivation

      1. count-2 3.1%

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

   8. Simplified 3.1%

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

   9. Taylor expanded in re around 0 41.3%

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

   if 1e152 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 97.3%

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

      1. *-lft-identity 97.3%

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

      2. associate-*r* 97.3%

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

      3. distribute-rgt-out 97.3%

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

   7. Simplified 97.3%

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

   8. Taylor expanded in im around inf 97.3%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 60000000000:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.55 \cdot 10^{+101}:\\ \;\;\;\;2 \cdot {\sin re}^{-2}\\ \mathbf{elif}\;im \leq 10^{+152}:\\ \;\;\;\;2 \cdot \left(re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sin re \cdot {im}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 60000000000:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+101}:\\ \;\;\;\;2 \cdot {\sin re}^{-2}\\ \mathbf{elif}\;im \leq 10^{+152}:\\ \;\;\;\;2 \cdot \left(re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + 0.5 \cdot {im}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 60000000000.0)
   (sin re)
   (if (<= im 1.9e+101)
     (* 2.0 (pow (sin re) -2.0))
     (if (<= im 1e+152)
       (* 2.0 (* re (+ 1.0 (* -0.16666666666666666 (pow re 2.0)))))
       (* re (+ 1.0 (* 0.5 (pow im 2.0))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 60000000000.0) {
		tmp = sin(re);
	} else if (im <= 1.9e+101) {
		tmp = 2.0 * pow(sin(re), -2.0);
	} else if (im <= 1e+152) {
		tmp = 2.0 * (re * (1.0 + (-0.16666666666666666 * pow(re, 2.0))));
	} else {
		tmp = re * (1.0 + (0.5 * 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 <= 60000000000.0d0) then
        tmp = sin(re)
    else if (im <= 1.9d+101) then
        tmp = 2.0d0 * (sin(re) ** (-2.0d0))
    else if (im <= 1d+152) then
        tmp = 2.0d0 * (re * (1.0d0 + ((-0.16666666666666666d0) * (re ** 2.0d0))))
    else
        tmp = re * (1.0d0 + (0.5d0 * (im ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 60000000000.0) {
		tmp = Math.sin(re);
	} else if (im <= 1.9e+101) {
		tmp = 2.0 * Math.pow(Math.sin(re), -2.0);
	} else if (im <= 1e+152) {
		tmp = 2.0 * (re * (1.0 + (-0.16666666666666666 * Math.pow(re, 2.0))));
	} else {
		tmp = re * (1.0 + (0.5 * Math.pow(im, 2.0)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 60000000000.0:
		tmp = math.sin(re)
	elif im <= 1.9e+101:
		tmp = 2.0 * math.pow(math.sin(re), -2.0)
	elif im <= 1e+152:
		tmp = 2.0 * (re * (1.0 + (-0.16666666666666666 * math.pow(re, 2.0))))
	else:
		tmp = re * (1.0 + (0.5 * math.pow(im, 2.0)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 60000000000.0)
		tmp = sin(re);
	elseif (im <= 1.9e+101)
		tmp = Float64(2.0 * (sin(re) ^ -2.0));
	elseif (im <= 1e+152)
		tmp = Float64(2.0 * Float64(re * Float64(1.0 + Float64(-0.16666666666666666 * (re ^ 2.0)))));
	else
		tmp = Float64(re * Float64(1.0 + Float64(0.5 * (im ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 60000000000.0)
		tmp = sin(re);
	elseif (im <= 1.9e+101)
		tmp = 2.0 * (sin(re) ^ -2.0);
	elseif (im <= 1e+152)
		tmp = 2.0 * (re * (1.0 + (-0.16666666666666666 * (re ^ 2.0))));
	else
		tmp = re * (1.0 + (0.5 * (im ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 60000000000.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.9e+101], N[(2.0 * N[Power[N[Sin[re], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1e+152], N[(2.0 * N[(re * N[(1.0 + N[(-0.16666666666666666 * N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(1.0 + N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < 6e10

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 64.1%

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

   if 6e10 < im < 1.8999999999999999e101

   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 2.5%

      \[\leadsto \color{blue}{\sin re + \sin re} \]
   7. Step-by-step derivation

      1. count-2 2.5%

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

   8. Simplified 2.5%

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

   10. Applied egg-rr 12.2%

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

   if 1.8999999999999999e101 < im < 1e152

   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 3.1%

      \[\leadsto \color{blue}{\sin re + \sin re} \]
   7. Step-by-step derivation

      1. count-2 3.1%

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

   8. Simplified 3.1%

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

   9. Taylor expanded in re around 0 41.3%

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

   if 1e152 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 97.3%

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

      1. *-lft-identity 97.3%

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

      2. associate-*r* 97.3%

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

      3. distribute-rgt-out 97.3%

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

   7. Simplified 97.3%

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

   8. Taylor expanded in re around 0 77.9%

      \[\leadsto \color{blue}{re \cdot \left(1 + 0.5 \cdot {im}^{2}\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 11: 62.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 11500000000000:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2.05 \cdot 10^{+101}:\\ \;\;\;\;{\sin re}^{-2}\\ \mathbf{elif}\;im \leq 9.2 \cdot 10^{+151}:\\ \;\;\;\;2 \cdot \left(re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + 0.5 \cdot {im}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 11500000000000.0)
   (sin re)
   (if (<= im 2.05e+101)
     (pow (sin re) -2.0)
     (if (<= im 9.2e+151)
       (* 2.0 (* re (+ 1.0 (* -0.16666666666666666 (pow re 2.0)))))
       (* re (+ 1.0 (* 0.5 (pow im 2.0))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 11500000000000.0) {
		tmp = sin(re);
	} else if (im <= 2.05e+101) {
		tmp = pow(sin(re), -2.0);
	} else if (im <= 9.2e+151) {
		tmp = 2.0 * (re * (1.0 + (-0.16666666666666666 * pow(re, 2.0))));
	} else {
		tmp = re * (1.0 + (0.5 * 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 <= 11500000000000.0d0) then
        tmp = sin(re)
    else if (im <= 2.05d+101) then
        tmp = sin(re) ** (-2.0d0)
    else if (im <= 9.2d+151) then
        tmp = 2.0d0 * (re * (1.0d0 + ((-0.16666666666666666d0) * (re ** 2.0d0))))
    else
        tmp = re * (1.0d0 + (0.5d0 * (im ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 11500000000000.0) {
		tmp = Math.sin(re);
	} else if (im <= 2.05e+101) {
		tmp = Math.pow(Math.sin(re), -2.0);
	} else if (im <= 9.2e+151) {
		tmp = 2.0 * (re * (1.0 + (-0.16666666666666666 * Math.pow(re, 2.0))));
	} else {
		tmp = re * (1.0 + (0.5 * Math.pow(im, 2.0)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 11500000000000.0:
		tmp = math.sin(re)
	elif im <= 2.05e+101:
		tmp = math.pow(math.sin(re), -2.0)
	elif im <= 9.2e+151:
		tmp = 2.0 * (re * (1.0 + (-0.16666666666666666 * math.pow(re, 2.0))))
	else:
		tmp = re * (1.0 + (0.5 * math.pow(im, 2.0)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 11500000000000.0)
		tmp = sin(re);
	elseif (im <= 2.05e+101)
		tmp = sin(re) ^ -2.0;
	elseif (im <= 9.2e+151)
		tmp = Float64(2.0 * Float64(re * Float64(1.0 + Float64(-0.16666666666666666 * (re ^ 2.0)))));
	else
		tmp = Float64(re * Float64(1.0 + Float64(0.5 * (im ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 11500000000000.0)
		tmp = sin(re);
	elseif (im <= 2.05e+101)
		tmp = sin(re) ^ -2.0;
	elseif (im <= 9.2e+151)
		tmp = 2.0 * (re * (1.0 + (-0.16666666666666666 * (re ^ 2.0))));
	else
		tmp = re * (1.0 + (0.5 * (im ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 11500000000000.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 2.05e+101], N[Power[N[Sin[re], $MachinePrecision], -2.0], $MachinePrecision], If[LessEqual[im, 9.2e+151], N[(2.0 * N[(re * N[(1.0 + N[(-0.16666666666666666 * N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(1.0 + N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < 1.15e13

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 64.1%

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

   if 1.15e13 < im < 2.05e101

   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 2.5%

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

   7. Applied egg-rr 12.2%

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

   if 2.05e101 < im < 9.2000000000000003e151

   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 3.1%

      \[\leadsto \color{blue}{\sin re + \sin re} \]
   7. Step-by-step derivation

      1. count-2 3.1%

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

   8. Simplified 3.1%

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

   9. Taylor expanded in re around 0 41.3%

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

   if 9.2000000000000003e151 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 97.3%

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

      1. *-lft-identity 97.3%

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

      2. associate-*r* 97.3%

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

      3. distribute-rgt-out 97.3%

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

   7. Simplified 97.3%

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

   8. Taylor expanded in re around 0 77.9%

      \[\leadsto \color{blue}{re \cdot \left(1 + 0.5 \cdot {im}^{2}\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 12: 62.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.1 \cdot 10^{+15}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+101}:\\ \;\;\;\;2 \cdot {re}^{-2}\\ \mathbf{elif}\;im \leq 10^{+152}:\\ \;\;\;\;2 \cdot \left(re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + 0.5 \cdot {im}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.1e+15)
   (sin re)
   (if (<= im 1.4e+101)
     (* 2.0 (pow re -2.0))
     (if (<= im 1e+152)
       (* 2.0 (* re (+ 1.0 (* -0.16666666666666666 (pow re 2.0)))))
       (* re (+ 1.0 (* 0.5 (pow im 2.0))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.1e+15) {
		tmp = sin(re);
	} else if (im <= 1.4e+101) {
		tmp = 2.0 * pow(re, -2.0);
	} else if (im <= 1e+152) {
		tmp = 2.0 * (re * (1.0 + (-0.16666666666666666 * pow(re, 2.0))));
	} else {
		tmp = re * (1.0 + (0.5 * 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 <= 2.1d+15) then
        tmp = sin(re)
    else if (im <= 1.4d+101) then
        tmp = 2.0d0 * (re ** (-2.0d0))
    else if (im <= 1d+152) then
        tmp = 2.0d0 * (re * (1.0d0 + ((-0.16666666666666666d0) * (re ** 2.0d0))))
    else
        tmp = re * (1.0d0 + (0.5d0 * (im ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.1e+15) {
		tmp = Math.sin(re);
	} else if (im <= 1.4e+101) {
		tmp = 2.0 * Math.pow(re, -2.0);
	} else if (im <= 1e+152) {
		tmp = 2.0 * (re * (1.0 + (-0.16666666666666666 * Math.pow(re, 2.0))));
	} else {
		tmp = re * (1.0 + (0.5 * Math.pow(im, 2.0)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.1e+15:
		tmp = math.sin(re)
	elif im <= 1.4e+101:
		tmp = 2.0 * math.pow(re, -2.0)
	elif im <= 1e+152:
		tmp = 2.0 * (re * (1.0 + (-0.16666666666666666 * math.pow(re, 2.0))))
	else:
		tmp = re * (1.0 + (0.5 * math.pow(im, 2.0)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.1e+15)
		tmp = sin(re);
	elseif (im <= 1.4e+101)
		tmp = Float64(2.0 * (re ^ -2.0));
	elseif (im <= 1e+152)
		tmp = Float64(2.0 * Float64(re * Float64(1.0 + Float64(-0.16666666666666666 * (re ^ 2.0)))));
	else
		tmp = Float64(re * Float64(1.0 + Float64(0.5 * (im ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.1e+15)
		tmp = sin(re);
	elseif (im <= 1.4e+101)
		tmp = 2.0 * (re ^ -2.0);
	elseif (im <= 1e+152)
		tmp = 2.0 * (re * (1.0 + (-0.16666666666666666 * (re ^ 2.0))));
	else
		tmp = re * (1.0 + (0.5 * (im ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.1e+15], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.4e+101], N[(2.0 * N[Power[re, -2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1e+152], N[(2.0 * N[(re * N[(1.0 + N[(-0.16666666666666666 * N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(1.0 + N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < 2.1e15

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 64.1%

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

   if 2.1e15 < im < 1.39999999999999991e101

   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 2.5%

      \[\leadsto \color{blue}{\sin re + \sin re} \]
   7. Step-by-step derivation

      1. count-2 2.5%

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

   8. Simplified 2.5%

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

   10. Applied egg-rr 12.2%

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

   11. Taylor expanded in re around 0 12.1%

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

   if 1.39999999999999991e101 < im < 1e152

   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 3.1%

      \[\leadsto \color{blue}{\sin re + \sin re} \]
   7. Step-by-step derivation

      1. count-2 3.1%

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

   8. Simplified 3.1%

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

   9. Taylor expanded in re around 0 41.3%

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

   if 1e152 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 97.3%

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

      1. *-lft-identity 97.3%

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

      2. associate-*r* 97.3%

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

      3. distribute-rgt-out 97.3%

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

   7. Simplified 97.3%

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

   8. Taylor expanded in re around 0 77.9%

      \[\leadsto \color{blue}{re \cdot \left(1 + 0.5 \cdot {im}^{2}\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 13: 62.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.1 \cdot 10^{+15}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2.05 \cdot 10^{+101}:\\ \;\;\;\;2 \cdot {re}^{-2}\\ \mathbf{elif}\;im \leq 9.2 \cdot 10^{+151}:\\ \;\;\;\;re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + 0.5 \cdot {im}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.1e+15)
   (sin re)
   (if (<= im 2.05e+101)
     (* 2.0 (pow re -2.0))
     (if (<= im 9.2e+151)
       (* re (+ 1.0 (* -0.16666666666666666 (pow re 2.0))))
       (* re (+ 1.0 (* 0.5 (pow im 2.0))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.1e+15) {
		tmp = sin(re);
	} else if (im <= 2.05e+101) {
		tmp = 2.0 * pow(re, -2.0);
	} else if (im <= 9.2e+151) {
		tmp = re * (1.0 + (-0.16666666666666666 * pow(re, 2.0)));
	} else {
		tmp = re * (1.0 + (0.5 * 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 <= 2.1d+15) then
        tmp = sin(re)
    else if (im <= 2.05d+101) then
        tmp = 2.0d0 * (re ** (-2.0d0))
    else if (im <= 9.2d+151) then
        tmp = re * (1.0d0 + ((-0.16666666666666666d0) * (re ** 2.0d0)))
    else
        tmp = re * (1.0d0 + (0.5d0 * (im ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.1e+15) {
		tmp = Math.sin(re);
	} else if (im <= 2.05e+101) {
		tmp = 2.0 * Math.pow(re, -2.0);
	} else if (im <= 9.2e+151) {
		tmp = re * (1.0 + (-0.16666666666666666 * Math.pow(re, 2.0)));
	} else {
		tmp = re * (1.0 + (0.5 * Math.pow(im, 2.0)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.1e+15:
		tmp = math.sin(re)
	elif im <= 2.05e+101:
		tmp = 2.0 * math.pow(re, -2.0)
	elif im <= 9.2e+151:
		tmp = re * (1.0 + (-0.16666666666666666 * math.pow(re, 2.0)))
	else:
		tmp = re * (1.0 + (0.5 * math.pow(im, 2.0)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.1e+15)
		tmp = sin(re);
	elseif (im <= 2.05e+101)
		tmp = Float64(2.0 * (re ^ -2.0));
	elseif (im <= 9.2e+151)
		tmp = Float64(re * Float64(1.0 + Float64(-0.16666666666666666 * (re ^ 2.0))));
	else
		tmp = Float64(re * Float64(1.0 + Float64(0.5 * (im ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.1e+15)
		tmp = sin(re);
	elseif (im <= 2.05e+101)
		tmp = 2.0 * (re ^ -2.0);
	elseif (im <= 9.2e+151)
		tmp = re * (1.0 + (-0.16666666666666666 * (re ^ 2.0)));
	else
		tmp = re * (1.0 + (0.5 * (im ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.1e+15], N[Sin[re], $MachinePrecision], If[LessEqual[im, 2.05e+101], N[(2.0 * N[Power[re, -2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 9.2e+151], N[(re * N[(1.0 + N[(-0.16666666666666666 * N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(1.0 + N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < 2.1e15

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 64.1%

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

   if 2.1e15 < im < 2.05e101

   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 2.5%

      \[\leadsto \color{blue}{\sin re + \sin re} \]
   7. Step-by-step derivation

      1. count-2 2.5%

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

   8. Simplified 2.5%

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

   10. Applied egg-rr 12.2%

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

   11. Taylor expanded in re around 0 12.1%

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

   if 2.05e101 < im < 9.2000000000000003e151

   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 3.1%

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

   6. Taylor expanded in re around 0 41.3%

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

   if 9.2000000000000003e151 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 97.3%

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

      1. *-lft-identity 97.3%

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

      2. associate-*r* 97.3%

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

      3. distribute-rgt-out 97.3%

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

   7. Simplified 97.3%

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

   8. Taylor expanded in re around 0 77.9%

      \[\leadsto \color{blue}{re \cdot \left(1 + 0.5 \cdot {im}^{2}\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 14: 62.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.1 \cdot 10^{+15}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.3 \cdot 10^{+101}:\\ \;\;\;\;2 \cdot {re}^{-2}\\ \mathbf{elif}\;im \leq 9.2 \cdot 10^{+151}:\\ \;\;\;\;re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.1e+15)
   (sin re)
   (if (<= im 1.3e+101)
     (* 2.0 (pow re -2.0))
     (if (<= im 9.2e+151)
       (* re (+ 1.0 (* -0.16666666666666666 (pow re 2.0))))
       (* re (* 0.5 (pow im 2.0)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.1e+15) {
		tmp = sin(re);
	} else if (im <= 1.3e+101) {
		tmp = 2.0 * pow(re, -2.0);
	} else if (im <= 9.2e+151) {
		tmp = re * (1.0 + (-0.16666666666666666 * pow(re, 2.0)));
	} else {
		tmp = re * (0.5 * 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 <= 2.1d+15) then
        tmp = sin(re)
    else if (im <= 1.3d+101) then
        tmp = 2.0d0 * (re ** (-2.0d0))
    else if (im <= 9.2d+151) then
        tmp = re * (1.0d0 + ((-0.16666666666666666d0) * (re ** 2.0d0)))
    else
        tmp = re * (0.5d0 * (im ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.1e+15) {
		tmp = Math.sin(re);
	} else if (im <= 1.3e+101) {
		tmp = 2.0 * Math.pow(re, -2.0);
	} else if (im <= 9.2e+151) {
		tmp = re * (1.0 + (-0.16666666666666666 * Math.pow(re, 2.0)));
	} else {
		tmp = re * (0.5 * Math.pow(im, 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.1e+15:
		tmp = math.sin(re)
	elif im <= 1.3e+101:
		tmp = 2.0 * math.pow(re, -2.0)
	elif im <= 9.2e+151:
		tmp = re * (1.0 + (-0.16666666666666666 * math.pow(re, 2.0)))
	else:
		tmp = re * (0.5 * math.pow(im, 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.1e+15)
		tmp = sin(re);
	elseif (im <= 1.3e+101)
		tmp = Float64(2.0 * (re ^ -2.0));
	elseif (im <= 9.2e+151)
		tmp = Float64(re * Float64(1.0 + Float64(-0.16666666666666666 * (re ^ 2.0))));
	else
		tmp = Float64(re * Float64(0.5 * (im ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.1e+15)
		tmp = sin(re);
	elseif (im <= 1.3e+101)
		tmp = 2.0 * (re ^ -2.0);
	elseif (im <= 9.2e+151)
		tmp = re * (1.0 + (-0.16666666666666666 * (re ^ 2.0)));
	else
		tmp = re * (0.5 * (im ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.1e+15], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.3e+101], N[(2.0 * N[Power[re, -2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 9.2e+151], N[(re * N[(1.0 + N[(-0.16666666666666666 * N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < 2.1e15

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 64.1%

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

   if 2.1e15 < im < 1.3e101

   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 2.5%

      \[\leadsto \color{blue}{\sin re + \sin re} \]
   7. Step-by-step derivation

      1. count-2 2.5%

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

   8. Simplified 2.5%

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

   10. Applied egg-rr 12.2%

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

   11. Taylor expanded in re around 0 12.1%

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

   if 1.3e101 < im < 9.2000000000000003e151

   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 3.1%

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

   6. Taylor expanded in re around 0 41.3%

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

   if 9.2000000000000003e151 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 97.3%

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

      1. *-lft-identity 97.3%

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

      2. associate-*r* 97.3%

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

      3. distribute-rgt-out 97.3%

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

   7. Simplified 97.3%

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

   8. Taylor expanded in re around 0 77.9%

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

   9. Taylor expanded in im around inf 77.9%

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

       1. associate-*r* 77.9%

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

       2. *-commutative 77.9%

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

   11. Simplified 77.9%

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

Alternative 15: 62.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.1 \cdot 10^{+15}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{+101}:\\ \;\;\;\;2 \cdot {re}^{-2}\\ \mathbf{elif}\;im \leq 9.2 \cdot 10^{+151}:\\ \;\;\;\;-0.16666666666666666 \cdot {re}^{3}\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.1e+15)
   (sin re)
   (if (<= im 2.5e+101)
     (* 2.0 (pow re -2.0))
     (if (<= im 9.2e+151)
       (* -0.16666666666666666 (pow re 3.0))
       (* re (* 0.5 (pow im 2.0)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.1e+15) {
		tmp = sin(re);
	} else if (im <= 2.5e+101) {
		tmp = 2.0 * pow(re, -2.0);
	} else if (im <= 9.2e+151) {
		tmp = -0.16666666666666666 * pow(re, 3.0);
	} else {
		tmp = re * (0.5 * 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 <= 2.1d+15) then
        tmp = sin(re)
    else if (im <= 2.5d+101) then
        tmp = 2.0d0 * (re ** (-2.0d0))
    else if (im <= 9.2d+151) then
        tmp = (-0.16666666666666666d0) * (re ** 3.0d0)
    else
        tmp = re * (0.5d0 * (im ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.1e+15) {
		tmp = Math.sin(re);
	} else if (im <= 2.5e+101) {
		tmp = 2.0 * Math.pow(re, -2.0);
	} else if (im <= 9.2e+151) {
		tmp = -0.16666666666666666 * Math.pow(re, 3.0);
	} else {
		tmp = re * (0.5 * Math.pow(im, 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.1e+15:
		tmp = math.sin(re)
	elif im <= 2.5e+101:
		tmp = 2.0 * math.pow(re, -2.0)
	elif im <= 9.2e+151:
		tmp = -0.16666666666666666 * math.pow(re, 3.0)
	else:
		tmp = re * (0.5 * math.pow(im, 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.1e+15)
		tmp = sin(re);
	elseif (im <= 2.5e+101)
		tmp = Float64(2.0 * (re ^ -2.0));
	elseif (im <= 9.2e+151)
		tmp = Float64(-0.16666666666666666 * (re ^ 3.0));
	else
		tmp = Float64(re * Float64(0.5 * (im ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.1e+15)
		tmp = sin(re);
	elseif (im <= 2.5e+101)
		tmp = 2.0 * (re ^ -2.0);
	elseif (im <= 9.2e+151)
		tmp = -0.16666666666666666 * (re ^ 3.0);
	else
		tmp = re * (0.5 * (im ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.1e+15], N[Sin[re], $MachinePrecision], If[LessEqual[im, 2.5e+101], N[(2.0 * N[Power[re, -2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 9.2e+151], N[(-0.16666666666666666 * N[Power[re, 3.0], $MachinePrecision]), $MachinePrecision], N[(re * N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < 2.1e15

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 64.1%

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

   if 2.1e15 < im < 2.49999999999999994e101

   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 2.5%

      \[\leadsto \color{blue}{\sin re + \sin re} \]
   7. Step-by-step derivation

      1. count-2 2.5%

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

   8. Simplified 2.5%

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

   10. Applied egg-rr 12.2%

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

   11. Taylor expanded in re around 0 12.1%

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

   if 2.49999999999999994e101 < im < 9.2000000000000003e151

   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 3.1%

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

   6. Taylor expanded in re around 0 41.3%

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

   7. Taylor expanded in re around inf 40.7%

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

   if 9.2000000000000003e151 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 97.3%

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

      1. *-lft-identity 97.3%

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

      2. associate-*r* 97.3%

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

      3. distribute-rgt-out 97.3%

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

   7. Simplified 97.3%

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

   8. Taylor expanded in re around 0 77.9%

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

   9. Taylor expanded in im around inf 77.9%

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

       1. associate-*r* 77.9%

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

       2. *-commutative 77.9%

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

   11. Simplified 77.9%

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

Alternative 16: 55.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.1 \cdot 10^{+15}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.05 \cdot 10^{+101} \lor \neg \left(im \leq 7.5 \cdot 10^{+259}\right):\\ \;\;\;\;2 \cdot {re}^{-2}\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot {re}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.1e+15)
   (sin re)
   (if (or (<= im 1.05e+101) (not (<= im 7.5e+259)))
     (* 2.0 (pow re -2.0))
     (* -0.16666666666666666 (pow re 3.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.1e+15) {
		tmp = sin(re);
	} else if ((im <= 1.05e+101) || !(im <= 7.5e+259)) {
		tmp = 2.0 * pow(re, -2.0);
	} 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 <= 2.1d+15) then
        tmp = sin(re)
    else if ((im <= 1.05d+101) .or. (.not. (im <= 7.5d+259))) then
        tmp = 2.0d0 * (re ** (-2.0d0))
    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 <= 2.1e+15) {
		tmp = Math.sin(re);
	} else if ((im <= 1.05e+101) || !(im <= 7.5e+259)) {
		tmp = 2.0 * Math.pow(re, -2.0);
	} else {
		tmp = -0.16666666666666666 * Math.pow(re, 3.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.1e+15:
		tmp = math.sin(re)
	elif (im <= 1.05e+101) or not (im <= 7.5e+259):
		tmp = 2.0 * math.pow(re, -2.0)
	else:
		tmp = -0.16666666666666666 * math.pow(re, 3.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.1e+15)
		tmp = sin(re);
	elseif ((im <= 1.05e+101) || !(im <= 7.5e+259))
		tmp = Float64(2.0 * (re ^ -2.0));
	else
		tmp = Float64(-0.16666666666666666 * (re ^ 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.1e+15)
		tmp = sin(re);
	elseif ((im <= 1.05e+101) || ~((im <= 7.5e+259)))
		tmp = 2.0 * (re ^ -2.0);
	else
		tmp = -0.16666666666666666 * (re ^ 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.1e+15], N[Sin[re], $MachinePrecision], If[Or[LessEqual[im, 1.05e+101], N[Not[LessEqual[im, 7.5e+259]], $MachinePrecision]], N[(2.0 * N[Power[re, -2.0], $MachinePrecision]), $MachinePrecision], N[(-0.16666666666666666 * N[Power[re, 3.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 2.1e15

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 64.1%

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

   if 2.1e15 < im < 1.05e101 or 7.4999999999999995e259 < 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

   6. Applied egg-rr 2.5%

      \[\leadsto \color{blue}{\sin re + \sin re} \]
   7. Step-by-step derivation

      1. count-2 2.5%

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

   8. Simplified 2.5%

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

   10. Applied egg-rr 8.2%

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

   11. Taylor expanded in re around 0 8.0%

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

   if 1.05e101 < im < 7.4999999999999995e259

   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 27.8%

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

   7. Taylor expanded in re around inf 27.3%

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

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.1 \cdot 10^{+15}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.05 \cdot 10^{+101} \lor \neg \left(im \leq 7.5 \cdot 10^{+259}\right):\\ \;\;\;\;2 \cdot {re}^{-2}\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot {re}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 55.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.1 \cdot 10^{+15}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+101} \lor \neg \left(im \leq 3.4 \cdot 10^{+260}\right):\\ \;\;\;\;{re}^{-2}\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot {re}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.1e+15)
   (sin re)
   (if (or (<= im 1.35e+101) (not (<= im 3.4e+260)))
     (pow re -2.0)
     (* -0.16666666666666666 (pow re 3.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.1e+15) {
		tmp = sin(re);
	} else if ((im <= 1.35e+101) || !(im <= 3.4e+260)) {
		tmp = pow(re, -2.0);
	} 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 <= 2.1d+15) then
        tmp = sin(re)
    else if ((im <= 1.35d+101) .or. (.not. (im <= 3.4d+260))) then
        tmp = re ** (-2.0d0)
    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 <= 2.1e+15) {
		tmp = Math.sin(re);
	} else if ((im <= 1.35e+101) || !(im <= 3.4e+260)) {
		tmp = Math.pow(re, -2.0);
	} else {
		tmp = -0.16666666666666666 * Math.pow(re, 3.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.1e+15:
		tmp = math.sin(re)
	elif (im <= 1.35e+101) or not (im <= 3.4e+260):
		tmp = math.pow(re, -2.0)
	else:
		tmp = -0.16666666666666666 * math.pow(re, 3.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.1e+15)
		tmp = sin(re);
	elseif ((im <= 1.35e+101) || !(im <= 3.4e+260))
		tmp = re ^ -2.0;
	else
		tmp = Float64(-0.16666666666666666 * (re ^ 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.1e+15)
		tmp = sin(re);
	elseif ((im <= 1.35e+101) || ~((im <= 3.4e+260)))
		tmp = re ^ -2.0;
	else
		tmp = -0.16666666666666666 * (re ^ 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.1e+15], N[Sin[re], $MachinePrecision], If[Or[LessEqual[im, 1.35e+101], N[Not[LessEqual[im, 3.4e+260]], $MachinePrecision]], N[Power[re, -2.0], $MachinePrecision], N[(-0.16666666666666666 * N[Power[re, 3.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 2.1e15

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 64.1%

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

   if 2.1e15 < im < 1.35000000000000003e101 or 3.3999999999999998e260 < 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 2.5%

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

   7. Applied egg-rr 8.2%

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

   8. Taylor expanded in re around 0 8.0%

      \[\leadsto {\color{blue}{re}}^{-2} \]

   if 1.35000000000000003e101 < im < 3.3999999999999998e260

   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 27.8%

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

   7. Taylor expanded in re around inf 27.3%

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

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.1 \cdot 10^{+15}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+101} \lor \neg \left(im \leq 3.4 \cdot 10^{+260}\right):\\ \;\;\;\;{re}^{-2}\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot {re}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 54.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.1 \cdot 10^{+15}:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;{re}^{-2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.1e+15) (sin re) (pow re -2.0)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.1e+15) {
		tmp = sin(re);
	} else {
		tmp = pow(re, -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 <= 2.1d+15) then
        tmp = sin(re)
    else
        tmp = re ** (-2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.1e+15) {
		tmp = Math.sin(re);
	} else {
		tmp = Math.pow(re, -2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.1e+15:
		tmp = math.sin(re)
	else:
		tmp = math.pow(re, -2.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.1e+15)
		tmp = sin(re);
	else
		tmp = re ^ -2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.1e+15)
		tmp = sin(re);
	else
		tmp = re ^ -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.1e+15], N[Sin[re], $MachinePrecision], N[Power[re, -2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2.1e15

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 64.1%

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

   if 2.1e15 < 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 2.7%

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

   7. Applied egg-rr 9.7%

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

   8. Taylor expanded in re around 0 9.5%

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

Alternative 19: 52.0% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[Sin[re], $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. distribute-rgt-in 100.0%

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

   2. cancel-sign-sub 100.0%

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

   3. distribute-rgt-out-- 100.0%

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

   4. sub-neg 100.0%

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

   5. remove-double-neg 100.0%

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

   6. neg-sub0 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in im around 0 48.5%

   \[\leadsto \color{blue}{\sin re} \]
6. Add Preprocessing

Alternative 20: 27.3% accurate, 309.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 re)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return re

Julia

function code(re, im)
	return re
end

MATLAB

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

Wolfram

code[re_, im_] := re

Derivation

1. Initial program 100.0%

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

   1. distribute-rgt-in 100.0%

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

   2. cancel-sign-sub 100.0%

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

   3. distribute-rgt-out-- 100.0%

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

   4. sub-neg 100.0%

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

   5. remove-double-neg 100.0%

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

   6. neg-sub0 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in im around 0 48.5%

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

6. Taylor expanded in re around 0 25.0%

   \[\leadsto \color{blue}{re} \]
7. Add Preprocessing

Alternative 21: 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

6. Applied egg-rr 4.8%

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

   1. +-inverses 4.8%

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

   2. +-rgt-identity 4.8%

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

   3. *-inverses 4.8%

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

8. Simplified 4.8%

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

Alternative 22: 2.9% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.0)

C

double code(double re, double im) {
	return 0.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.0d0
end function

Java

public static double code(double re, double im) {
	return 0.0;
}

Python

def code(re, im):
	return 0.0

Julia

function code(re, im)
	return 0.0
end

MATLAB

function tmp = code(re, im)
	tmp = 0.0;
end

Wolfram

code[re_, im_] := 0.0

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 2.8%

   \[\leadsto \color{blue}{\log \left({1}^{\sin re}\right)} \]
7. Step-by-step derivation

   1. pow-base-1 2.8%

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

   2. metadata-eval 2.8%

      \[\leadsto \color{blue}{0} \]

8. Simplified 2.8%

   \[\leadsto \color{blue}{0} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024101 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))