math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 9.9s

Alternatives: 23

Speedup: 1.0×

• 49.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. *-commutative 100.0%

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

   2. associate-*l* 100.0%

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

   3. sub0-neg 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 85.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.03:\\ \;\;\;\;\sin re + 0.5 \cdot \left(\sin re \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 1.75 \cdot 10^{+50}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(re \cdot 0.5\right)\\ \mathbf{elif}\;im \leq 5 \cdot 10^{+76}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\sin re \cdot im\right)\right) \cdot \left(0.5 \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.03)
   (+ (sin re) (* 0.5 (* (sin re) (* im im))))
   (if (<= im 1.75e+50)
     (* (+ (exp (- im)) (exp im)) (* re 0.5))
     (if (<= im 5e+76)
       (* (log1p (expm1 (* (sin re) im))) (* 0.5 im))
       (*
        (sin re)
        (+ 1.0 (* (* im im) (+ 0.5 (* (* im im) 0.041666666666666664)))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.03) {
		tmp = sin(re) + (0.5 * (sin(re) * (im * im)));
	} else if (im <= 1.75e+50) {
		tmp = (exp(-im) + exp(im)) * (re * 0.5);
	} else if (im <= 5e+76) {
		tmp = log1p(expm1((sin(re) * im))) * (0.5 * im);
	} else {
		tmp = sin(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.03) {
		tmp = Math.sin(re) + (0.5 * (Math.sin(re) * (im * im)));
	} else if (im <= 1.75e+50) {
		tmp = (Math.exp(-im) + Math.exp(im)) * (re * 0.5);
	} else if (im <= 5e+76) {
		tmp = Math.log1p(Math.expm1((Math.sin(re) * im))) * (0.5 * im);
	} else {
		tmp = Math.sin(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.03:
		tmp = math.sin(re) + (0.5 * (math.sin(re) * (im * im)))
	elif im <= 1.75e+50:
		tmp = (math.exp(-im) + math.exp(im)) * (re * 0.5)
	elif im <= 5e+76:
		tmp = math.log1p(math.expm1((math.sin(re) * im))) * (0.5 * im)
	else:
		tmp = math.sin(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.03)
		tmp = Float64(sin(re) + Float64(0.5 * Float64(sin(re) * Float64(im * im))));
	elseif (im <= 1.75e+50)
		tmp = Float64(Float64(exp(Float64(-im)) + exp(im)) * Float64(re * 0.5));
	elseif (im <= 5e+76)
		tmp = Float64(log1p(expm1(Float64(sin(re) * im))) * Float64(0.5 * im));
	else
		tmp = Float64(sin(re) * Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(Float64(im * im) * 0.041666666666666664)))));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.03], N[(N[Sin[re], $MachinePrecision] + N[(0.5 * N[(N[Sin[re], $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.75e+50], N[(N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[(re * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 5e+76], N[(N[Log[1 + N[(Exp[N[(N[Sin[re], $MachinePrecision] * im), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision] * N[(0.5 * im), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < 0.029999999999999999

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 84.8%

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

      1. unpow2 84.8%

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

   6. Simplified 84.8%

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

   if 0.029999999999999999 < im < 1.75000000000000003e50

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 73.3%

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

      1. associate-*r* 73.3%

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

   6. Simplified 73.3%

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

   if 1.75000000000000003e50 < im < 4.99999999999999991e76

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 7.9%

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

      1. *-rgt-identity 7.9%

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

      2. *-commutative 7.9%

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

      3. associate-*r* 7.9%

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

      4. *-commutative 7.9%

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

      5. associate-*r* 7.9%

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

      6. distribute-rgt-out 7.9%

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

      7. distribute-lft-out 7.9%

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

      8. metadata-eval 7.9%

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

      9. pow-sqr 7.9%

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

      10. associate-*r* 7.9%

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

      11. distribute-rgt-out 7.9%

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

      12. unpow2 7.9%

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

      13. unpow2 7.9%

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

   6. Simplified 7.9%

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

   7. Taylor expanded in im around 0 4.0%

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

      1. *-commutative 4.0%

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

      2. unpow2 4.0%

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

   9. Simplified 4.0%

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

   10. Taylor expanded in im around inf 4.0%

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

       1. *-commutative 4.0%

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

       2. unpow2 4.0%

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

       3. associate-*r* 4.0%

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

       4. associate-*l* 4.0%

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

       5. *-commutative 4.0%

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

   12. Simplified 4.0%

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

       1. log1p-expm1-u 75.6%

          \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot \sin re\right)\right)} \cdot \left(im \cdot 0.5\right) \]

   14. Applied egg-rr 75.6%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot \sin re\right)\right)} \cdot \left(im \cdot 0.5\right) \]

   if 4.99999999999999991e76 < im

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 98.3%

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

      1. *-rgt-identity 98.3%

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

      2. *-commutative 98.3%

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

      3. associate-*r* 98.3%

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

      4. *-commutative 98.3%

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

      5. associate-*r* 98.3%

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

      6. distribute-rgt-out 98.3%

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

      7. distribute-lft-out 98.3%

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

      8. metadata-eval 98.3%

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

      9. pow-sqr 98.3%

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

      10. associate-*r* 98.3%

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

      11. distribute-rgt-out 98.3%

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

      12. unpow2 98.3%

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

      13. unpow2 98.3%

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

   6. Simplified 98.3%

      \[\leadsto \color{blue}{\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.03:\\ \;\;\;\;\sin re + 0.5 \cdot \left(\sin re \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 1.75 \cdot 10^{+50}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(re \cdot 0.5\right)\\ \mathbf{elif}\;im \leq 5 \cdot 10^{+76}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\sin re \cdot im\right)\right) \cdot \left(0.5 \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \end{array} \]

Alternative 3: 85.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.08:\\ \;\;\;\;\sin re \cdot \left(1 + im \cdot \left(0.5 \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{+77}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(re \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.08)
   (* (sin re) (+ 1.0 (* im (* 0.5 im))))
   (if (<= im 2.5e+77)
     (* (+ (exp (- im)) (exp im)) (* re 0.5))
     (*
      (sin re)
      (+ 1.0 (* (* im im) (+ 0.5 (* (* im im) 0.041666666666666664))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.08) {
		tmp = sin(re) * (1.0 + (im * (0.5 * im)));
	} else if (im <= 2.5e+77) {
		tmp = (exp(-im) + exp(im)) * (re * 0.5);
	} else {
		tmp = sin(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.08d0) then
        tmp = sin(re) * (1.0d0 + (im * (0.5d0 * im)))
    else if (im <= 2.5d+77) then
        tmp = (exp(-im) + exp(im)) * (re * 0.5d0)
    else
        tmp = sin(re) * (1.0d0 + ((im * im) * (0.5d0 + ((im * im) * 0.041666666666666664d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.08) {
		tmp = Math.sin(re) * (1.0 + (im * (0.5 * im)));
	} else if (im <= 2.5e+77) {
		tmp = (Math.exp(-im) + Math.exp(im)) * (re * 0.5);
	} else {
		tmp = Math.sin(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.08:
		tmp = math.sin(re) * (1.0 + (im * (0.5 * im)))
	elif im <= 2.5e+77:
		tmp = (math.exp(-im) + math.exp(im)) * (re * 0.5)
	else:
		tmp = math.sin(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.08)
		tmp = Float64(sin(re) * Float64(1.0 + Float64(im * Float64(0.5 * im))));
	elseif (im <= 2.5e+77)
		tmp = Float64(Float64(exp(Float64(-im)) + exp(im)) * Float64(re * 0.5));
	else
		tmp = Float64(sin(re) * Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(Float64(im * im) * 0.041666666666666664)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.08)
		tmp = sin(re) * (1.0 + (im * (0.5 * im)));
	elseif (im <= 2.5e+77)
		tmp = (exp(-im) + exp(im)) * (re * 0.5);
	else
		tmp = sin(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.08], N[(N[Sin[re], $MachinePrecision] * N[(1.0 + N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.5e+77], N[(N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[(re * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.0800000000000000017

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 84.8%

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

      1. *-commutative 84.8%

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

      2. associate-*r* 84.8%

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

      3. distribute-rgt1-in 84.8%

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

      4. *-commutative 84.8%

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

      5. unpow2 84.8%

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

      6. associate-*l* 84.8%

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

   6. Simplified 84.8%

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

   if 0.0800000000000000017 < im < 2.50000000000000002e77

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 65.0%

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

      1. associate-*r* 65.0%

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

   6. Simplified 65.0%

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

   if 2.50000000000000002e77 < im

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

      1. *-rgt-identity 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*r* 100.0%

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

      6. distribute-rgt-out 100.0%

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

      7. distribute-lft-out 100.0%

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

      8. metadata-eval 100.0%

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

      9. pow-sqr 100.0%

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

      10. associate-*r* 100.0%

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

      11. distribute-rgt-out 100.0%

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

      12. unpow2 100.0%

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

      13. unpow2 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.08:\\ \;\;\;\;\sin re \cdot \left(1 + im \cdot \left(0.5 \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{+77}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(re \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \end{array} \]

Alternative 4: 85.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.058:\\ \;\;\;\;\sin re + 0.5 \cdot \left(\sin re \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{+77}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(re \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.058)
   (+ (sin re) (* 0.5 (* (sin re) (* im im))))
   (if (<= im 2.5e+77)
     (* (+ (exp (- im)) (exp im)) (* re 0.5))
     (*
      (sin re)
      (+ 1.0 (* (* im im) (+ 0.5 (* (* im im) 0.041666666666666664))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.058) {
		tmp = sin(re) + (0.5 * (sin(re) * (im * im)));
	} else if (im <= 2.5e+77) {
		tmp = (exp(-im) + exp(im)) * (re * 0.5);
	} else {
		tmp = sin(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.058d0) then
        tmp = sin(re) + (0.5d0 * (sin(re) * (im * im)))
    else if (im <= 2.5d+77) then
        tmp = (exp(-im) + exp(im)) * (re * 0.5d0)
    else
        tmp = sin(re) * (1.0d0 + ((im * im) * (0.5d0 + ((im * im) * 0.041666666666666664d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.058) {
		tmp = Math.sin(re) + (0.5 * (Math.sin(re) * (im * im)));
	} else if (im <= 2.5e+77) {
		tmp = (Math.exp(-im) + Math.exp(im)) * (re * 0.5);
	} else {
		tmp = Math.sin(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.058:
		tmp = math.sin(re) + (0.5 * (math.sin(re) * (im * im)))
	elif im <= 2.5e+77:
		tmp = (math.exp(-im) + math.exp(im)) * (re * 0.5)
	else:
		tmp = math.sin(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.058)
		tmp = Float64(sin(re) + Float64(0.5 * Float64(sin(re) * Float64(im * im))));
	elseif (im <= 2.5e+77)
		tmp = Float64(Float64(exp(Float64(-im)) + exp(im)) * Float64(re * 0.5));
	else
		tmp = Float64(sin(re) * Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(Float64(im * im) * 0.041666666666666664)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.058)
		tmp = sin(re) + (0.5 * (sin(re) * (im * im)));
	elseif (im <= 2.5e+77)
		tmp = (exp(-im) + exp(im)) * (re * 0.5);
	else
		tmp = sin(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.058], N[(N[Sin[re], $MachinePrecision] + N[(0.5 * N[(N[Sin[re], $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.5e+77], N[(N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[(re * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.0580000000000000029

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 84.8%

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

      1. unpow2 84.8%

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

   6. Simplified 84.8%

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

   if 0.0580000000000000029 < im < 2.50000000000000002e77

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 65.0%

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

      1. associate-*r* 65.0%

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

   6. Simplified 65.0%

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

   if 2.50000000000000002e77 < im

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

      1. *-rgt-identity 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*r* 100.0%

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

      6. distribute-rgt-out 100.0%

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

      7. distribute-lft-out 100.0%

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

      8. metadata-eval 100.0%

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

      9. pow-sqr 100.0%

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

      10. associate-*r* 100.0%

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

      11. distribute-rgt-out 100.0%

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

      12. unpow2 100.0%

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

      13. unpow2 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.058:\\ \;\;\;\;\sin re + 0.5 \cdot \left(\sin re \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{+77}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(re \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \end{array} \]

Alternative 5: 88.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 480 \lor \neg \left(im \leq 5 \cdot 10^{+76}\right):\\ \;\;\;\;\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(re \cdot 0.5 + {re}^{3} \cdot -0.08333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 480.0) (not (<= im 5e+76)))
   (*
    (sin re)
    (+ 1.0 (* (* im im) (+ 0.5 (* (* im im) 0.041666666666666664)))))
   (* (* im im) (+ (* re 0.5) (* (pow re 3.0) -0.08333333333333333)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 480.0) || !(im <= 5e+76)) {
		tmp = sin(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	} else {
		tmp = (im * im) * ((re * 0.5) + (pow(re, 3.0) * -0.08333333333333333));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= 480.0d0) .or. (.not. (im <= 5d+76))) then
        tmp = sin(re) * (1.0d0 + ((im * im) * (0.5d0 + ((im * im) * 0.041666666666666664d0))))
    else
        tmp = (im * im) * ((re * 0.5d0) + ((re ** 3.0d0) * (-0.08333333333333333d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= 480.0) || !(im <= 5e+76)) {
		tmp = Math.sin(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	} else {
		tmp = (im * im) * ((re * 0.5) + (Math.pow(re, 3.0) * -0.08333333333333333));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= 480.0) or not (im <= 5e+76):
		tmp = math.sin(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))))
	else:
		tmp = (im * im) * ((re * 0.5) + (math.pow(re, 3.0) * -0.08333333333333333))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 480.0) || !(im <= 5e+76))
		tmp = Float64(sin(re) * Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(Float64(im * im) * 0.041666666666666664)))));
	else
		tmp = Float64(Float64(im * im) * Float64(Float64(re * 0.5) + Float64((re ^ 3.0) * -0.08333333333333333)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= 480.0) || ~((im <= 5e+76)))
		tmp = sin(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	else
		tmp = (im * im) * ((re * 0.5) + ((re ^ 3.0) * -0.08333333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 480.0], N[Not[LessEqual[im, 5e+76]], $MachinePrecision]], N[(N[Sin[re], $MachinePrecision] * N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im * im), $MachinePrecision] * N[(N[(re * 0.5), $MachinePrecision] + N[(N[Power[re, 3.0], $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 480 or 4.99999999999999991e76 < im

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 92.0%

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

      1. *-rgt-identity 92.0%

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

      2. *-commutative 92.0%

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

      3. associate-*r* 92.0%

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

      4. *-commutative 92.0%

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

      5. associate-*r* 92.0%

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

      6. distribute-rgt-out 92.0%

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

      7. distribute-lft-out 92.0%

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

      8. metadata-eval 92.0%

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

      9. pow-sqr 92.0%

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

      10. associate-*r* 92.0%

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

      11. distribute-rgt-out 92.0%

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

      12. unpow2 92.0%

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

      13. unpow2 92.0%

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

   6. Simplified 92.0%

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

   if 480 < im < 4.99999999999999991e76

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 4.6%

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

      1. *-rgt-identity 4.6%

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

      2. *-commutative 4.6%

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

      3. associate-*r* 4.6%

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

      4. *-commutative 4.6%

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

      5. associate-*r* 4.6%

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

      6. distribute-rgt-out 4.6%

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

      7. distribute-lft-out 4.6%

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

      8. metadata-eval 4.6%

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

      9. pow-sqr 4.6%

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

      10. associate-*r* 4.6%

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

      11. distribute-rgt-out 4.6%

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

      12. unpow2 4.6%

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

      13. unpow2 4.6%

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

   6. Simplified 4.6%

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

   7. Taylor expanded in im around 0 3.4%

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

      1. *-commutative 3.4%

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

      2. unpow2 3.4%

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

   9. Simplified 3.4%

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

   10. Taylor expanded in im around inf 3.4%

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

       1. *-commutative 3.4%

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

       2. unpow2 3.4%

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

       3. associate-*r* 3.4%

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

       4. associate-*l* 3.4%

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

       5. *-commutative 3.4%

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

   12. Simplified 3.4%

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

   13. Taylor expanded in re around 0 24.1%

       \[\leadsto \color{blue}{-0.08333333333333333 \cdot \left({re}^{3} \cdot {im}^{2}\right) + 0.5 \cdot \left(re \cdot {im}^{2}\right)} \]
   14. Step-by-step derivation

       1. +-commutative 24.1%

          \[\leadsto \color{blue}{0.5 \cdot \left(re \cdot {im}^{2}\right) + -0.08333333333333333 \cdot \left({re}^{3} \cdot {im}^{2}\right)} \]

       2. associate-*r* 24.1%

          \[\leadsto \color{blue}{\left(0.5 \cdot re\right) \cdot {im}^{2}} + -0.08333333333333333 \cdot \left({re}^{3} \cdot {im}^{2}\right) \]

       3. associate-*r* 24.1%

          \[\leadsto \left(0.5 \cdot re\right) \cdot {im}^{2} + \color{blue}{\left(-0.08333333333333333 \cdot {re}^{3}\right) \cdot {im}^{2}} \]

       4. distribute-rgt-out 29.6%

          \[\leadsto \color{blue}{{im}^{2} \cdot \left(0.5 \cdot re + -0.08333333333333333 \cdot {re}^{3}\right)} \]

       5. unpow2 29.6%

          \[\leadsto \color{blue}{\left(im \cdot im\right)} \cdot \left(0.5 \cdot re + -0.08333333333333333 \cdot {re}^{3}\right) \]

       6. *-commutative 29.6%

          \[\leadsto \left(im \cdot im\right) \cdot \left(\color{blue}{re \cdot 0.5} + -0.08333333333333333 \cdot {re}^{3}\right) \]

       7. *-commutative 29.6%

          \[\leadsto \left(im \cdot im\right) \cdot \left(re \cdot 0.5 + \color{blue}{{re}^{3} \cdot -0.08333333333333333}\right) \]

   15. Simplified 29.6%

       \[\leadsto \color{blue}{\left(im \cdot im\right) \cdot \left(re \cdot 0.5 + {re}^{3} \cdot -0.08333333333333333\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 480 \lor \neg \left(im \leq 5 \cdot 10^{+76}\right):\\ \;\;\;\;\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(re \cdot 0.5 + {re}^{3} \cdot -0.08333333333333333\right)\\ \end{array} \]

Alternative 6: 80.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin re \cdot \left(1 + im \cdot \left(0.5 \cdot im\right)\right)\\ \mathbf{if}\;im \leq 400:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 3.3 \cdot 10^{+78}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(re \cdot 0.5 + {re}^{3} \cdot -0.08333333333333333\right)\\ \mathbf{elif}\;im \leq 1.85 \cdot 10^{+154}:\\ \;\;\;\;re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (sin re) (+ 1.0 (* im (* 0.5 im))))))
   (if (<= im 400.0)
     t_0
     (if (<= im 3.3e+78)
       (* (* im im) (+ (* re 0.5) (* (pow re 3.0) -0.08333333333333333)))
       (if (<= im 1.85e+154)
         (*
          re
          (+ 1.0 (* (* im im) (+ 0.5 (* (* im im) 0.041666666666666664)))))
         t_0)))))

C

double code(double re, double im) {
	double t_0 = sin(re) * (1.0 + (im * (0.5 * im)));
	double tmp;
	if (im <= 400.0) {
		tmp = t_0;
	} else if (im <= 3.3e+78) {
		tmp = (im * im) * ((re * 0.5) + (pow(re, 3.0) * -0.08333333333333333));
	} else if (im <= 1.85e+154) {
		tmp = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(re) * (1.0d0 + (im * (0.5d0 * im)))
    if (im <= 400.0d0) then
        tmp = t_0
    else if (im <= 3.3d+78) then
        tmp = (im * im) * ((re * 0.5d0) + ((re ** 3.0d0) * (-0.08333333333333333d0)))
    else if (im <= 1.85d+154) then
        tmp = re * (1.0d0 + ((im * im) * (0.5d0 + ((im * im) * 0.041666666666666664d0))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.sin(re) * (1.0 + (im * (0.5 * im)));
	double tmp;
	if (im <= 400.0) {
		tmp = t_0;
	} else if (im <= 3.3e+78) {
		tmp = (im * im) * ((re * 0.5) + (Math.pow(re, 3.0) * -0.08333333333333333));
	} else if (im <= 1.85e+154) {
		tmp = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.sin(re) * (1.0 + (im * (0.5 * im)))
	tmp = 0
	if im <= 400.0:
		tmp = t_0
	elif im <= 3.3e+78:
		tmp = (im * im) * ((re * 0.5) + (math.pow(re, 3.0) * -0.08333333333333333))
	elif im <= 1.85e+154:
		tmp = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(sin(re) * Float64(1.0 + Float64(im * Float64(0.5 * im))))
	tmp = 0.0
	if (im <= 400.0)
		tmp = t_0;
	elseif (im <= 3.3e+78)
		tmp = Float64(Float64(im * im) * Float64(Float64(re * 0.5) + Float64((re ^ 3.0) * -0.08333333333333333)));
	elseif (im <= 1.85e+154)
		tmp = Float64(re * Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(Float64(im * im) * 0.041666666666666664)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = sin(re) * (1.0 + (im * (0.5 * im)));
	tmp = 0.0;
	if (im <= 400.0)
		tmp = t_0;
	elseif (im <= 3.3e+78)
		tmp = (im * im) * ((re * 0.5) + ((re ^ 3.0) * -0.08333333333333333));
	elseif (im <= 1.85e+154)
		tmp = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Sin[re], $MachinePrecision] * N[(1.0 + N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 400.0], t$95$0, If[LessEqual[im, 3.3e+78], N[(N[(im * im), $MachinePrecision] * N[(N[(re * 0.5), $MachinePrecision] + N[(N[Power[re, 3.0], $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.85e+154], N[(re * N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < 400 or 1.84999999999999997e154 < im

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 86.7%

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

      1. *-commutative 86.7%

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

      2. associate-*r* 86.7%

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

      3. distribute-rgt1-in 86.7%

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

      4. *-commutative 86.7%

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

      5. unpow2 86.7%

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

      6. associate-*l* 86.7%

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

   6. Simplified 86.7%

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

   if 400 < im < 3.3e78

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 9.4%

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

      1. *-rgt-identity 9.4%

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

      2. *-commutative 9.4%

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

      3. associate-*r* 9.4%

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

      4. *-commutative 9.4%

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

      5. associate-*r* 9.4%

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

      6. distribute-rgt-out 9.4%

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

      7. distribute-lft-out 9.4%

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

      8. metadata-eval 9.4%

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

      9. pow-sqr 9.4%

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

      10. associate-*r* 9.4%

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

      11. distribute-rgt-out 9.4%

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

      12. unpow2 9.4%

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

      13. unpow2 9.4%

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

   6. Simplified 9.4%

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

   7. Taylor expanded in im around 0 3.4%

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

      1. *-commutative 3.4%

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

      2. unpow2 3.4%

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

   9. Simplified 3.4%

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

   10. Taylor expanded in im around inf 3.4%

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

       1. *-commutative 3.4%

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

       2. unpow2 3.4%

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

       3. associate-*r* 3.4%

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

       4. associate-*l* 3.4%

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

       5. *-commutative 3.4%

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

   12. Simplified 3.4%

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

   13. Taylor expanded in re around 0 26.8%

       \[\leadsto \color{blue}{-0.08333333333333333 \cdot \left({re}^{3} \cdot {im}^{2}\right) + 0.5 \cdot \left(re \cdot {im}^{2}\right)} \]
   14. Step-by-step derivation

       1. +-commutative 26.8%

          \[\leadsto \color{blue}{0.5 \cdot \left(re \cdot {im}^{2}\right) + -0.08333333333333333 \cdot \left({re}^{3} \cdot {im}^{2}\right)} \]

       2. associate-*r* 26.8%

          \[\leadsto \color{blue}{\left(0.5 \cdot re\right) \cdot {im}^{2}} + -0.08333333333333333 \cdot \left({re}^{3} \cdot {im}^{2}\right) \]

       3. associate-*r* 26.8%

          \[\leadsto \left(0.5 \cdot re\right) \cdot {im}^{2} + \color{blue}{\left(-0.08333333333333333 \cdot {re}^{3}\right) \cdot {im}^{2}} \]

       4. distribute-rgt-out 31.8%

          \[\leadsto \color{blue}{{im}^{2} \cdot \left(0.5 \cdot re + -0.08333333333333333 \cdot {re}^{3}\right)} \]

       5. unpow2 31.8%

          \[\leadsto \color{blue}{\left(im \cdot im\right)} \cdot \left(0.5 \cdot re + -0.08333333333333333 \cdot {re}^{3}\right) \]

       6. *-commutative 31.8%

          \[\leadsto \left(im \cdot im\right) \cdot \left(\color{blue}{re \cdot 0.5} + -0.08333333333333333 \cdot {re}^{3}\right) \]

       7. *-commutative 31.8%

          \[\leadsto \left(im \cdot im\right) \cdot \left(re \cdot 0.5 + \color{blue}{{re}^{3} \cdot -0.08333333333333333}\right) \]

   15. Simplified 31.8%

       \[\leadsto \color{blue}{\left(im \cdot im\right) \cdot \left(re \cdot 0.5 + {re}^{3} \cdot -0.08333333333333333\right)} \]

   if 3.3e78 < im < 1.84999999999999997e154

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

      1. *-rgt-identity 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*r* 100.0%

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

      6. distribute-rgt-out 100.0%

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

      7. distribute-lft-out 100.0%

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

      8. metadata-eval 100.0%

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

      9. pow-sqr 100.0%

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

      10. associate-*r* 100.0%

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

      11. distribute-rgt-out 100.0%

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

      12. unpow2 100.0%

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

      13. unpow2 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around 0 80.0%

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

      1. *-commutative 80.0%

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

      2. unpow2 80.0%

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

      3. unpow2 80.0%

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

   9. Simplified 80.0%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 400:\\ \;\;\;\;\sin re \cdot \left(1 + im \cdot \left(0.5 \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 3.3 \cdot 10^{+78}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(re \cdot 0.5 + {re}^{3} \cdot -0.08333333333333333\right)\\ \mathbf{elif}\;im \leq 1.85 \cdot 10^{+154}:\\ \;\;\;\;re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(1 + im \cdot \left(0.5 \cdot im\right)\right)\\ \end{array} \]

Alternative 7: 80.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 7 \cdot 10^{+24} \lor \neg \left(im \leq 1.85 \cdot 10^{+154}\right):\\ \;\;\;\;\sin re \cdot \left(1 + im \cdot \left(0.5 \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 7e+24) (not (<= im 1.85e+154)))
   (* (sin re) (+ 1.0 (* im (* 0.5 im))))
   (* re (+ 1.0 (* (* im im) (+ 0.5 (* (* im im) 0.041666666666666664)))))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 7e+24) || !(im <= 1.85e+154)) {
		tmp = sin(re) * (1.0 + (im * (0.5 * im)));
	} else {
		tmp = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= 7d+24) .or. (.not. (im <= 1.85d+154))) then
        tmp = sin(re) * (1.0d0 + (im * (0.5d0 * im)))
    else
        tmp = re * (1.0d0 + ((im * im) * (0.5d0 + ((im * im) * 0.041666666666666664d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= 7e+24) || !(im <= 1.85e+154)) {
		tmp = Math.sin(re) * (1.0 + (im * (0.5 * im)));
	} else {
		tmp = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= 7e+24) or not (im <= 1.85e+154):
		tmp = math.sin(re) * (1.0 + (im * (0.5 * im)))
	else:
		tmp = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 7e+24) || !(im <= 1.85e+154))
		tmp = Float64(sin(re) * Float64(1.0 + Float64(im * Float64(0.5 * im))));
	else
		tmp = Float64(re * Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(Float64(im * im) * 0.041666666666666664)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= 7e+24) || ~((im <= 1.85e+154)))
		tmp = sin(re) * (1.0 + (im * (0.5 * im)));
	else
		tmp = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 7e+24], N[Not[LessEqual[im, 1.85e+154]], $MachinePrecision]], N[(N[Sin[re], $MachinePrecision] * N[(1.0 + N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 7.0000000000000004e24 or 1.84999999999999997e154 < im

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 85.2%

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

      1. *-commutative 85.2%

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

      2. associate-*r* 85.2%

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

      3. distribute-rgt1-in 85.2%

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

      4. *-commutative 85.2%

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

      5. unpow2 85.2%

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

      6. associate-*l* 85.2%

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

   6. Simplified 85.2%

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

   if 7.0000000000000004e24 < im < 1.84999999999999997e154

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 60.4%

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

      1. *-rgt-identity 60.4%

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

      2. *-commutative 60.4%

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

      3. associate-*r* 60.4%

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

      4. *-commutative 60.4%

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

      5. associate-*r* 60.4%

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

      6. distribute-rgt-out 60.4%

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

      7. distribute-lft-out 60.4%

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

      8. metadata-eval 60.4%

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

      9. pow-sqr 60.4%

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

      10. associate-*r* 60.4%

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

      11. distribute-rgt-out 60.4%

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

      12. unpow2 60.4%

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

      13. unpow2 60.4%

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

   6. Simplified 60.4%

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

   7. Taylor expanded in re around 0 51.1%

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

      1. *-commutative 51.1%

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

      2. unpow2 51.1%

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

      3. unpow2 51.1%

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

   9. Simplified 51.1%

      \[\leadsto \color{blue}{\left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right) \cdot re} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 7 \cdot 10^{+24} \lor \neg \left(im \leq 1.85 \cdot 10^{+154}\right):\\ \;\;\;\;\sin re \cdot \left(1 + im \cdot \left(0.5 \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \end{array} \]

Alternative 8: 66.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 6 \cdot 10^{+24}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2.8 \cdot 10^{+182}:\\ \;\;\;\;re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(\sin re \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 6e+24)
   (sin re)
   (if (<= im 2.8e+182)
     (* re (+ 1.0 (* (* im im) (+ 0.5 (* (* im im) 0.041666666666666664)))))
     (* 0.5 (* im (* (sin re) im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 6e+24) {
		tmp = sin(re);
	} else if (im <= 2.8e+182) {
		tmp = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	} else {
		tmp = 0.5 * (im * (sin(re) * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 6d+24) then
        tmp = sin(re)
    else if (im <= 2.8d+182) then
        tmp = re * (1.0d0 + ((im * im) * (0.5d0 + ((im * im) * 0.041666666666666664d0))))
    else
        tmp = 0.5d0 * (im * (sin(re) * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 6e+24) {
		tmp = Math.sin(re);
	} else if (im <= 2.8e+182) {
		tmp = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	} else {
		tmp = 0.5 * (im * (Math.sin(re) * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 6e+24:
		tmp = math.sin(re)
	elif im <= 2.8e+182:
		tmp = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))))
	else:
		tmp = 0.5 * (im * (math.sin(re) * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 6e+24)
		tmp = sin(re);
	elseif (im <= 2.8e+182)
		tmp = Float64(re * Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(Float64(im * im) * 0.041666666666666664)))));
	else
		tmp = Float64(0.5 * Float64(im * Float64(sin(re) * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 6e+24)
		tmp = sin(re);
	elseif (im <= 2.8e+182)
		tmp = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	else
		tmp = 0.5 * (im * (sin(re) * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 6e+24], N[Sin[re], $MachinePrecision], If[LessEqual[im, 2.8e+182], N[(re * N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[(N[Sin[re], $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 5.9999999999999999e24

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 66.6%

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

   if 5.9999999999999999e24 < im < 2.80000000000000006e182

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 66.9%

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

      1. *-rgt-identity 66.9%

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

      2. *-commutative 66.9%

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

      3. associate-*r* 66.9%

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

      4. *-commutative 66.9%

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

      5. associate-*r* 66.9%

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

      6. distribute-rgt-out 66.9%

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

      7. distribute-lft-out 66.9%

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

      8. metadata-eval 66.9%

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

      9. pow-sqr 66.9%

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

      10. associate-*r* 66.9%

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

      11. distribute-rgt-out 66.9%

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

      12. unpow2 66.9%

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

      13. unpow2 66.9%

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

   6. Simplified 66.9%

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

   7. Taylor expanded in re around 0 56.7%

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

      1. *-commutative 56.7%

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

      2. unpow2 56.7%

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

      3. unpow2 56.7%

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

   9. Simplified 56.7%

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

   if 2.80000000000000006e182 < im

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

      3. distribute-rgt1-in 100.0%

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

      4. *-commutative 100.0%

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

      5. unpow2 100.0%

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

      6. associate-*l* 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in im around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. unpow2 100.0%

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

      3. associate-*l* 84.9%

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

   9. Simplified 84.9%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 6 \cdot 10^{+24}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2.8 \cdot 10^{+182}:\\ \;\;\;\;re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(\sin re \cdot im\right)\right)\\ \end{array} \]

Alternative 9: 65.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 7 \cdot 10^{+24}:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 7e+24)
   (sin re)
   (* re (+ 1.0 (* (* im im) (+ 0.5 (* (* im im) 0.041666666666666664)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 7e+24) {
		tmp = sin(re);
	} else {
		tmp = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 7d+24) then
        tmp = sin(re)
    else
        tmp = re * (1.0d0 + ((im * im) * (0.5d0 + ((im * im) * 0.041666666666666664d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 7e+24) {
		tmp = Math.sin(re);
	} else {
		tmp = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 7e+24:
		tmp = math.sin(re)
	else:
		tmp = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 7e+24)
		tmp = sin(re);
	else
		tmp = Float64(re * Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(Float64(im * im) * 0.041666666666666664)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 7e+24)
		tmp = sin(re);
	else
		tmp = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 7e+24], N[Sin[re], $MachinePrecision], N[(re * N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 7.0000000000000004e24

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 66.6%

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

   if 7.0000000000000004e24 < im

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 79.0%

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

      1. *-rgt-identity 79.0%

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

      2. *-commutative 79.0%

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

      3. associate-*r* 79.0%

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

      4. *-commutative 79.0%

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

      5. associate-*r* 79.0%

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

      6. distribute-rgt-out 79.0%

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

      7. distribute-lft-out 79.0%

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

      8. metadata-eval 79.0%

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

      9. pow-sqr 79.0%

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

      10. associate-*r* 79.0%

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

      11. distribute-rgt-out 79.0%

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

      12. unpow2 79.0%

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

      13. unpow2 79.0%

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

   6. Simplified 79.0%

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

   7. Taylor expanded in re around 0 62.3%

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

      1. *-commutative 62.3%

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

      2. unpow2 62.3%

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

      3. unpow2 62.3%

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

   9. Simplified 62.3%

      \[\leadsto \color{blue}{\left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right) \cdot re} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 7 \cdot 10^{+24}:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \end{array} \]

Alternative 10: 55.1% accurate, 20.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(re * N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. *-commutative 100.0%

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

   2. associate-*l* 100.0%

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

   3. sub0-neg 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in im around 0 85.8%

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

   1. *-rgt-identity 85.8%

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

   2. *-commutative 85.8%

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

   3. associate-*r* 85.8%

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

   4. *-commutative 85.8%

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

   5. associate-*r* 85.8%

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

   6. distribute-rgt-out 85.8%

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

   7. distribute-lft-out 85.8%

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

   8. metadata-eval 85.8%

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

   9. pow-sqr 85.8%

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

   10. associate-*r* 85.8%

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

   11. distribute-rgt-out 85.8%

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

   12. unpow2 85.8%

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

   13. unpow2 85.8%

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

6. Simplified 85.8%

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

7. Taylor expanded in re around 0 53.0%

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

   1. *-commutative 53.0%

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

   2. unpow2 53.0%

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

   3. unpow2 53.0%

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

9. Simplified 53.0%

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

10. Final simplification 53.0%

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

Alternative 11: 37.2% accurate, 27.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.5 \cdot 10^{-15}:\\ \;\;\;\;re\\ \mathbf{elif}\;im \leq 1.52 \cdot 10^{+100}:\\ \;\;\;\;0.08333333333333333 + \frac{0.25}{re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.5e-15)
   re
   (if (<= im 1.52e+100)
     (+ 0.08333333333333333 (/ 0.25 (* re re)))
     (* (* im im) (* re 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.5e-15) {
		tmp = re;
	} else if (im <= 1.52e+100) {
		tmp = 0.08333333333333333 + (0.25 / (re * re));
	} else {
		tmp = (im * im) * (re * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.5d-15) then
        tmp = re
    else if (im <= 1.52d+100) then
        tmp = 0.08333333333333333d0 + (0.25d0 / (re * re))
    else
        tmp = (im * im) * (re * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.5e-15) {
		tmp = re;
	} else if (im <= 1.52e+100) {
		tmp = 0.08333333333333333 + (0.25 / (re * re));
	} else {
		tmp = (im * im) * (re * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.5e-15:
		tmp = re
	elif im <= 1.52e+100:
		tmp = 0.08333333333333333 + (0.25 / (re * re))
	else:
		tmp = (im * im) * (re * 0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.5e-15)
		tmp = re;
	elseif (im <= 1.52e+100)
		tmp = Float64(0.08333333333333333 + Float64(0.25 / Float64(re * re)));
	else
		tmp = Float64(Float64(im * im) * Float64(re * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.5e-15)
		tmp = re;
	elseif (im <= 1.52e+100)
		tmp = 0.08333333333333333 + (0.25 / (re * re));
	else
		tmp = (im * im) * (re * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.5e-15], re, If[LessEqual[im, 1.52e+100], N[(0.08333333333333333 + N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im * im), $MachinePrecision] * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.5e-15

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 58.2%

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

      1. associate-*r* 58.2%

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

   6. Simplified 58.2%

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

   7. Taylor expanded in im around 0 35.0%

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

   if 1.5e-15 < im < 1.52e100

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*l* 99.9%

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

      3. sub0-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Applied egg-rr 13.8%

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

   6. Taylor expanded in re around 0 13.4%

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

      1. associate-*r/ 13.4%

         \[\leadsto 0.08333333333333333 + \color{blue}{\frac{0.25 \cdot 1}{{re}^{2}}} \]

      2. metadata-eval 13.4%

         \[\leadsto 0.08333333333333333 + \frac{\color{blue}{0.25}}{{re}^{2}} \]

      3. unpow2 13.4%

         \[\leadsto 0.08333333333333333 + \frac{0.25}{\color{blue}{re \cdot re}} \]

   8. Simplified 13.4%

      \[\leadsto \color{blue}{0.08333333333333333 + \frac{0.25}{re \cdot re}} \]

   if 1.52e100 < im

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

      1. *-rgt-identity 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*r* 100.0%

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

      6. distribute-rgt-out 100.0%

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

      7. distribute-lft-out 100.0%

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

      8. metadata-eval 100.0%

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

      9. pow-sqr 100.0%

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

      10. associate-*r* 100.0%

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

      11. distribute-rgt-out 100.0%

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

      12. unpow2 100.0%

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

      13. unpow2 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in im around 0 69.9%

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

      1. *-commutative 69.9%

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

      2. unpow2 69.9%

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

   9. Simplified 69.9%

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

   10. Taylor expanded in im around inf 69.9%

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

       1. *-commutative 69.9%

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

       2. unpow2 69.9%

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

       3. associate-*r* 51.7%

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

       4. associate-*l* 51.7%

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

       5. *-commutative 51.7%

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

   12. Simplified 51.7%

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

   13. Taylor expanded in re around 0 56.1%

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

       1. *-commutative 56.1%

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

       2. *-commutative 56.1%

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

       3. associate-*l* 56.1%

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

       4. unpow2 56.1%

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

   15. Simplified 56.1%

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

4. Final simplification 36.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.5 \cdot 10^{-15}:\\ \;\;\;\;re\\ \mathbf{elif}\;im \leq 1.52 \cdot 10^{+100}:\\ \;\;\;\;0.08333333333333333 + \frac{0.25}{re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(re \cdot 0.5\right)\\ \end{array} \]

Alternative 12: 36.9% accurate, 34.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.5 \cdot 10^{-15}:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.5e-15) re (* (* im im) (* re 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.5e-15) {
		tmp = re;
	} else {
		tmp = (im * im) * (re * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.5d-15) then
        tmp = re
    else
        tmp = (im * im) * (re * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.5e-15) {
		tmp = re;
	} else {
		tmp = (im * im) * (re * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.5e-15:
		tmp = re
	else:
		tmp = (im * im) * (re * 0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.5e-15)
		tmp = re;
	else
		tmp = Float64(Float64(im * im) * Float64(re * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.5e-15)
		tmp = re;
	else
		tmp = (im * im) * (re * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.5e-15], re, N[(N[(im * im), $MachinePrecision] * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.5e-15

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 58.2%

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

      1. associate-*r* 58.2%

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

   6. Simplified 58.2%

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

   7. Taylor expanded in im around 0 35.0%

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

   if 1.5e-15 < im

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 75.0%

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

      1. *-rgt-identity 75.0%

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

      2. *-commutative 75.0%

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

      3. associate-*r* 75.0%

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

      4. *-commutative 75.0%

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

      5. associate-*r* 75.0%

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

      6. distribute-rgt-out 75.0%

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

      7. distribute-lft-out 74.9%

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

      8. metadata-eval 74.9%

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

      9. pow-sqr 74.9%

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

      10. associate-*r* 74.9%

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

      11. distribute-rgt-out 74.9%

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

      12. unpow2 74.9%

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

      13. unpow2 74.9%

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

   6. Simplified 74.9%

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

   7. Taylor expanded in im around 0 48.4%

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

      1. *-commutative 48.4%

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

      2. unpow2 48.4%

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

   9. Simplified 48.4%

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

   10. Taylor expanded in im around inf 44.9%

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

       1. *-commutative 44.9%

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

       2. unpow2 44.9%

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

       3. associate-*r* 33.6%

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

       4. associate-*l* 33.6%

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

       5. *-commutative 33.6%

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

   12. Simplified 33.6%

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

   13. Taylor expanded in re around 0 38.3%

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

       1. *-commutative 38.3%

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

       2. *-commutative 38.3%

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

       3. associate-*l* 38.3%

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

       4. unpow2 38.3%

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

   15. Simplified 38.3%

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

4. Final simplification 36.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.5 \cdot 10^{-15}:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(re \cdot 0.5\right)\\ \end{array} \]

Alternative 13: 46.9% accurate, 34.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(re * N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. *-commutative 100.0%

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

   2. associate-*l* 100.0%

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

   3. sub0-neg 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in im around 0 73.8%

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

   1. *-commutative 73.8%

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

   2. associate-*r* 73.8%

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

   3. distribute-rgt1-in 73.8%

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

   4. *-commutative 73.8%

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

   5. unpow2 73.8%

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

   6. associate-*l* 73.8%

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

6. Simplified 73.8%

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

7. Taylor expanded in re around 0 45.5%

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

   1. *-commutative 45.5%

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

   2. unpow2 45.5%

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

9. Simplified 45.5%

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

10. Final simplification 45.5%

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

Alternative 14: 29.1% accurate, 43.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 850:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{re \cdot re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 850.0) re (/ 0.25 (* re re))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 850.0) {
		tmp = re;
	} else {
		tmp = 0.25 / (re * re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 850.0d0) then
        tmp = re
    else
        tmp = 0.25d0 / (re * re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 850.0) {
		tmp = re;
	} else {
		tmp = 0.25 / (re * re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 850.0:
		tmp = re
	else:
		tmp = 0.25 / (re * re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 850.0)
		tmp = re;
	else
		tmp = Float64(0.25 / Float64(re * re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 850.0)
		tmp = re;
	else
		tmp = 0.25 / (re * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 850.0], re, N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 850

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 57.5%

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

      1. associate-*r* 57.5%

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

   6. Simplified 57.5%

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

   7. Taylor expanded in im around 0 34.3%

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

   if 850 < im

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 12.7%

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

   6. Taylor expanded in re around 0 12.6%

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

      1. unpow2 12.6%

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

   8. Simplified 12.6%

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

4. Final simplification 28.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 850:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{re \cdot re}\\ \end{array} \]

Alternative 15: 29.1% accurate, 43.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 850:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25}{re}}{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 850.0) re (/ (/ 0.25 re) re)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 850.0) {
		tmp = re;
	} else {
		tmp = (0.25 / re) / re;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 850.0d0) then
        tmp = re
    else
        tmp = (0.25d0 / re) / re
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 850.0) {
		tmp = re;
	} else {
		tmp = (0.25 / re) / re;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 850.0:
		tmp = re
	else:
		tmp = (0.25 / re) / re
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 850.0)
		tmp = re;
	else
		tmp = Float64(Float64(0.25 / re) / re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 850.0)
		tmp = re;
	else
		tmp = (0.25 / re) / re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 850.0], re, N[(N[(0.25 / re), $MachinePrecision] / re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 850

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 57.5%

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

      1. associate-*r* 57.5%

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

   6. Simplified 57.5%

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

   7. Taylor expanded in im around 0 34.3%

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

   if 850 < im

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 12.7%

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

   6. Taylor expanded in re around 0 12.6%

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

      1. unpow2 12.6%

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

   8. Simplified 12.6%

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

      1. add-sqr-sqrt 12.6%

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

      2. sqrt-div 12.6%

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

      3. metadata-eval 12.6%

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

      4. sqrt-prod 12.1%

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

      5. add-sqr-sqrt 28.4%

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

      6. sqrt-div 28.4%

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

      7. metadata-eval 28.4%

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

      8. sqrt-prod 12.1%

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

      9. add-sqr-sqrt 12.6%

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

   10. Applied egg-rr 12.6%

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

   11. Taylor expanded in re around 0 12.6%

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

       1. unpow2 12.6%

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

       2. associate-/r* 12.6%

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

   13. Simplified 12.6%

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

4. Final simplification 28.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 850:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25}{re}}{re}\\ \end{array} \]

Alternative 16: 3.9% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ -512 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 -512.0)

C

double code(double re, double im) {
	return -512.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = -512.0d0
end function

Java

public static double code(double re, double im) {
	return -512.0;
}

Python

def code(re, im):
	return -512.0

Julia

function code(re, im)
	return -512.0
end

MATLAB

function tmp = code(re, im)
	tmp = -512.0;
end

Wolfram

code[re_, im_] := -512.0

Derivation

1. Initial program 100.0%

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

   1. *-commutative 100.0%

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

   2. associate-*l* 100.0%

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

   3. sub0-neg 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in im around 0 85.8%

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

   1. *-rgt-identity 85.8%

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

   2. *-commutative 85.8%

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

   3. associate-*r* 85.8%

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

   4. *-commutative 85.8%

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

   5. associate-*r* 85.8%

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

   6. distribute-rgt-out 85.8%

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

   7. distribute-lft-out 85.8%

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

   8. metadata-eval 85.8%

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

   9. pow-sqr 85.8%

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

   10. associate-*r* 85.8%

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

   11. distribute-rgt-out 85.8%

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

   12. unpow2 85.8%

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

   13. unpow2 85.8%

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

6. Simplified 85.8%

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

8. Applied egg-rr 3.5%

   \[\leadsto \color{blue}{-512} \]

9. Final simplification 3.5%

   \[\leadsto -512 \]

Alternative 17: 4.9% accurate, 309.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 -1.0)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return -1.0

Julia

function code(re, im)
	return -1.0
end

MATLAB

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

Wolfram

code[re_, im_] := -1.0

Derivation

1. Initial program 100.0%

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

   1. *-commutative 100.0%

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

   2. associate-*l* 100.0%

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

   3. sub0-neg 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in im around 0 85.8%

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

   1. *-rgt-identity 85.8%

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

   2. *-commutative 85.8%

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

   3. associate-*r* 85.8%

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

   4. *-commutative 85.8%

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

   5. associate-*r* 85.8%

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

   6. distribute-rgt-out 85.8%

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

   7. distribute-lft-out 85.8%

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

   8. metadata-eval 85.8%

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

   9. pow-sqr 85.8%

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

   10. associate-*r* 85.8%

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

   11. distribute-rgt-out 85.8%

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

   12. unpow2 85.8%

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

   13. unpow2 85.8%

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

6. Simplified 85.8%

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

8. Applied egg-rr 4.3%

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

9. Final simplification 4.3%

   \[\leadsto -1 \]

Alternative 18: 4.0% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ 0.015625 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.015625)

C

double code(double re, double im) {
	return 0.015625;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.015625d0
end function

Java

public static double code(double re, double im) {
	return 0.015625;
}

Python

def code(re, im):
	return 0.015625

Julia

function code(re, im)
	return 0.015625
end

MATLAB

function tmp = code(re, im)
	tmp = 0.015625;
end

Wolfram

code[re_, im_] := 0.015625

Derivation

1. Initial program 100.0%

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

   1. *-commutative 100.0%

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

   2. associate-*l* 100.0%

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

   3. sub0-neg 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in im around 0 85.8%

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

   1. *-rgt-identity 85.8%

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

   2. *-commutative 85.8%

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

   3. associate-*r* 85.8%

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

   4. *-commutative 85.8%

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

   5. associate-*r* 85.8%

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

   6. distribute-rgt-out 85.8%

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

   7. distribute-lft-out 85.8%

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

   8. metadata-eval 85.8%

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

   9. pow-sqr 85.8%

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

   10. associate-*r* 85.8%

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

   11. distribute-rgt-out 85.8%

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

   12. unpow2 85.8%

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

   13. unpow2 85.8%

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

6. Simplified 85.8%

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

8. Applied egg-rr 4.3%

   \[\leadsto \color{blue}{0.015625} \]

9. Final simplification 4.3%

   \[\leadsto 0.015625 \]

Alternative 19: 4.1% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ 0.0625 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.0625)

C

double code(double re, double im) {
	return 0.0625;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.0625d0
end function

Java

public static double code(double re, double im) {
	return 0.0625;
}

Python

def code(re, im):
	return 0.0625

Julia

function code(re, im)
	return 0.0625
end

MATLAB

function tmp = code(re, im)
	tmp = 0.0625;
end

Wolfram

code[re_, im_] := 0.0625

Derivation

1. Initial program 100.0%

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

   1. *-commutative 100.0%

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

   2. associate-*l* 100.0%

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

   3. sub0-neg 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in im around 0 85.8%

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

   1. *-rgt-identity 85.8%

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

   2. *-commutative 85.8%

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

   3. associate-*r* 85.8%

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

   4. *-commutative 85.8%

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

   5. associate-*r* 85.8%

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

   6. distribute-rgt-out 85.8%

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

   7. distribute-lft-out 85.8%

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

   8. metadata-eval 85.8%

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

   9. pow-sqr 85.8%

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

   10. associate-*r* 85.8%

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

   11. distribute-rgt-out 85.8%

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

   12. unpow2 85.8%

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

   13. unpow2 85.8%

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

6. Simplified 85.8%

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

8. Applied egg-rr 4.5%

   \[\leadsto \color{blue}{0.0625} \]

9. Final simplification 4.5%

   \[\leadsto 0.0625 \]

Alternative 20: 4.4% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ 0.25 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.25)

C

double code(double re, double im) {
	return 0.25;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.25d0
end function

Java

public static double code(double re, double im) {
	return 0.25;
}

Python

def code(re, im):
	return 0.25

Julia

function code(re, im)
	return 0.25
end

MATLAB

function tmp = code(re, im)
	tmp = 0.25;
end

Wolfram

code[re_, im_] := 0.25

Derivation

1. Initial program 100.0%

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

   1. *-commutative 100.0%

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

   2. associate-*l* 100.0%

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

   3. sub0-neg 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in im around 0 85.8%

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

   1. *-rgt-identity 85.8%

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

   2. *-commutative 85.8%

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

   3. associate-*r* 85.8%

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

   4. *-commutative 85.8%

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

   5. associate-*r* 85.8%

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

   6. distribute-rgt-out 85.8%

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

   7. distribute-lft-out 85.8%

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

   8. metadata-eval 85.8%

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

   9. pow-sqr 85.8%

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

   10. associate-*r* 85.8%

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

   11. distribute-rgt-out 85.8%

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

   12. unpow2 85.8%

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

   13. unpow2 85.8%

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

6. Simplified 85.8%

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

8. Applied egg-rr 4.7%

   \[\leadsto \color{blue}{0.25} \]

9. Final simplification 4.7%

   \[\leadsto 0.25 \]

Alternative 21: 4.6% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.5)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return 0.5

Julia

function code(re, im)
	return 0.5
end

MATLAB

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

Wolfram

code[re_, im_] := 0.5

Derivation

1. Initial program 100.0%

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

   1. *-commutative 100.0%

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

   2. associate-*l* 100.0%

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

   3. sub0-neg 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in im around 0 85.8%

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

   1. *-rgt-identity 85.8%

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

   2. *-commutative 85.8%

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

   3. associate-*r* 85.8%

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

   4. *-commutative 85.8%

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

   5. associate-*r* 85.8%

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

   6. distribute-rgt-out 85.8%

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

   7. distribute-lft-out 85.8%

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

   8. metadata-eval 85.8%

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

   9. pow-sqr 85.8%

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

   10. associate-*r* 85.8%

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

   11. distribute-rgt-out 85.8%

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

   12. unpow2 85.8%

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

   13. unpow2 85.8%

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

6. Simplified 85.8%

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

8. Applied egg-rr 5.2%

   \[\leadsto \color{blue}{0.5} \]

9. Final simplification 5.2%

   \[\leadsto 0.5 \]

Alternative 22: 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. *-commutative 100.0%

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

   2. associate-*l* 100.0%

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

   3. sub0-neg 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in im around 0 85.8%

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

   1. *-rgt-identity 85.8%

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

   2. *-commutative 85.8%

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

   3. associate-*r* 85.8%

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

   4. *-commutative 85.8%

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

   5. associate-*r* 85.8%

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

   6. distribute-rgt-out 85.8%

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

   7. distribute-lft-out 85.8%

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

   8. metadata-eval 85.8%

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

   9. pow-sqr 85.8%

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

   10. associate-*r* 85.8%

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

   11. distribute-rgt-out 85.8%

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

   12. unpow2 85.8%

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

   13. unpow2 85.8%

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

6. Simplified 85.8%

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

8. Applied egg-rr 5.2%

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

9. Final simplification 5.2%

   \[\leadsto 1 \]

Alternative 23: 26.5% accurate, 309.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 re)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return re

Julia

function code(re, im)
	return re
end

MATLAB

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

Wolfram

code[re_, im_] := re

Derivation

1. Initial program 100.0%

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

   1. *-commutative 100.0%

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

   2. associate-*l* 100.0%

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

   3. sub0-neg 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in re around 0 61.6%

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

   1. associate-*r* 61.6%

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

6. Simplified 61.6%

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

7. Taylor expanded in im around 0 25.4%

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

8. Final simplification 25.4%

   \[\leadsto re \]

Reproduce

herbie shell --seed 2023242 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))