math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 9.5s

Alternatives: 20

Speedup: 1.0×

• 50.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (cos re) (fma 0.5 (exp im) (/ 0.5 (exp im)))))

C

double code(double re, double im) {
	return cos(re) * fma(0.5, exp(im), (0.5 / exp(im)));
}

Julia

function code(re, im)
	return Float64(cos(re) * fma(0.5, exp(im), Float64(0.5 / exp(im))))
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. *-commutative 100.0%

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

   2. associate-*l* 100.0%

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

   3. +-commutative 100.0%

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

   4. distribute-lft-in 100.0%

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

   5. distribute-lft-in 100.0%

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

   6. distribute-rgt-in 100.0%

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

   7. *-commutative 100.0%

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

   8. fma-def 100.0%

      \[\leadsto \cos re \cdot \color{blue}{\mathsf{fma}\left(0.5, e^{im}, e^{-im} \cdot 0.5\right)} \]

   9. exp-neg 100.0%

      \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

   10. associate-*l/ 100.0%

       \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

   11. metadata-eval 100.0%

       \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{\color{blue}{0.5}}{e^{im}}\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

4. Final simplification 100.0%

   \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right) \]

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -365:\\ \;\;\;\;\cos re \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot \left(0.5 \cdot im\right)\right)\right)\\ \mathbf{elif}\;im \leq 1.3:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot e^{im} + 0.001953125\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -365.0)
   (* (cos re) (log1p (expm1 (* im (* 0.5 im)))))
   (if (<= im 1.3)
     (* (* (cos re) 0.5) (+ 2.0 (* im im)))
     (* (cos re) (+ (* 0.5 (exp im)) 0.001953125)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -365.0) {
		tmp = cos(re) * log1p(expm1((im * (0.5 * im))));
	} else if (im <= 1.3) {
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	} else {
		tmp = cos(re) * ((0.5 * exp(im)) + 0.001953125);
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -365.0) {
		tmp = Math.cos(re) * Math.log1p(Math.expm1((im * (0.5 * im))));
	} else if (im <= 1.3) {
		tmp = (Math.cos(re) * 0.5) * (2.0 + (im * im));
	} else {
		tmp = Math.cos(re) * ((0.5 * Math.exp(im)) + 0.001953125);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -365.0:
		tmp = math.cos(re) * math.log1p(math.expm1((im * (0.5 * im))))
	elif im <= 1.3:
		tmp = (math.cos(re) * 0.5) * (2.0 + (im * im))
	else:
		tmp = math.cos(re) * ((0.5 * math.exp(im)) + 0.001953125)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -365.0)
		tmp = Float64(cos(re) * log1p(expm1(Float64(im * Float64(0.5 * im)))));
	elseif (im <= 1.3)
		tmp = Float64(Float64(cos(re) * 0.5) * Float64(2.0 + Float64(im * im)));
	else
		tmp = Float64(cos(re) * Float64(Float64(0.5 * exp(im)) + 0.001953125));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, -365.0], N[(N[Cos[re], $MachinePrecision] * N[Log[1 + N[(Exp[N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.3], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision] + 0.001953125), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < -365

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 44.4%

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

      1. unpow2 44.4%

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

   4. Simplified 44.4%

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

   5. Taylor expanded in im around inf 44.4%

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

      1. unpow2 44.4%

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

      2. associate-*r* 44.4%

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

      3. *-commutative 44.4%

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

      4. associate-*r* 44.4%

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

   7. Simplified 44.4%

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

      1. log1p-expm1-u 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

   9. Applied egg-rr 100.0%

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

   if -365 < im < 1.30000000000000004

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

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

      1. unpow2 100.0%

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

   4. Simplified 100.0%

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

   if 1.30000000000000004 < im

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-def 100.0%

         \[\leadsto \cos re \cdot \color{blue}{\mathsf{fma}\left(0.5, e^{im}, e^{-im} \cdot 0.5\right)} \]

      9. exp-neg 100.0%

         \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

      10. associate-*l/ 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

      11. metadata-eval 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{\color{blue}{0.5}}{e^{im}}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

   5. Applied egg-rr 100.0%

      \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{0.001953125}\right) \]
   6. Step-by-step derivation

      1. fma-udef 100.0%

         \[\leadsto \cos re \cdot \color{blue}{\left(0.5 \cdot e^{im} + 0.001953125\right)} \]

   7. Applied egg-rr 100.0%

      \[\leadsto \cos re \cdot \color{blue}{\left(0.5 \cdot e^{im} + 0.001953125\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -365:\\ \;\;\;\;\cos re \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot \left(0.5 \cdot im\right)\right)\right)\\ \mathbf{elif}\;im \leq 1.3:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot e^{im} + 0.001953125\right)\\ \end{array} \]

Alternative 3: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1.5 \cdot 10^{+85}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \mathbf{elif}\;im \leq -430:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(0.041666666666666664 \cdot {im}^{4}\right)\right)\\ \mathbf{elif}\;im \leq 1.3:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot e^{im} + 0.001953125\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -1.5e+85)
   (* 0.041666666666666664 (* (cos re) (pow im 4.0)))
   (if (<= im -430.0)
     (log1p (expm1 (* 0.041666666666666664 (pow im 4.0))))
     (if (<= im 1.3)
       (* (* (cos re) 0.5) (+ 2.0 (* im im)))
       (* (cos re) (+ (* 0.5 (exp im)) 0.001953125))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -1.5e+85) {
		tmp = 0.041666666666666664 * (cos(re) * pow(im, 4.0));
	} else if (im <= -430.0) {
		tmp = log1p(expm1((0.041666666666666664 * pow(im, 4.0))));
	} else if (im <= 1.3) {
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	} else {
		tmp = cos(re) * ((0.5 * exp(im)) + 0.001953125);
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -1.5e+85) {
		tmp = 0.041666666666666664 * (Math.cos(re) * Math.pow(im, 4.0));
	} else if (im <= -430.0) {
		tmp = Math.log1p(Math.expm1((0.041666666666666664 * Math.pow(im, 4.0))));
	} else if (im <= 1.3) {
		tmp = (Math.cos(re) * 0.5) * (2.0 + (im * im));
	} else {
		tmp = Math.cos(re) * ((0.5 * Math.exp(im)) + 0.001953125);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -1.5e+85:
		tmp = 0.041666666666666664 * (math.cos(re) * math.pow(im, 4.0))
	elif im <= -430.0:
		tmp = math.log1p(math.expm1((0.041666666666666664 * math.pow(im, 4.0))))
	elif im <= 1.3:
		tmp = (math.cos(re) * 0.5) * (2.0 + (im * im))
	else:
		tmp = math.cos(re) * ((0.5 * math.exp(im)) + 0.001953125)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -1.5e+85)
		tmp = Float64(0.041666666666666664 * Float64(cos(re) * (im ^ 4.0)));
	elseif (im <= -430.0)
		tmp = log1p(expm1(Float64(0.041666666666666664 * (im ^ 4.0))));
	elseif (im <= 1.3)
		tmp = Float64(Float64(cos(re) * 0.5) * Float64(2.0 + Float64(im * im)));
	else
		tmp = Float64(cos(re) * Float64(Float64(0.5 * exp(im)) + 0.001953125));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, -1.5e+85], N[(0.041666666666666664 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, -430.0], N[Log[1 + N[(Exp[N[(0.041666666666666664 * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision], If[LessEqual[im, 1.3], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision] + 0.001953125), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < -1.5e85

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]
   3. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(\color{blue}{im \cdot im} + 0.08333333333333333 \cdot {im}^{4}\right)\right) \]

      2. *-commutative 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(im \cdot im + \color{blue}{{im}^{4} \cdot 0.08333333333333333}\right)\right) \]

   4. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left(im \cdot im + {im}^{4} \cdot 0.08333333333333333\right)\right)} \]

   5. Taylor expanded in im around inf 100.0%

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

   if -1.5e85 < im < -430

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 21.9%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]
   3. Step-by-step derivation

      1. unpow2 21.9%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(\color{blue}{im \cdot im} + 0.08333333333333333 \cdot {im}^{4}\right)\right) \]

      2. *-commutative 21.9%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(im \cdot im + \color{blue}{{im}^{4} \cdot 0.08333333333333333}\right)\right) \]

   4. Simplified 21.9%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left(im \cdot im + {im}^{4} \cdot 0.08333333333333333\right)\right)} \]

   5. Taylor expanded in im around inf 21.9%

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

   6. Taylor expanded in re around 0 21.3%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot {im}^{4}} \]
   7. Step-by-step derivation

      1. *-commutative 21.3%

         \[\leadsto \color{blue}{{im}^{4} \cdot 0.041666666666666664} \]

   8. Simplified 21.3%

      \[\leadsto \color{blue}{{im}^{4} \cdot 0.041666666666666664} \]
   9. Step-by-step derivation

      1. log1p-expm1-u 82.4%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left({im}^{4} \cdot 0.041666666666666664\right)\right)} \]

   10. Applied egg-rr 82.4%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left({im}^{4} \cdot 0.041666666666666664\right)\right)} \]

   if -430 < im < 1.30000000000000004

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

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

      1. unpow2 100.0%

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

   4. Simplified 100.0%

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

   if 1.30000000000000004 < im

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-def 100.0%

         \[\leadsto \cos re \cdot \color{blue}{\mathsf{fma}\left(0.5, e^{im}, e^{-im} \cdot 0.5\right)} \]

      9. exp-neg 100.0%

         \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

      10. associate-*l/ 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

      11. metadata-eval 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{\color{blue}{0.5}}{e^{im}}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

   5. Applied egg-rr 100.0%

      \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{0.001953125}\right) \]
   6. Step-by-step derivation

      1. fma-udef 100.0%

         \[\leadsto \cos re \cdot \color{blue}{\left(0.5 \cdot e^{im} + 0.001953125\right)} \]

   7. Applied egg-rr 100.0%

      \[\leadsto \cos re \cdot \color{blue}{\left(0.5 \cdot e^{im} + 0.001953125\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.5 \cdot 10^{+85}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \mathbf{elif}\;im \leq -430:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(0.041666666666666664 \cdot {im}^{4}\right)\right)\\ \mathbf{elif}\;im \leq 1.3:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot e^{im} + 0.001953125\right)\\ \end{array} \]

Alternative 4: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	return (cos(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 = (cos(re) * 0.5d0) * (exp(im) + exp(-im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Final simplification 100.0%

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

Alternative 5: 96.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1.5 \cdot 10^{+85}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \mathbf{elif}\;im \leq -4.9 \cdot 10^{+25}:\\ \;\;\;\;0.5 \cdot \left(2 + \left(im \cdot im + \sqrt[3]{{im}^{12} \cdot 0.0005787037037037037}\right)\right)\\ \mathbf{elif}\;im \leq 1.92:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + \left(im \cdot im + {im}^{4} \cdot 0.08333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot e^{im} + 0.001953125\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -1.5e+85)
   (* 0.041666666666666664 (* (cos re) (pow im 4.0)))
   (if (<= im -4.9e+25)
     (*
      0.5
      (+ 2.0 (+ (* im im) (cbrt (* (pow im 12.0) 0.0005787037037037037)))))
     (if (<= im 1.92)
       (*
        (* (cos re) 0.5)
        (+ 2.0 (+ (* im im) (* (pow im 4.0) 0.08333333333333333))))
       (* (cos re) (+ (* 0.5 (exp im)) 0.001953125))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -1.5e+85) {
		tmp = 0.041666666666666664 * (cos(re) * pow(im, 4.0));
	} else if (im <= -4.9e+25) {
		tmp = 0.5 * (2.0 + ((im * im) + cbrt((pow(im, 12.0) * 0.0005787037037037037))));
	} else if (im <= 1.92) {
		tmp = (cos(re) * 0.5) * (2.0 + ((im * im) + (pow(im, 4.0) * 0.08333333333333333)));
	} else {
		tmp = cos(re) * ((0.5 * exp(im)) + 0.001953125);
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -1.5e+85) {
		tmp = 0.041666666666666664 * (Math.cos(re) * Math.pow(im, 4.0));
	} else if (im <= -4.9e+25) {
		tmp = 0.5 * (2.0 + ((im * im) + Math.cbrt((Math.pow(im, 12.0) * 0.0005787037037037037))));
	} else if (im <= 1.92) {
		tmp = (Math.cos(re) * 0.5) * (2.0 + ((im * im) + (Math.pow(im, 4.0) * 0.08333333333333333)));
	} else {
		tmp = Math.cos(re) * ((0.5 * Math.exp(im)) + 0.001953125);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -1.5e+85)
		tmp = Float64(0.041666666666666664 * Float64(cos(re) * (im ^ 4.0)));
	elseif (im <= -4.9e+25)
		tmp = Float64(0.5 * Float64(2.0 + Float64(Float64(im * im) + cbrt(Float64((im ^ 12.0) * 0.0005787037037037037)))));
	elseif (im <= 1.92)
		tmp = Float64(Float64(cos(re) * 0.5) * Float64(2.0 + Float64(Float64(im * im) + Float64((im ^ 4.0) * 0.08333333333333333))));
	else
		tmp = Float64(cos(re) * Float64(Float64(0.5 * exp(im)) + 0.001953125));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, -1.5e+85], N[(0.041666666666666664 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, -4.9e+25], N[(0.5 * N[(2.0 + N[(N[(im * im), $MachinePrecision] + N[Power[N[(N[Power[im, 12.0], $MachinePrecision] * 0.0005787037037037037), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.92], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(2.0 + N[(N[(im * im), $MachinePrecision] + N[(N[Power[im, 4.0], $MachinePrecision] * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision] + 0.001953125), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < -1.5e85

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]
   3. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(\color{blue}{im \cdot im} + 0.08333333333333333 \cdot {im}^{4}\right)\right) \]

      2. *-commutative 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(im \cdot im + \color{blue}{{im}^{4} \cdot 0.08333333333333333}\right)\right) \]

   4. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left(im \cdot im + {im}^{4} \cdot 0.08333333333333333\right)\right)} \]

   5. Taylor expanded in im around inf 100.0%

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

   if -1.5e85 < im < -4.9000000000000001e25

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 31.9%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]
   3. Step-by-step derivation

      1. unpow2 31.9%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(\color{blue}{im \cdot im} + 0.08333333333333333 \cdot {im}^{4}\right)\right) \]

      2. *-commutative 31.9%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(im \cdot im + \color{blue}{{im}^{4} \cdot 0.08333333333333333}\right)\right) \]

   4. Simplified 31.9%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left(im \cdot im + {im}^{4} \cdot 0.08333333333333333\right)\right)} \]

   5. Taylor expanded in re around 0 31.5%

      \[\leadsto \color{blue}{0.5 \cdot \left(2 + \left(0.08333333333333333 \cdot {im}^{4} + {im}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 31.5%

         \[\leadsto 0.5 \cdot \left(2 + \left(0.08333333333333333 \cdot {im}^{4} + \color{blue}{1 \cdot {im}^{2}}\right)\right) \]

      2. *-commutative 31.5%

         \[\leadsto 0.5 \cdot \left(2 + \left(0.08333333333333333 \cdot {im}^{4} + \color{blue}{{im}^{2} \cdot 1}\right)\right) \]

      3. unpow2 31.5%

         \[\leadsto 0.5 \cdot \left(2 + \left(0.08333333333333333 \cdot {im}^{4} + \color{blue}{\left(im \cdot im\right)} \cdot 1\right)\right) \]

   7. Applied egg-rr 31.5%

      \[\leadsto 0.5 \cdot \left(2 + \left(0.08333333333333333 \cdot {im}^{4} + \color{blue}{\left(im \cdot im\right) \cdot 1}\right)\right) \]
   8. Step-by-step derivation

      1. add-cbrt-cube 90.9%

         \[\leadsto 0.5 \cdot \left(2 + \left(\color{blue}{\sqrt[3]{\left(\left(0.08333333333333333 \cdot {im}^{4}\right) \cdot \left(0.08333333333333333 \cdot {im}^{4}\right)\right) \cdot \left(0.08333333333333333 \cdot {im}^{4}\right)}} + \left(im \cdot im\right) \cdot 1\right)\right) \]

      2. pow1/3 90.9%

         \[\leadsto 0.5 \cdot \left(2 + \left(\color{blue}{{\left(\left(\left(0.08333333333333333 \cdot {im}^{4}\right) \cdot \left(0.08333333333333333 \cdot {im}^{4}\right)\right) \cdot \left(0.08333333333333333 \cdot {im}^{4}\right)\right)}^{0.3333333333333333}} + \left(im \cdot im\right) \cdot 1\right)\right) \]

      3. pow3 90.9%

         \[\leadsto 0.5 \cdot \left(2 + \left({\color{blue}{\left({\left(0.08333333333333333 \cdot {im}^{4}\right)}^{3}\right)}}^{0.3333333333333333} + \left(im \cdot im\right) \cdot 1\right)\right) \]

      4. *-commutative 90.9%

         \[\leadsto 0.5 \cdot \left(2 + \left({\left({\color{blue}{\left({im}^{4} \cdot 0.08333333333333333\right)}}^{3}\right)}^{0.3333333333333333} + \left(im \cdot im\right) \cdot 1\right)\right) \]

      5. unpow-prod-down 90.9%

         \[\leadsto 0.5 \cdot \left(2 + \left({\color{blue}{\left({\left({im}^{4}\right)}^{3} \cdot {0.08333333333333333}^{3}\right)}}^{0.3333333333333333} + \left(im \cdot im\right) \cdot 1\right)\right) \]

      6. metadata-eval 90.9%

         \[\leadsto 0.5 \cdot \left(2 + \left({\left({\left({im}^{4}\right)}^{3} \cdot \color{blue}{0.0005787037037037037}\right)}^{0.3333333333333333} + \left(im \cdot im\right) \cdot 1\right)\right) \]

   9. Applied egg-rr 90.9%

      \[\leadsto 0.5 \cdot \left(2 + \left(\color{blue}{{\left({\left({im}^{4}\right)}^{3} \cdot 0.0005787037037037037\right)}^{0.3333333333333333}} + \left(im \cdot im\right) \cdot 1\right)\right) \]
   10. Step-by-step derivation

       1. unpow1/3 90.9%

          \[\leadsto 0.5 \cdot \left(2 + \left(\color{blue}{\sqrt[3]{{\left({im}^{4}\right)}^{3} \cdot 0.0005787037037037037}} + \left(im \cdot im\right) \cdot 1\right)\right) \]

       2. metadata-eval 90.9%

          \[\leadsto 0.5 \cdot \left(2 + \left(\sqrt[3]{{\left({im}^{\color{blue}{\left(2 \cdot 2\right)}}\right)}^{3} \cdot 0.0005787037037037037} + \left(im \cdot im\right) \cdot 1\right)\right) \]

       3. pow-sqr 90.9%

          \[\leadsto 0.5 \cdot \left(2 + \left(\sqrt[3]{{\color{blue}{\left({im}^{2} \cdot {im}^{2}\right)}}^{3} \cdot 0.0005787037037037037} + \left(im \cdot im\right) \cdot 1\right)\right) \]

       4. unpow2 90.9%

          \[\leadsto 0.5 \cdot \left(2 + \left(\sqrt[3]{{\left(\color{blue}{\left(im \cdot im\right)} \cdot {im}^{2}\right)}^{3} \cdot 0.0005787037037037037} + \left(im \cdot im\right) \cdot 1\right)\right) \]

       5. unpow2 90.9%

          \[\leadsto 0.5 \cdot \left(2 + \left(\sqrt[3]{{\left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right)}^{3} \cdot 0.0005787037037037037} + \left(im \cdot im\right) \cdot 1\right)\right) \]

       6. cube-prod 90.9%

          \[\leadsto 0.5 \cdot \left(2 + \left(\sqrt[3]{\color{blue}{\left({\left(im \cdot im\right)}^{3} \cdot {\left(im \cdot im\right)}^{3}\right)} \cdot 0.0005787037037037037} + \left(im \cdot im\right) \cdot 1\right)\right) \]

       7. cube-prod 90.9%

          \[\leadsto 0.5 \cdot \left(2 + \left(\sqrt[3]{\left(\color{blue}{\left({im}^{3} \cdot {im}^{3}\right)} \cdot {\left(im \cdot im\right)}^{3}\right) \cdot 0.0005787037037037037} + \left(im \cdot im\right) \cdot 1\right)\right) \]

       8. pow-sqr 90.9%

          \[\leadsto 0.5 \cdot \left(2 + \left(\sqrt[3]{\left(\color{blue}{{im}^{\left(2 \cdot 3\right)}} \cdot {\left(im \cdot im\right)}^{3}\right) \cdot 0.0005787037037037037} + \left(im \cdot im\right) \cdot 1\right)\right) \]

       9. metadata-eval 90.9%

          \[\leadsto 0.5 \cdot \left(2 + \left(\sqrt[3]{\left({im}^{\color{blue}{6}} \cdot {\left(im \cdot im\right)}^{3}\right) \cdot 0.0005787037037037037} + \left(im \cdot im\right) \cdot 1\right)\right) \]

       10. cube-prod 90.9%

           \[\leadsto 0.5 \cdot \left(2 + \left(\sqrt[3]{\left({im}^{6} \cdot \color{blue}{\left({im}^{3} \cdot {im}^{3}\right)}\right) \cdot 0.0005787037037037037} + \left(im \cdot im\right) \cdot 1\right)\right) \]

       11. pow-sqr 90.9%

           \[\leadsto 0.5 \cdot \left(2 + \left(\sqrt[3]{\left({im}^{6} \cdot \color{blue}{{im}^{\left(2 \cdot 3\right)}}\right) \cdot 0.0005787037037037037} + \left(im \cdot im\right) \cdot 1\right)\right) \]

       12. metadata-eval 90.9%

           \[\leadsto 0.5 \cdot \left(2 + \left(\sqrt[3]{\left({im}^{6} \cdot {im}^{\color{blue}{6}}\right) \cdot 0.0005787037037037037} + \left(im \cdot im\right) \cdot 1\right)\right) \]

       13. pow-sqr 90.9%

           \[\leadsto 0.5 \cdot \left(2 + \left(\sqrt[3]{\color{blue}{{im}^{\left(2 \cdot 6\right)}} \cdot 0.0005787037037037037} + \left(im \cdot im\right) \cdot 1\right)\right) \]

       14. metadata-eval 90.9%

           \[\leadsto 0.5 \cdot \left(2 + \left(\sqrt[3]{{im}^{\color{blue}{12}} \cdot 0.0005787037037037037} + \left(im \cdot im\right) \cdot 1\right)\right) \]

   11. Simplified 90.9%

       \[\leadsto 0.5 \cdot \left(2 + \left(\color{blue}{\sqrt[3]{{im}^{12} \cdot 0.0005787037037037037}} + \left(im \cdot im\right) \cdot 1\right)\right) \]

   if -4.9000000000000001e25 < im < 1.9199999999999999

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 95.5%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]
   3. Step-by-step derivation

      1. unpow2 95.5%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(\color{blue}{im \cdot im} + 0.08333333333333333 \cdot {im}^{4}\right)\right) \]

      2. *-commutative 95.5%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(im \cdot im + \color{blue}{{im}^{4} \cdot 0.08333333333333333}\right)\right) \]

   4. Simplified 95.5%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left(im \cdot im + {im}^{4} \cdot 0.08333333333333333\right)\right)} \]

   if 1.9199999999999999 < im

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-def 100.0%

         \[\leadsto \cos re \cdot \color{blue}{\mathsf{fma}\left(0.5, e^{im}, e^{-im} \cdot 0.5\right)} \]

      9. exp-neg 100.0%

         \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

      10. associate-*l/ 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

      11. metadata-eval 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{\color{blue}{0.5}}{e^{im}}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

   5. Applied egg-rr 100.0%

      \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{0.001953125}\right) \]
   6. Step-by-step derivation

      1. fma-udef 100.0%

         \[\leadsto \cos re \cdot \color{blue}{\left(0.5 \cdot e^{im} + 0.001953125\right)} \]

   7. Applied egg-rr 100.0%

      \[\leadsto \cos re \cdot \color{blue}{\left(0.5 \cdot e^{im} + 0.001953125\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.5 \cdot 10^{+85}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \mathbf{elif}\;im \leq -4.9 \cdot 10^{+25}:\\ \;\;\;\;0.5 \cdot \left(2 + \left(im \cdot im + \sqrt[3]{{im}^{12} \cdot 0.0005787037037037037}\right)\right)\\ \mathbf{elif}\;im \leq 1.92:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + \left(im \cdot im + {im}^{4} \cdot 0.08333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot e^{im} + 0.001953125\right)\\ \end{array} \]

Alternative 6: 96.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1 \cdot 10^{+86}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \mathbf{elif}\;im \leq -4.9 \cdot 10^{+25}:\\ \;\;\;\;0.5 \cdot \left(2 + \left(im \cdot im + \sqrt[3]{{im}^{12} \cdot 0.0005787037037037037}\right)\right)\\ \mathbf{elif}\;im \leq 1.3:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot e^{im} + 0.001953125\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -1e+86)
   (* 0.041666666666666664 (* (cos re) (pow im 4.0)))
   (if (<= im -4.9e+25)
     (*
      0.5
      (+ 2.0 (+ (* im im) (cbrt (* (pow im 12.0) 0.0005787037037037037)))))
     (if (<= im 1.3)
       (* (* (cos re) 0.5) (+ 2.0 (* im im)))
       (* (cos re) (+ (* 0.5 (exp im)) 0.001953125))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -1e+86) {
		tmp = 0.041666666666666664 * (cos(re) * pow(im, 4.0));
	} else if (im <= -4.9e+25) {
		tmp = 0.5 * (2.0 + ((im * im) + cbrt((pow(im, 12.0) * 0.0005787037037037037))));
	} else if (im <= 1.3) {
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	} else {
		tmp = cos(re) * ((0.5 * exp(im)) + 0.001953125);
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -1e+86) {
		tmp = 0.041666666666666664 * (Math.cos(re) * Math.pow(im, 4.0));
	} else if (im <= -4.9e+25) {
		tmp = 0.5 * (2.0 + ((im * im) + Math.cbrt((Math.pow(im, 12.0) * 0.0005787037037037037))));
	} else if (im <= 1.3) {
		tmp = (Math.cos(re) * 0.5) * (2.0 + (im * im));
	} else {
		tmp = Math.cos(re) * ((0.5 * Math.exp(im)) + 0.001953125);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -1e+86)
		tmp = Float64(0.041666666666666664 * Float64(cos(re) * (im ^ 4.0)));
	elseif (im <= -4.9e+25)
		tmp = Float64(0.5 * Float64(2.0 + Float64(Float64(im * im) + cbrt(Float64((im ^ 12.0) * 0.0005787037037037037)))));
	elseif (im <= 1.3)
		tmp = Float64(Float64(cos(re) * 0.5) * Float64(2.0 + Float64(im * im)));
	else
		tmp = Float64(cos(re) * Float64(Float64(0.5 * exp(im)) + 0.001953125));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, -1e+86], N[(0.041666666666666664 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, -4.9e+25], N[(0.5 * N[(2.0 + N[(N[(im * im), $MachinePrecision] + N[Power[N[(N[Power[im, 12.0], $MachinePrecision] * 0.0005787037037037037), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.3], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision] + 0.001953125), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < -1e86

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]
   3. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(\color{blue}{im \cdot im} + 0.08333333333333333 \cdot {im}^{4}\right)\right) \]

      2. *-commutative 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(im \cdot im + \color{blue}{{im}^{4} \cdot 0.08333333333333333}\right)\right) \]

   4. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left(im \cdot im + {im}^{4} \cdot 0.08333333333333333\right)\right)} \]

   5. Taylor expanded in im around inf 100.0%

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

   if -1e86 < im < -4.9000000000000001e25

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 31.9%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]
   3. Step-by-step derivation

      1. unpow2 31.9%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(\color{blue}{im \cdot im} + 0.08333333333333333 \cdot {im}^{4}\right)\right) \]

      2. *-commutative 31.9%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(im \cdot im + \color{blue}{{im}^{4} \cdot 0.08333333333333333}\right)\right) \]

   4. Simplified 31.9%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left(im \cdot im + {im}^{4} \cdot 0.08333333333333333\right)\right)} \]

   5. Taylor expanded in re around 0 31.5%

      \[\leadsto \color{blue}{0.5 \cdot \left(2 + \left(0.08333333333333333 \cdot {im}^{4} + {im}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 31.5%

         \[\leadsto 0.5 \cdot \left(2 + \left(0.08333333333333333 \cdot {im}^{4} + \color{blue}{1 \cdot {im}^{2}}\right)\right) \]

      2. *-commutative 31.5%

         \[\leadsto 0.5 \cdot \left(2 + \left(0.08333333333333333 \cdot {im}^{4} + \color{blue}{{im}^{2} \cdot 1}\right)\right) \]

      3. unpow2 31.5%

         \[\leadsto 0.5 \cdot \left(2 + \left(0.08333333333333333 \cdot {im}^{4} + \color{blue}{\left(im \cdot im\right)} \cdot 1\right)\right) \]

   7. Applied egg-rr 31.5%

      \[\leadsto 0.5 \cdot \left(2 + \left(0.08333333333333333 \cdot {im}^{4} + \color{blue}{\left(im \cdot im\right) \cdot 1}\right)\right) \]
   8. Step-by-step derivation

      1. add-cbrt-cube 90.9%

         \[\leadsto 0.5 \cdot \left(2 + \left(\color{blue}{\sqrt[3]{\left(\left(0.08333333333333333 \cdot {im}^{4}\right) \cdot \left(0.08333333333333333 \cdot {im}^{4}\right)\right) \cdot \left(0.08333333333333333 \cdot {im}^{4}\right)}} + \left(im \cdot im\right) \cdot 1\right)\right) \]

      2. pow1/3 90.9%

         \[\leadsto 0.5 \cdot \left(2 + \left(\color{blue}{{\left(\left(\left(0.08333333333333333 \cdot {im}^{4}\right) \cdot \left(0.08333333333333333 \cdot {im}^{4}\right)\right) \cdot \left(0.08333333333333333 \cdot {im}^{4}\right)\right)}^{0.3333333333333333}} + \left(im \cdot im\right) \cdot 1\right)\right) \]

      3. pow3 90.9%

         \[\leadsto 0.5 \cdot \left(2 + \left({\color{blue}{\left({\left(0.08333333333333333 \cdot {im}^{4}\right)}^{3}\right)}}^{0.3333333333333333} + \left(im \cdot im\right) \cdot 1\right)\right) \]

      4. *-commutative 90.9%

         \[\leadsto 0.5 \cdot \left(2 + \left({\left({\color{blue}{\left({im}^{4} \cdot 0.08333333333333333\right)}}^{3}\right)}^{0.3333333333333333} + \left(im \cdot im\right) \cdot 1\right)\right) \]

      5. unpow-prod-down 90.9%

         \[\leadsto 0.5 \cdot \left(2 + \left({\color{blue}{\left({\left({im}^{4}\right)}^{3} \cdot {0.08333333333333333}^{3}\right)}}^{0.3333333333333333} + \left(im \cdot im\right) \cdot 1\right)\right) \]

      6. metadata-eval 90.9%

         \[\leadsto 0.5 \cdot \left(2 + \left({\left({\left({im}^{4}\right)}^{3} \cdot \color{blue}{0.0005787037037037037}\right)}^{0.3333333333333333} + \left(im \cdot im\right) \cdot 1\right)\right) \]

   9. Applied egg-rr 90.9%

      \[\leadsto 0.5 \cdot \left(2 + \left(\color{blue}{{\left({\left({im}^{4}\right)}^{3} \cdot 0.0005787037037037037\right)}^{0.3333333333333333}} + \left(im \cdot im\right) \cdot 1\right)\right) \]
   10. Step-by-step derivation

       1. unpow1/3 90.9%

          \[\leadsto 0.5 \cdot \left(2 + \left(\color{blue}{\sqrt[3]{{\left({im}^{4}\right)}^{3} \cdot 0.0005787037037037037}} + \left(im \cdot im\right) \cdot 1\right)\right) \]

       2. metadata-eval 90.9%

          \[\leadsto 0.5 \cdot \left(2 + \left(\sqrt[3]{{\left({im}^{\color{blue}{\left(2 \cdot 2\right)}}\right)}^{3} \cdot 0.0005787037037037037} + \left(im \cdot im\right) \cdot 1\right)\right) \]

       3. pow-sqr 90.9%

          \[\leadsto 0.5 \cdot \left(2 + \left(\sqrt[3]{{\color{blue}{\left({im}^{2} \cdot {im}^{2}\right)}}^{3} \cdot 0.0005787037037037037} + \left(im \cdot im\right) \cdot 1\right)\right) \]

       4. unpow2 90.9%

          \[\leadsto 0.5 \cdot \left(2 + \left(\sqrt[3]{{\left(\color{blue}{\left(im \cdot im\right)} \cdot {im}^{2}\right)}^{3} \cdot 0.0005787037037037037} + \left(im \cdot im\right) \cdot 1\right)\right) \]

       5. unpow2 90.9%

          \[\leadsto 0.5 \cdot \left(2 + \left(\sqrt[3]{{\left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right)}^{3} \cdot 0.0005787037037037037} + \left(im \cdot im\right) \cdot 1\right)\right) \]

       6. cube-prod 90.9%

          \[\leadsto 0.5 \cdot \left(2 + \left(\sqrt[3]{\color{blue}{\left({\left(im \cdot im\right)}^{3} \cdot {\left(im \cdot im\right)}^{3}\right)} \cdot 0.0005787037037037037} + \left(im \cdot im\right) \cdot 1\right)\right) \]

       7. cube-prod 90.9%

          \[\leadsto 0.5 \cdot \left(2 + \left(\sqrt[3]{\left(\color{blue}{\left({im}^{3} \cdot {im}^{3}\right)} \cdot {\left(im \cdot im\right)}^{3}\right) \cdot 0.0005787037037037037} + \left(im \cdot im\right) \cdot 1\right)\right) \]

       8. pow-sqr 90.9%

          \[\leadsto 0.5 \cdot \left(2 + \left(\sqrt[3]{\left(\color{blue}{{im}^{\left(2 \cdot 3\right)}} \cdot {\left(im \cdot im\right)}^{3}\right) \cdot 0.0005787037037037037} + \left(im \cdot im\right) \cdot 1\right)\right) \]

       9. metadata-eval 90.9%

          \[\leadsto 0.5 \cdot \left(2 + \left(\sqrt[3]{\left({im}^{\color{blue}{6}} \cdot {\left(im \cdot im\right)}^{3}\right) \cdot 0.0005787037037037037} + \left(im \cdot im\right) \cdot 1\right)\right) \]

       10. cube-prod 90.9%

           \[\leadsto 0.5 \cdot \left(2 + \left(\sqrt[3]{\left({im}^{6} \cdot \color{blue}{\left({im}^{3} \cdot {im}^{3}\right)}\right) \cdot 0.0005787037037037037} + \left(im \cdot im\right) \cdot 1\right)\right) \]

       11. pow-sqr 90.9%

           \[\leadsto 0.5 \cdot \left(2 + \left(\sqrt[3]{\left({im}^{6} \cdot \color{blue}{{im}^{\left(2 \cdot 3\right)}}\right) \cdot 0.0005787037037037037} + \left(im \cdot im\right) \cdot 1\right)\right) \]

       12. metadata-eval 90.9%

           \[\leadsto 0.5 \cdot \left(2 + \left(\sqrt[3]{\left({im}^{6} \cdot {im}^{\color{blue}{6}}\right) \cdot 0.0005787037037037037} + \left(im \cdot im\right) \cdot 1\right)\right) \]

       13. pow-sqr 90.9%

           \[\leadsto 0.5 \cdot \left(2 + \left(\sqrt[3]{\color{blue}{{im}^{\left(2 \cdot 6\right)}} \cdot 0.0005787037037037037} + \left(im \cdot im\right) \cdot 1\right)\right) \]

       14. metadata-eval 90.9%

           \[\leadsto 0.5 \cdot \left(2 + \left(\sqrt[3]{{im}^{\color{blue}{12}} \cdot 0.0005787037037037037} + \left(im \cdot im\right) \cdot 1\right)\right) \]

   11. Simplified 90.9%

       \[\leadsto 0.5 \cdot \left(2 + \left(\color{blue}{\sqrt[3]{{im}^{12} \cdot 0.0005787037037037037}} + \left(im \cdot im\right) \cdot 1\right)\right) \]

   if -4.9000000000000001e25 < im < 1.30000000000000004

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 95.5%

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

      1. unpow2 95.5%

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

   4. Simplified 95.5%

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

   if 1.30000000000000004 < im

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-def 100.0%

         \[\leadsto \cos re \cdot \color{blue}{\mathsf{fma}\left(0.5, e^{im}, e^{-im} \cdot 0.5\right)} \]

      9. exp-neg 100.0%

         \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

      10. associate-*l/ 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

      11. metadata-eval 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{\color{blue}{0.5}}{e^{im}}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

   5. Applied egg-rr 100.0%

      \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{0.001953125}\right) \]
   6. Step-by-step derivation

      1. fma-udef 100.0%

         \[\leadsto \cos re \cdot \color{blue}{\left(0.5 \cdot e^{im} + 0.001953125\right)} \]

   7. Applied egg-rr 100.0%

      \[\leadsto \cos re \cdot \color{blue}{\left(0.5 \cdot e^{im} + 0.001953125\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1 \cdot 10^{+86}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \mathbf{elif}\;im \leq -4.9 \cdot 10^{+25}:\\ \;\;\;\;0.5 \cdot \left(2 + \left(im \cdot im + \sqrt[3]{{im}^{12} \cdot 0.0005787037037037037}\right)\right)\\ \mathbf{elif}\;im \leq 1.3:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot e^{im} + 0.001953125\right)\\ \end{array} \]

Alternative 7: 93.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \mathbf{if}\;im \leq -2 \cdot 10^{+88}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -1.46 \cdot 10^{+41}:\\ \;\;\;\;\sqrt{{im}^{8} \cdot 0.001736111111111111}\\ \mathbf{elif}\;im \leq 2.2:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.16 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot e^{im} + 0.001953125\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.041666666666666664 (* (cos re) (pow im 4.0)))))
   (if (<= im -2e+88)
     t_0
     (if (<= im -1.46e+41)
       (sqrt (* (pow im 8.0) 0.001736111111111111))
       (if (<= im 2.2)
         (* (* (cos re) 0.5) (+ 2.0 (* im im)))
         (if (<= im 1.16e+77) (+ (* 0.5 (exp im)) 0.001953125) t_0))))))

C

double code(double re, double im) {
	double t_0 = 0.041666666666666664 * (cos(re) * pow(im, 4.0));
	double tmp;
	if (im <= -2e+88) {
		tmp = t_0;
	} else if (im <= -1.46e+41) {
		tmp = sqrt((pow(im, 8.0) * 0.001736111111111111));
	} else if (im <= 2.2) {
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	} else if (im <= 1.16e+77) {
		tmp = (0.5 * exp(im)) + 0.001953125;
	} 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 = 0.041666666666666664d0 * (cos(re) * (im ** 4.0d0))
    if (im <= (-2d+88)) then
        tmp = t_0
    else if (im <= (-1.46d+41)) then
        tmp = sqrt(((im ** 8.0d0) * 0.001736111111111111d0))
    else if (im <= 2.2d0) then
        tmp = (cos(re) * 0.5d0) * (2.0d0 + (im * im))
    else if (im <= 1.16d+77) then
        tmp = (0.5d0 * exp(im)) + 0.001953125d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.041666666666666664 * (Math.cos(re) * Math.pow(im, 4.0));
	double tmp;
	if (im <= -2e+88) {
		tmp = t_0;
	} else if (im <= -1.46e+41) {
		tmp = Math.sqrt((Math.pow(im, 8.0) * 0.001736111111111111));
	} else if (im <= 2.2) {
		tmp = (Math.cos(re) * 0.5) * (2.0 + (im * im));
	} else if (im <= 1.16e+77) {
		tmp = (0.5 * Math.exp(im)) + 0.001953125;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.041666666666666664 * (math.cos(re) * math.pow(im, 4.0))
	tmp = 0
	if im <= -2e+88:
		tmp = t_0
	elif im <= -1.46e+41:
		tmp = math.sqrt((math.pow(im, 8.0) * 0.001736111111111111))
	elif im <= 2.2:
		tmp = (math.cos(re) * 0.5) * (2.0 + (im * im))
	elif im <= 1.16e+77:
		tmp = (0.5 * math.exp(im)) + 0.001953125
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.041666666666666664 * Float64(cos(re) * (im ^ 4.0)))
	tmp = 0.0
	if (im <= -2e+88)
		tmp = t_0;
	elseif (im <= -1.46e+41)
		tmp = sqrt(Float64((im ^ 8.0) * 0.001736111111111111));
	elseif (im <= 2.2)
		tmp = Float64(Float64(cos(re) * 0.5) * Float64(2.0 + Float64(im * im)));
	elseif (im <= 1.16e+77)
		tmp = Float64(Float64(0.5 * exp(im)) + 0.001953125);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.041666666666666664 * (cos(re) * (im ^ 4.0));
	tmp = 0.0;
	if (im <= -2e+88)
		tmp = t_0;
	elseif (im <= -1.46e+41)
		tmp = sqrt(((im ^ 8.0) * 0.001736111111111111));
	elseif (im <= 2.2)
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	elseif (im <= 1.16e+77)
		tmp = (0.5 * exp(im)) + 0.001953125;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.041666666666666664 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -2e+88], t$95$0, If[LessEqual[im, -1.46e+41], N[Sqrt[N[(N[Power[im, 8.0], $MachinePrecision] * 0.001736111111111111), $MachinePrecision]], $MachinePrecision], If[LessEqual[im, 2.2], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.16e+77], N[(N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision] + 0.001953125), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -1.99999999999999992e88 or 1.1600000000000001e77 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]
   3. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(\color{blue}{im \cdot im} + 0.08333333333333333 \cdot {im}^{4}\right)\right) \]

      2. *-commutative 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(im \cdot im + \color{blue}{{im}^{4} \cdot 0.08333333333333333}\right)\right) \]

   4. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left(im \cdot im + {im}^{4} \cdot 0.08333333333333333\right)\right)} \]

   5. Taylor expanded in im around inf 100.0%

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

   if -1.99999999999999992e88 < im < -1.4600000000000001e41

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 38.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]
   3. Step-by-step derivation

      1. unpow2 38.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(\color{blue}{im \cdot im} + 0.08333333333333333 \cdot {im}^{4}\right)\right) \]

      2. *-commutative 38.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(im \cdot im + \color{blue}{{im}^{4} \cdot 0.08333333333333333}\right)\right) \]

   4. Simplified 38.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left(im \cdot im + {im}^{4} \cdot 0.08333333333333333\right)\right)} \]

   5. Taylor expanded in im around inf 38.0%

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

   6. Taylor expanded in re around 0 38.0%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot {im}^{4}} \]
   7. Step-by-step derivation

      1. *-commutative 38.0%

         \[\leadsto \color{blue}{{im}^{4} \cdot 0.041666666666666664} \]

   8. Simplified 38.0%

      \[\leadsto \color{blue}{{im}^{4} \cdot 0.041666666666666664} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 38.0%

         \[\leadsto \color{blue}{\sqrt{{im}^{4} \cdot 0.041666666666666664} \cdot \sqrt{{im}^{4} \cdot 0.041666666666666664}} \]

      2. sqrt-unprod 100.0%

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

      3. swap-sqr 100.0%

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

      4. pow-prod-up 100.0%

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

      5. metadata-eval 100.0%

         \[\leadsto \sqrt{{im}^{\color{blue}{8}} \cdot \left(0.041666666666666664 \cdot 0.041666666666666664\right)} \]

      6. metadata-eval 100.0%

         \[\leadsto \sqrt{{im}^{8} \cdot \color{blue}{0.001736111111111111}} \]

   10. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\sqrt{{im}^{8} \cdot 0.001736111111111111}} \]

   if -1.4600000000000001e41 < im < 2.2000000000000002

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 94.1%

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

      1. unpow2 94.1%

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

   4. Simplified 94.1%

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

   if 2.2000000000000002 < im < 1.1600000000000001e77

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-def 100.0%

         \[\leadsto \cos re \cdot \color{blue}{\mathsf{fma}\left(0.5, e^{im}, e^{-im} \cdot 0.5\right)} \]

      9. exp-neg 100.0%

         \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

      10. associate-*l/ 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

      11. metadata-eval 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{\color{blue}{0.5}}{e^{im}}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

   5. Applied egg-rr 100.0%

      \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{0.001953125}\right) \]

   6. Taylor expanded in re around 0 92.3%

      \[\leadsto \color{blue}{0.001953125 + 0.5 \cdot e^{im}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2 \cdot 10^{+88}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \mathbf{elif}\;im \leq -1.46 \cdot 10^{+41}:\\ \;\;\;\;\sqrt{{im}^{8} \cdot 0.001736111111111111}\\ \mathbf{elif}\;im \leq 2.2:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.16 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot e^{im} + 0.001953125\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \end{array} \]

Alternative 8: 95.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1.5 \cdot 10^{+85}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \mathbf{elif}\;im \leq -1.46 \cdot 10^{+41}:\\ \;\;\;\;\sqrt{{im}^{8} \cdot 0.001736111111111111}\\ \mathbf{elif}\;im \leq 1.3:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot e^{im} + 0.001953125\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -1.5e+85)
   (* 0.041666666666666664 (* (cos re) (pow im 4.0)))
   (if (<= im -1.46e+41)
     (sqrt (* (pow im 8.0) 0.001736111111111111))
     (if (<= im 1.3)
       (* (* (cos re) 0.5) (+ 2.0 (* im im)))
       (* (cos re) (+ (* 0.5 (exp im)) 0.001953125))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -1.5e+85) {
		tmp = 0.041666666666666664 * (cos(re) * pow(im, 4.0));
	} else if (im <= -1.46e+41) {
		tmp = sqrt((pow(im, 8.0) * 0.001736111111111111));
	} else if (im <= 1.3) {
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	} else {
		tmp = cos(re) * ((0.5 * exp(im)) + 0.001953125);
	}
	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+85)) then
        tmp = 0.041666666666666664d0 * (cos(re) * (im ** 4.0d0))
    else if (im <= (-1.46d+41)) then
        tmp = sqrt(((im ** 8.0d0) * 0.001736111111111111d0))
    else if (im <= 1.3d0) then
        tmp = (cos(re) * 0.5d0) * (2.0d0 + (im * im))
    else
        tmp = cos(re) * ((0.5d0 * exp(im)) + 0.001953125d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -1.5e+85) {
		tmp = 0.041666666666666664 * (Math.cos(re) * Math.pow(im, 4.0));
	} else if (im <= -1.46e+41) {
		tmp = Math.sqrt((Math.pow(im, 8.0) * 0.001736111111111111));
	} else if (im <= 1.3) {
		tmp = (Math.cos(re) * 0.5) * (2.0 + (im * im));
	} else {
		tmp = Math.cos(re) * ((0.5 * Math.exp(im)) + 0.001953125);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -1.5e+85:
		tmp = 0.041666666666666664 * (math.cos(re) * math.pow(im, 4.0))
	elif im <= -1.46e+41:
		tmp = math.sqrt((math.pow(im, 8.0) * 0.001736111111111111))
	elif im <= 1.3:
		tmp = (math.cos(re) * 0.5) * (2.0 + (im * im))
	else:
		tmp = math.cos(re) * ((0.5 * math.exp(im)) + 0.001953125)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -1.5e+85)
		tmp = Float64(0.041666666666666664 * Float64(cos(re) * (im ^ 4.0)));
	elseif (im <= -1.46e+41)
		tmp = sqrt(Float64((im ^ 8.0) * 0.001736111111111111));
	elseif (im <= 1.3)
		tmp = Float64(Float64(cos(re) * 0.5) * Float64(2.0 + Float64(im * im)));
	else
		tmp = Float64(cos(re) * Float64(Float64(0.5 * exp(im)) + 0.001953125));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -1.5e+85)
		tmp = 0.041666666666666664 * (cos(re) * (im ^ 4.0));
	elseif (im <= -1.46e+41)
		tmp = sqrt(((im ^ 8.0) * 0.001736111111111111));
	elseif (im <= 1.3)
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	else
		tmp = cos(re) * ((0.5 * exp(im)) + 0.001953125);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -1.5e+85], N[(0.041666666666666664 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, -1.46e+41], N[Sqrt[N[(N[Power[im, 8.0], $MachinePrecision] * 0.001736111111111111), $MachinePrecision]], $MachinePrecision], If[LessEqual[im, 1.3], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision] + 0.001953125), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < -1.5e85

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]
   3. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(\color{blue}{im \cdot im} + 0.08333333333333333 \cdot {im}^{4}\right)\right) \]

      2. *-commutative 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(im \cdot im + \color{blue}{{im}^{4} \cdot 0.08333333333333333}\right)\right) \]

   4. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left(im \cdot im + {im}^{4} \cdot 0.08333333333333333\right)\right)} \]

   5. Taylor expanded in im around inf 100.0%

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

   if -1.5e85 < im < -1.4600000000000001e41

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 38.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]
   3. Step-by-step derivation

      1. unpow2 38.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(\color{blue}{im \cdot im} + 0.08333333333333333 \cdot {im}^{4}\right)\right) \]

      2. *-commutative 38.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(im \cdot im + \color{blue}{{im}^{4} \cdot 0.08333333333333333}\right)\right) \]

   4. Simplified 38.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left(im \cdot im + {im}^{4} \cdot 0.08333333333333333\right)\right)} \]

   5. Taylor expanded in im around inf 38.0%

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

   6. Taylor expanded in re around 0 38.0%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot {im}^{4}} \]
   7. Step-by-step derivation

      1. *-commutative 38.0%

         \[\leadsto \color{blue}{{im}^{4} \cdot 0.041666666666666664} \]

   8. Simplified 38.0%

      \[\leadsto \color{blue}{{im}^{4} \cdot 0.041666666666666664} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 38.0%

         \[\leadsto \color{blue}{\sqrt{{im}^{4} \cdot 0.041666666666666664} \cdot \sqrt{{im}^{4} \cdot 0.041666666666666664}} \]

      2. sqrt-unprod 100.0%

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

      3. swap-sqr 100.0%

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

      4. pow-prod-up 100.0%

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

      5. metadata-eval 100.0%

         \[\leadsto \sqrt{{im}^{\color{blue}{8}} \cdot \left(0.041666666666666664 \cdot 0.041666666666666664\right)} \]

      6. metadata-eval 100.0%

         \[\leadsto \sqrt{{im}^{8} \cdot \color{blue}{0.001736111111111111}} \]

   10. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\sqrt{{im}^{8} \cdot 0.001736111111111111}} \]

   if -1.4600000000000001e41 < im < 1.30000000000000004

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 94.1%

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

      1. unpow2 94.1%

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

   4. Simplified 94.1%

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

   if 1.30000000000000004 < im

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-def 100.0%

         \[\leadsto \cos re \cdot \color{blue}{\mathsf{fma}\left(0.5, e^{im}, e^{-im} \cdot 0.5\right)} \]

      9. exp-neg 100.0%

         \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

      10. associate-*l/ 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

      11. metadata-eval 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{\color{blue}{0.5}}{e^{im}}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

   5. Applied egg-rr 100.0%

      \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{0.001953125}\right) \]
   6. Step-by-step derivation

      1. fma-udef 100.0%

         \[\leadsto \cos re \cdot \color{blue}{\left(0.5 \cdot e^{im} + 0.001953125\right)} \]

   7. Applied egg-rr 100.0%

      \[\leadsto \cos re \cdot \color{blue}{\left(0.5 \cdot e^{im} + 0.001953125\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.5 \cdot 10^{+85}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \mathbf{elif}\;im \leq -1.46 \cdot 10^{+41}:\\ \;\;\;\;\sqrt{{im}^{8} \cdot 0.001736111111111111}\\ \mathbf{elif}\;im \leq 1.3:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot e^{im} + 0.001953125\right)\\ \end{array} \]

Alternative 9: 90.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{if}\;im \leq -5 \cdot 10^{+167}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -1.46 \cdot 10^{+41}:\\ \;\;\;\;\sqrt{{im}^{8} \cdot 0.001736111111111111}\\ \mathbf{elif}\;im \leq 2.35:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot e^{im} + 0.001953125\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (cos re) (* 0.5 (* im im)))))
   (if (<= im -5e+167)
     t_0
     (if (<= im -1.46e+41)
       (sqrt (* (pow im 8.0) 0.001736111111111111))
       (if (<= im 2.35)
         (* (* (cos re) 0.5) (+ 2.0 (* im im)))
         (if (<= im 1.35e+154) (+ (* 0.5 (exp im)) 0.001953125) t_0))))))

C

double code(double re, double im) {
	double t_0 = cos(re) * (0.5 * (im * im));
	double tmp;
	if (im <= -5e+167) {
		tmp = t_0;
	} else if (im <= -1.46e+41) {
		tmp = sqrt((pow(im, 8.0) * 0.001736111111111111));
	} else if (im <= 2.35) {
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	} else if (im <= 1.35e+154) {
		tmp = (0.5 * exp(im)) + 0.001953125;
	} 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 = cos(re) * (0.5d0 * (im * im))
    if (im <= (-5d+167)) then
        tmp = t_0
    else if (im <= (-1.46d+41)) then
        tmp = sqrt(((im ** 8.0d0) * 0.001736111111111111d0))
    else if (im <= 2.35d0) then
        tmp = (cos(re) * 0.5d0) * (2.0d0 + (im * im))
    else if (im <= 1.35d+154) then
        tmp = (0.5d0 * exp(im)) + 0.001953125d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.cos(re) * (0.5 * (im * im));
	double tmp;
	if (im <= -5e+167) {
		tmp = t_0;
	} else if (im <= -1.46e+41) {
		tmp = Math.sqrt((Math.pow(im, 8.0) * 0.001736111111111111));
	} else if (im <= 2.35) {
		tmp = (Math.cos(re) * 0.5) * (2.0 + (im * im));
	} else if (im <= 1.35e+154) {
		tmp = (0.5 * Math.exp(im)) + 0.001953125;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.cos(re) * (0.5 * (im * im))
	tmp = 0
	if im <= -5e+167:
		tmp = t_0
	elif im <= -1.46e+41:
		tmp = math.sqrt((math.pow(im, 8.0) * 0.001736111111111111))
	elif im <= 2.35:
		tmp = (math.cos(re) * 0.5) * (2.0 + (im * im))
	elif im <= 1.35e+154:
		tmp = (0.5 * math.exp(im)) + 0.001953125
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(cos(re) * Float64(0.5 * Float64(im * im)))
	tmp = 0.0
	if (im <= -5e+167)
		tmp = t_0;
	elseif (im <= -1.46e+41)
		tmp = sqrt(Float64((im ^ 8.0) * 0.001736111111111111));
	elseif (im <= 2.35)
		tmp = Float64(Float64(cos(re) * 0.5) * Float64(2.0 + Float64(im * im)));
	elseif (im <= 1.35e+154)
		tmp = Float64(Float64(0.5 * exp(im)) + 0.001953125);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = cos(re) * (0.5 * (im * im));
	tmp = 0.0;
	if (im <= -5e+167)
		tmp = t_0;
	elseif (im <= -1.46e+41)
		tmp = sqrt(((im ^ 8.0) * 0.001736111111111111));
	elseif (im <= 2.35)
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	elseif (im <= 1.35e+154)
		tmp = (0.5 * exp(im)) + 0.001953125;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Cos[re], $MachinePrecision] * N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -5e+167], t$95$0, If[LessEqual[im, -1.46e+41], N[Sqrt[N[(N[Power[im, 8.0], $MachinePrecision] * 0.001736111111111111), $MachinePrecision]], $MachinePrecision], If[LessEqual[im, 2.35], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision] + 0.001953125), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -4.9999999999999997e167 or 1.35000000000000003e154 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

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

      1. unpow2 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around inf 100.0%

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

      1. unpow2 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*r* 100.0%

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

   7. Simplified 100.0%

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

   if -4.9999999999999997e167 < im < -1.4600000000000001e41

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 75.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]
   3. Step-by-step derivation

      1. unpow2 75.7%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(\color{blue}{im \cdot im} + 0.08333333333333333 \cdot {im}^{4}\right)\right) \]

      2. *-commutative 75.7%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(im \cdot im + \color{blue}{{im}^{4} \cdot 0.08333333333333333}\right)\right) \]

   4. Simplified 75.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left(im \cdot im + {im}^{4} \cdot 0.08333333333333333\right)\right)} \]

   5. Taylor expanded in im around inf 75.7%

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

   6. Taylor expanded in re around 0 62.7%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot {im}^{4}} \]
   7. Step-by-step derivation

      1. *-commutative 62.7%

         \[\leadsto \color{blue}{{im}^{4} \cdot 0.041666666666666664} \]

   8. Simplified 62.7%

      \[\leadsto \color{blue}{{im}^{4} \cdot 0.041666666666666664} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 62.7%

         \[\leadsto \color{blue}{\sqrt{{im}^{4} \cdot 0.041666666666666664} \cdot \sqrt{{im}^{4} \cdot 0.041666666666666664}} \]

      2. sqrt-unprod 87.0%

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

      3. swap-sqr 87.0%

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

      4. pow-prod-up 87.0%

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

      5. metadata-eval 87.0%

         \[\leadsto \sqrt{{im}^{\color{blue}{8}} \cdot \left(0.041666666666666664 \cdot 0.041666666666666664\right)} \]

      6. metadata-eval 87.0%

         \[\leadsto \sqrt{{im}^{8} \cdot \color{blue}{0.001736111111111111}} \]

   10. Applied egg-rr 87.0%

       \[\leadsto \color{blue}{\sqrt{{im}^{8} \cdot 0.001736111111111111}} \]

   if -1.4600000000000001e41 < im < 2.35000000000000009

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 94.1%

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

      1. unpow2 94.1%

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

   4. Simplified 94.1%

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

   if 2.35000000000000009 < im < 1.35000000000000003e154

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-def 100.0%

         \[\leadsto \cos re \cdot \color{blue}{\mathsf{fma}\left(0.5, e^{im}, e^{-im} \cdot 0.5\right)} \]

      9. exp-neg 100.0%

         \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

      10. associate-*l/ 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

      11. metadata-eval 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{\color{blue}{0.5}}{e^{im}}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

   5. Applied egg-rr 100.0%

      \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{0.001953125}\right) \]

   6. Taylor expanded in re around 0 93.2%

      \[\leadsto \color{blue}{0.001953125 + 0.5 \cdot e^{im}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -5 \cdot 10^{+167}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq -1.46 \cdot 10^{+41}:\\ \;\;\;\;\sqrt{{im}^{8} \cdot 0.001736111111111111}\\ \mathbf{elif}\;im \leq 2.35:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot e^{im} + 0.001953125\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 10: 87.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{if}\;im \leq -1.6 \cdot 10^{+166}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -1.46 \cdot 10^{+41}:\\ \;\;\;\;0.041666666666666664 \cdot {im}^{4}\\ \mathbf{elif}\;im \leq 1.3:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot e^{im} + 0.001953125\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (cos re) (* 0.5 (* im im)))))
   (if (<= im -1.6e+166)
     t_0
     (if (<= im -1.46e+41)
       (* 0.041666666666666664 (pow im 4.0))
       (if (<= im 1.3)
         (cos re)
         (if (<= im 1.35e+154) (+ (* 0.5 (exp im)) 0.001953125) t_0))))))

C

double code(double re, double im) {
	double t_0 = cos(re) * (0.5 * (im * im));
	double tmp;
	if (im <= -1.6e+166) {
		tmp = t_0;
	} else if (im <= -1.46e+41) {
		tmp = 0.041666666666666664 * pow(im, 4.0);
	} else if (im <= 1.3) {
		tmp = cos(re);
	} else if (im <= 1.35e+154) {
		tmp = (0.5 * exp(im)) + 0.001953125;
	} 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 = cos(re) * (0.5d0 * (im * im))
    if (im <= (-1.6d+166)) then
        tmp = t_0
    else if (im <= (-1.46d+41)) then
        tmp = 0.041666666666666664d0 * (im ** 4.0d0)
    else if (im <= 1.3d0) then
        tmp = cos(re)
    else if (im <= 1.35d+154) then
        tmp = (0.5d0 * exp(im)) + 0.001953125d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.cos(re) * (0.5 * (im * im));
	double tmp;
	if (im <= -1.6e+166) {
		tmp = t_0;
	} else if (im <= -1.46e+41) {
		tmp = 0.041666666666666664 * Math.pow(im, 4.0);
	} else if (im <= 1.3) {
		tmp = Math.cos(re);
	} else if (im <= 1.35e+154) {
		tmp = (0.5 * Math.exp(im)) + 0.001953125;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.cos(re) * (0.5 * (im * im))
	tmp = 0
	if im <= -1.6e+166:
		tmp = t_0
	elif im <= -1.46e+41:
		tmp = 0.041666666666666664 * math.pow(im, 4.0)
	elif im <= 1.3:
		tmp = math.cos(re)
	elif im <= 1.35e+154:
		tmp = (0.5 * math.exp(im)) + 0.001953125
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(cos(re) * Float64(0.5 * Float64(im * im)))
	tmp = 0.0
	if (im <= -1.6e+166)
		tmp = t_0;
	elseif (im <= -1.46e+41)
		tmp = Float64(0.041666666666666664 * (im ^ 4.0));
	elseif (im <= 1.3)
		tmp = cos(re);
	elseif (im <= 1.35e+154)
		tmp = Float64(Float64(0.5 * exp(im)) + 0.001953125);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = cos(re) * (0.5 * (im * im));
	tmp = 0.0;
	if (im <= -1.6e+166)
		tmp = t_0;
	elseif (im <= -1.46e+41)
		tmp = 0.041666666666666664 * (im ^ 4.0);
	elseif (im <= 1.3)
		tmp = cos(re);
	elseif (im <= 1.35e+154)
		tmp = (0.5 * exp(im)) + 0.001953125;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Cos[re], $MachinePrecision] * N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.6e+166], t$95$0, If[LessEqual[im, -1.46e+41], N[(0.041666666666666664 * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.3], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision] + 0.001953125), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -1.59999999999999984e166 or 1.35000000000000003e154 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

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

      1. unpow2 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around inf 100.0%

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

      1. unpow2 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*r* 100.0%

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

   7. Simplified 100.0%

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

   if -1.59999999999999984e166 < im < -1.4600000000000001e41

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 75.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]
   3. Step-by-step derivation

      1. unpow2 75.7%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(\color{blue}{im \cdot im} + 0.08333333333333333 \cdot {im}^{4}\right)\right) \]

      2. *-commutative 75.7%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(im \cdot im + \color{blue}{{im}^{4} \cdot 0.08333333333333333}\right)\right) \]

   4. Simplified 75.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left(im \cdot im + {im}^{4} \cdot 0.08333333333333333\right)\right)} \]

   5. Taylor expanded in im around inf 75.7%

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

   6. Taylor expanded in re around 0 62.7%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot {im}^{4}} \]
   7. Step-by-step derivation

      1. *-commutative 62.7%

         \[\leadsto \color{blue}{{im}^{4} \cdot 0.041666666666666664} \]

   8. Simplified 62.7%

      \[\leadsto \color{blue}{{im}^{4} \cdot 0.041666666666666664} \]

   if -1.4600000000000001e41 < im < 1.30000000000000004

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 94.1%

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

      1. unpow2 94.1%

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

   4. Simplified 94.1%

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

   5. Taylor expanded in im around 0 93.8%

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

   if 1.30000000000000004 < im < 1.35000000000000003e154

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-def 100.0%

         \[\leadsto \cos re \cdot \color{blue}{\mathsf{fma}\left(0.5, e^{im}, e^{-im} \cdot 0.5\right)} \]

      9. exp-neg 100.0%

         \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

      10. associate-*l/ 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

      11. metadata-eval 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{\color{blue}{0.5}}{e^{im}}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

   5. Applied egg-rr 100.0%

      \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{0.001953125}\right) \]

   6. Taylor expanded in re around 0 93.2%

      \[\leadsto \color{blue}{0.001953125 + 0.5 \cdot e^{im}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.6 \cdot 10^{+166}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq -1.46 \cdot 10^{+41}:\\ \;\;\;\;0.041666666666666664 \cdot {im}^{4}\\ \mathbf{elif}\;im \leq 1.3:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot e^{im} + 0.001953125\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 11: 88.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{if}\;im \leq -1.6 \cdot 10^{+166}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -1.85 \cdot 10^{+42}:\\ \;\;\;\;0.041666666666666664 \cdot {im}^{4}\\ \mathbf{elif}\;im \leq 2.1:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot e^{im} + 0.001953125\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (cos re) (* 0.5 (* im im)))))
   (if (<= im -1.6e+166)
     t_0
     (if (<= im -1.85e+42)
       (* 0.041666666666666664 (pow im 4.0))
       (if (<= im 2.1)
         (* (* (cos re) 0.5) (+ 2.0 (* im im)))
         (if (<= im 1.35e+154) (+ (* 0.5 (exp im)) 0.001953125) t_0))))))

C

double code(double re, double im) {
	double t_0 = cos(re) * (0.5 * (im * im));
	double tmp;
	if (im <= -1.6e+166) {
		tmp = t_0;
	} else if (im <= -1.85e+42) {
		tmp = 0.041666666666666664 * pow(im, 4.0);
	} else if (im <= 2.1) {
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	} else if (im <= 1.35e+154) {
		tmp = (0.5 * exp(im)) + 0.001953125;
	} 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 = cos(re) * (0.5d0 * (im * im))
    if (im <= (-1.6d+166)) then
        tmp = t_0
    else if (im <= (-1.85d+42)) then
        tmp = 0.041666666666666664d0 * (im ** 4.0d0)
    else if (im <= 2.1d0) then
        tmp = (cos(re) * 0.5d0) * (2.0d0 + (im * im))
    else if (im <= 1.35d+154) then
        tmp = (0.5d0 * exp(im)) + 0.001953125d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.cos(re) * (0.5 * (im * im));
	double tmp;
	if (im <= -1.6e+166) {
		tmp = t_0;
	} else if (im <= -1.85e+42) {
		tmp = 0.041666666666666664 * Math.pow(im, 4.0);
	} else if (im <= 2.1) {
		tmp = (Math.cos(re) * 0.5) * (2.0 + (im * im));
	} else if (im <= 1.35e+154) {
		tmp = (0.5 * Math.exp(im)) + 0.001953125;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.cos(re) * (0.5 * (im * im))
	tmp = 0
	if im <= -1.6e+166:
		tmp = t_0
	elif im <= -1.85e+42:
		tmp = 0.041666666666666664 * math.pow(im, 4.0)
	elif im <= 2.1:
		tmp = (math.cos(re) * 0.5) * (2.0 + (im * im))
	elif im <= 1.35e+154:
		tmp = (0.5 * math.exp(im)) + 0.001953125
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(cos(re) * Float64(0.5 * Float64(im * im)))
	tmp = 0.0
	if (im <= -1.6e+166)
		tmp = t_0;
	elseif (im <= -1.85e+42)
		tmp = Float64(0.041666666666666664 * (im ^ 4.0));
	elseif (im <= 2.1)
		tmp = Float64(Float64(cos(re) * 0.5) * Float64(2.0 + Float64(im * im)));
	elseif (im <= 1.35e+154)
		tmp = Float64(Float64(0.5 * exp(im)) + 0.001953125);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = cos(re) * (0.5 * (im * im));
	tmp = 0.0;
	if (im <= -1.6e+166)
		tmp = t_0;
	elseif (im <= -1.85e+42)
		tmp = 0.041666666666666664 * (im ^ 4.0);
	elseif (im <= 2.1)
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	elseif (im <= 1.35e+154)
		tmp = (0.5 * exp(im)) + 0.001953125;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Cos[re], $MachinePrecision] * N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.6e+166], t$95$0, If[LessEqual[im, -1.85e+42], N[(0.041666666666666664 * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.1], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision] + 0.001953125), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -1.59999999999999984e166 or 1.35000000000000003e154 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

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

      1. unpow2 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around inf 100.0%

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

      1. unpow2 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*r* 100.0%

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

   7. Simplified 100.0%

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

   if -1.59999999999999984e166 < im < -1.84999999999999998e42

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 75.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]
   3. Step-by-step derivation

      1. unpow2 75.7%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(\color{blue}{im \cdot im} + 0.08333333333333333 \cdot {im}^{4}\right)\right) \]

      2. *-commutative 75.7%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(im \cdot im + \color{blue}{{im}^{4} \cdot 0.08333333333333333}\right)\right) \]

   4. Simplified 75.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left(im \cdot im + {im}^{4} \cdot 0.08333333333333333\right)\right)} \]

   5. Taylor expanded in im around inf 75.7%

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

   6. Taylor expanded in re around 0 62.7%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot {im}^{4}} \]
   7. Step-by-step derivation

      1. *-commutative 62.7%

         \[\leadsto \color{blue}{{im}^{4} \cdot 0.041666666666666664} \]

   8. Simplified 62.7%

      \[\leadsto \color{blue}{{im}^{4} \cdot 0.041666666666666664} \]

   if -1.84999999999999998e42 < im < 2.10000000000000009

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 94.1%

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

      1. unpow2 94.1%

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

   4. Simplified 94.1%

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

   if 2.10000000000000009 < im < 1.35000000000000003e154

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-def 100.0%

         \[\leadsto \cos re \cdot \color{blue}{\mathsf{fma}\left(0.5, e^{im}, e^{-im} \cdot 0.5\right)} \]

      9. exp-neg 100.0%

         \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

      10. associate-*l/ 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

      11. metadata-eval 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{\color{blue}{0.5}}{e^{im}}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

   5. Applied egg-rr 100.0%

      \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{0.001953125}\right) \]

   6. Taylor expanded in re around 0 93.2%

      \[\leadsto \color{blue}{0.001953125 + 0.5 \cdot e^{im}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.6 \cdot 10^{+166}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq -1.85 \cdot 10^{+42}:\\ \;\;\;\;0.041666666666666664 \cdot {im}^{4}\\ \mathbf{elif}\;im \leq 2.1:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot e^{im} + 0.001953125\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 12: 89.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{if}\;im \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -750:\\ \;\;\;\;{im}^{4} \cdot \left(0.041666666666666664 + -0.020833333333333332 \cdot \left(re \cdot re\right)\right)\\ \mathbf{elif}\;im \leq 1.82:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot e^{im} + 0.001953125\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (cos re) (* 0.5 (* im im)))))
   (if (<= im -1.32e+154)
     t_0
     (if (<= im -750.0)
       (*
        (pow im 4.0)
        (+ 0.041666666666666664 (* -0.020833333333333332 (* re re))))
       (if (<= im 1.82)
         (* (* (cos re) 0.5) (+ 2.0 (* im im)))
         (if (<= im 1.35e+154) (+ (* 0.5 (exp im)) 0.001953125) t_0))))))

C

double code(double re, double im) {
	double t_0 = cos(re) * (0.5 * (im * im));
	double tmp;
	if (im <= -1.32e+154) {
		tmp = t_0;
	} else if (im <= -750.0) {
		tmp = pow(im, 4.0) * (0.041666666666666664 + (-0.020833333333333332 * (re * re)));
	} else if (im <= 1.82) {
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	} else if (im <= 1.35e+154) {
		tmp = (0.5 * exp(im)) + 0.001953125;
	} 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 = cos(re) * (0.5d0 * (im * im))
    if (im <= (-1.32d+154)) then
        tmp = t_0
    else if (im <= (-750.0d0)) then
        tmp = (im ** 4.0d0) * (0.041666666666666664d0 + ((-0.020833333333333332d0) * (re * re)))
    else if (im <= 1.82d0) then
        tmp = (cos(re) * 0.5d0) * (2.0d0 + (im * im))
    else if (im <= 1.35d+154) then
        tmp = (0.5d0 * exp(im)) + 0.001953125d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.cos(re) * (0.5 * (im * im));
	double tmp;
	if (im <= -1.32e+154) {
		tmp = t_0;
	} else if (im <= -750.0) {
		tmp = Math.pow(im, 4.0) * (0.041666666666666664 + (-0.020833333333333332 * (re * re)));
	} else if (im <= 1.82) {
		tmp = (Math.cos(re) * 0.5) * (2.0 + (im * im));
	} else if (im <= 1.35e+154) {
		tmp = (0.5 * Math.exp(im)) + 0.001953125;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.cos(re) * (0.5 * (im * im))
	tmp = 0
	if im <= -1.32e+154:
		tmp = t_0
	elif im <= -750.0:
		tmp = math.pow(im, 4.0) * (0.041666666666666664 + (-0.020833333333333332 * (re * re)))
	elif im <= 1.82:
		tmp = (math.cos(re) * 0.5) * (2.0 + (im * im))
	elif im <= 1.35e+154:
		tmp = (0.5 * math.exp(im)) + 0.001953125
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(cos(re) * Float64(0.5 * Float64(im * im)))
	tmp = 0.0
	if (im <= -1.32e+154)
		tmp = t_0;
	elseif (im <= -750.0)
		tmp = Float64((im ^ 4.0) * Float64(0.041666666666666664 + Float64(-0.020833333333333332 * Float64(re * re))));
	elseif (im <= 1.82)
		tmp = Float64(Float64(cos(re) * 0.5) * Float64(2.0 + Float64(im * im)));
	elseif (im <= 1.35e+154)
		tmp = Float64(Float64(0.5 * exp(im)) + 0.001953125);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = cos(re) * (0.5 * (im * im));
	tmp = 0.0;
	if (im <= -1.32e+154)
		tmp = t_0;
	elseif (im <= -750.0)
		tmp = (im ^ 4.0) * (0.041666666666666664 + (-0.020833333333333332 * (re * re)));
	elseif (im <= 1.82)
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	elseif (im <= 1.35e+154)
		tmp = (0.5 * exp(im)) + 0.001953125;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Cos[re], $MachinePrecision] * N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.32e+154], t$95$0, If[LessEqual[im, -750.0], N[(N[Power[im, 4.0], $MachinePrecision] * N[(0.041666666666666664 + N[(-0.020833333333333332 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.82], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision] + 0.001953125), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -1.31999999999999998e154 or 1.35000000000000003e154 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

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

      1. unpow2 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around inf 100.0%

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

      1. unpow2 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*r* 100.0%

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

   7. Simplified 100.0%

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

   if -1.31999999999999998e154 < im < -750

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 55.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]
   3. Step-by-step derivation

      1. unpow2 55.7%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(\color{blue}{im \cdot im} + 0.08333333333333333 \cdot {im}^{4}\right)\right) \]

      2. *-commutative 55.7%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(im \cdot im + \color{blue}{{im}^{4} \cdot 0.08333333333333333}\right)\right) \]

   4. Simplified 55.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left(im \cdot im + {im}^{4} \cdot 0.08333333333333333\right)\right)} \]

   5. Taylor expanded in im around inf 55.7%

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

   6. Taylor expanded in re around 0 5.3%

      \[\leadsto \color{blue}{-0.020833333333333332 \cdot \left({re}^{2} \cdot {im}^{4}\right) + 0.041666666666666664 \cdot {im}^{4}} \]
   7. Step-by-step derivation

      1. +-commutative 5.3%

         \[\leadsto \color{blue}{0.041666666666666664 \cdot {im}^{4} + -0.020833333333333332 \cdot \left({re}^{2} \cdot {im}^{4}\right)} \]

      2. associate-*r* 5.3%

         \[\leadsto 0.041666666666666664 \cdot {im}^{4} + \color{blue}{\left(-0.020833333333333332 \cdot {re}^{2}\right) \cdot {im}^{4}} \]

      3. distribute-rgt-out 52.0%

         \[\leadsto \color{blue}{{im}^{4} \cdot \left(0.041666666666666664 + -0.020833333333333332 \cdot {re}^{2}\right)} \]

      4. unpow2 52.0%

         \[\leadsto {im}^{4} \cdot \left(0.041666666666666664 + -0.020833333333333332 \cdot \color{blue}{\left(re \cdot re\right)}\right) \]

   8. Simplified 52.0%

      \[\leadsto \color{blue}{{im}^{4} \cdot \left(0.041666666666666664 + -0.020833333333333332 \cdot \left(re \cdot re\right)\right)} \]

   if -750 < im < 1.82000000000000006

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

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

      1. unpow2 100.0%

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

   4. Simplified 100.0%

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

   if 1.82000000000000006 < im < 1.35000000000000003e154

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-def 100.0%

         \[\leadsto \cos re \cdot \color{blue}{\mathsf{fma}\left(0.5, e^{im}, e^{-im} \cdot 0.5\right)} \]

      9. exp-neg 100.0%

         \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

      10. associate-*l/ 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

      11. metadata-eval 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{\color{blue}{0.5}}{e^{im}}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

   5. Applied egg-rr 100.0%

      \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{0.001953125}\right) \]

   6. Taylor expanded in re around 0 93.2%

      \[\leadsto \color{blue}{0.001953125 + 0.5 \cdot e^{im}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq -750:\\ \;\;\;\;{im}^{4} \cdot \left(0.041666666666666664 + -0.020833333333333332 \cdot \left(re \cdot re\right)\right)\\ \mathbf{elif}\;im \leq 1.82:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot e^{im} + 0.001953125\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 13: 81.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1.46 \cdot 10^{+41}:\\ \;\;\;\;0.041666666666666664 \cdot {im}^{4}\\ \mathbf{elif}\;im \leq 2:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot e^{im} + 0.001953125\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -1.46e+41)
   (* 0.041666666666666664 (pow im 4.0))
   (if (<= im 2.0) (cos re) (+ (* 0.5 (exp im)) 0.001953125))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -1.46e+41) {
		tmp = 0.041666666666666664 * pow(im, 4.0);
	} else if (im <= 2.0) {
		tmp = cos(re);
	} else {
		tmp = (0.5 * exp(im)) + 0.001953125;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= (-1.46d+41)) then
        tmp = 0.041666666666666664d0 * (im ** 4.0d0)
    else if (im <= 2.0d0) then
        tmp = cos(re)
    else
        tmp = (0.5d0 * exp(im)) + 0.001953125d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -1.46e+41) {
		tmp = 0.041666666666666664 * Math.pow(im, 4.0);
	} else if (im <= 2.0) {
		tmp = Math.cos(re);
	} else {
		tmp = (0.5 * Math.exp(im)) + 0.001953125;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -1.46e+41:
		tmp = 0.041666666666666664 * math.pow(im, 4.0)
	elif im <= 2.0:
		tmp = math.cos(re)
	else:
		tmp = (0.5 * math.exp(im)) + 0.001953125
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -1.46e+41)
		tmp = Float64(0.041666666666666664 * (im ^ 4.0));
	elseif (im <= 2.0)
		tmp = cos(re);
	else
		tmp = Float64(Float64(0.5 * exp(im)) + 0.001953125);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -1.46e+41)
		tmp = 0.041666666666666664 * (im ^ 4.0);
	elseif (im <= 2.0)
		tmp = cos(re);
	else
		tmp = (0.5 * exp(im)) + 0.001953125;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -1.46e+41], N[(0.041666666666666664 * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.0], N[Cos[re], $MachinePrecision], N[(N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision] + 0.001953125), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.4600000000000001e41

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 87.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]
   3. Step-by-step derivation

      1. unpow2 87.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(\color{blue}{im \cdot im} + 0.08333333333333333 \cdot {im}^{4}\right)\right) \]

      2. *-commutative 87.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(im \cdot im + \color{blue}{{im}^{4} \cdot 0.08333333333333333}\right)\right) \]

   4. Simplified 87.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left(im \cdot im + {im}^{4} \cdot 0.08333333333333333\right)\right)} \]

   5. Taylor expanded in im around inf 87.0%

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

   6. Taylor expanded in re around 0 68.4%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot {im}^{4}} \]
   7. Step-by-step derivation

      1. *-commutative 68.4%

         \[\leadsto \color{blue}{{im}^{4} \cdot 0.041666666666666664} \]

   8. Simplified 68.4%

      \[\leadsto \color{blue}{{im}^{4} \cdot 0.041666666666666664} \]

   if -1.4600000000000001e41 < im < 2

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 94.1%

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

      1. unpow2 94.1%

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

   4. Simplified 94.1%

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

   5. Taylor expanded in im around 0 93.8%

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

   if 2 < im

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-def 100.0%

         \[\leadsto \cos re \cdot \color{blue}{\mathsf{fma}\left(0.5, e^{im}, e^{-im} \cdot 0.5\right)} \]

      9. exp-neg 100.0%

         \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

      10. associate-*l/ 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

      11. metadata-eval 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{\color{blue}{0.5}}{e^{im}}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

   5. Applied egg-rr 100.0%

      \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{0.001953125}\right) \]

   6. Taylor expanded in re around 0 81.9%

      \[\leadsto \color{blue}{0.001953125 + 0.5 \cdot e^{im}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.46 \cdot 10^{+41}:\\ \;\;\;\;0.041666666666666664 \cdot {im}^{4}\\ \mathbf{elif}\;im \leq 2:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot e^{im} + 0.001953125\\ \end{array} \]

Alternative 14: 77.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1.7 \cdot 10^{+41} \lor \neg \left(im \leq 12.6\right):\\ \;\;\;\;0.041666666666666664 \cdot {im}^{4}\\ \mathbf{else}:\\ \;\;\;\;\cos re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -1.7e+41) (not (<= im 12.6)))
   (* 0.041666666666666664 (pow im 4.0))
   (cos re)))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -1.7e+41) || !(im <= 12.6)) {
		tmp = 0.041666666666666664 * pow(im, 4.0);
	} else {
		tmp = cos(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= (-1.7d+41)) .or. (.not. (im <= 12.6d0))) then
        tmp = 0.041666666666666664d0 * (im ** 4.0d0)
    else
        tmp = cos(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -1.7e+41) || !(im <= 12.6)) {
		tmp = 0.041666666666666664 * Math.pow(im, 4.0);
	} else {
		tmp = Math.cos(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -1.7e+41) or not (im <= 12.6):
		tmp = 0.041666666666666664 * math.pow(im, 4.0)
	else:
		tmp = math.cos(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -1.7e+41) || !(im <= 12.6))
		tmp = Float64(0.041666666666666664 * (im ^ 4.0));
	else
		tmp = cos(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -1.7e+41) || ~((im <= 12.6)))
		tmp = 0.041666666666666664 * (im ^ 4.0);
	else
		tmp = cos(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -1.7e+41], N[Not[LessEqual[im, 12.6]], $MachinePrecision]], N[(0.041666666666666664 * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision], N[Cos[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -1.69999999999999999e41 or 12.5999999999999996 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 76.2%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]
   3. Step-by-step derivation

      1. unpow2 76.2%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(\color{blue}{im \cdot im} + 0.08333333333333333 \cdot {im}^{4}\right)\right) \]

      2. *-commutative 76.2%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(im \cdot im + \color{blue}{{im}^{4} \cdot 0.08333333333333333}\right)\right) \]

   4. Simplified 76.2%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left(im \cdot im + {im}^{4} \cdot 0.08333333333333333\right)\right)} \]

   5. Taylor expanded in im around inf 76.2%

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

   6. Taylor expanded in re around 0 59.5%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot {im}^{4}} \]
   7. Step-by-step derivation

      1. *-commutative 59.5%

         \[\leadsto \color{blue}{{im}^{4} \cdot 0.041666666666666664} \]

   8. Simplified 59.5%

      \[\leadsto \color{blue}{{im}^{4} \cdot 0.041666666666666664} \]

   if -1.69999999999999999e41 < im < 12.5999999999999996

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 94.1%

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

      1. unpow2 94.1%

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

   4. Simplified 94.1%

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

   5. Taylor expanded in im around 0 93.8%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.7 \cdot 10^{+41} \lor \neg \left(im \leq 12.6\right):\\ \;\;\;\;0.041666666666666664 \cdot {im}^{4}\\ \mathbf{else}:\\ \;\;\;\;\cos re\\ \end{array} \]

Alternative 15: 73.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -510 \lor \neg \left(im \leq 2.2 \cdot 10^{+14}\right):\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -510.0) (not (<= im 2.2e+14)))
   (* (* im im) (+ 0.5 (* (* re re) -0.25)))
   (cos re)))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -510.0) || !(im <= 2.2e+14)) {
		tmp = (im * im) * (0.5 + ((re * re) * -0.25));
	} else {
		tmp = cos(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 <= (-510.0d0)) .or. (.not. (im <= 2.2d+14))) then
        tmp = (im * im) * (0.5d0 + ((re * re) * (-0.25d0)))
    else
        tmp = cos(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -510.0) || !(im <= 2.2e+14)) {
		tmp = (im * im) * (0.5 + ((re * re) * -0.25));
	} else {
		tmp = Math.cos(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -510.0) or not (im <= 2.2e+14):
		tmp = (im * im) * (0.5 + ((re * re) * -0.25))
	else:
		tmp = math.cos(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -510.0) || !(im <= 2.2e+14))
		tmp = Float64(Float64(im * im) * Float64(0.5 + Float64(Float64(re * re) * -0.25)));
	else
		tmp = cos(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -510.0) || ~((im <= 2.2e+14)))
		tmp = (im * im) * (0.5 + ((re * re) * -0.25));
	else
		tmp = cos(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -510.0], N[Not[LessEqual[im, 2.2e+14]], $MachinePrecision]], N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(N[(re * re), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Cos[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -510 or 2.2e14 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 48.8%

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

      1. unpow2 48.8%

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

   4. Simplified 48.8%

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

   5. Taylor expanded in im around inf 48.8%

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

      1. unpow2 48.8%

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

      2. associate-*r* 48.8%

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

      3. *-commutative 48.8%

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

      4. associate-*r* 48.8%

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

   7. Simplified 48.8%

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

   8. Taylor expanded in re around 0 6.0%

      \[\leadsto \color{blue}{0.5 \cdot {im}^{2} + -0.25 \cdot \left({re}^{2} \cdot {im}^{2}\right)} \]
   9. Step-by-step derivation

      1. +-commutative 6.0%

         \[\leadsto \color{blue}{-0.25 \cdot \left({re}^{2} \cdot {im}^{2}\right) + 0.5 \cdot {im}^{2}} \]

      2. unpow2 6.0%

         \[\leadsto -0.25 \cdot \left({re}^{2} \cdot \color{blue}{\left(im \cdot im\right)}\right) + 0.5 \cdot {im}^{2} \]

      3. associate-*r* 6.0%

         \[\leadsto \color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot \left(im \cdot im\right)} + 0.5 \cdot {im}^{2} \]

      4. unpow2 6.0%

         \[\leadsto \left(-0.25 \cdot {re}^{2}\right) \cdot \left(im \cdot im\right) + 0.5 \cdot \color{blue}{\left(im \cdot im\right)} \]

      5. distribute-rgt-out 38.8%

         \[\leadsto \color{blue}{\left(im \cdot im\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right)} \]

      6. unpow2 38.8%

         \[\leadsto \left(im \cdot im\right) \cdot \left(-0.25 \cdot \color{blue}{\left(re \cdot re\right)} + 0.5\right) \]

   10. Simplified 38.8%

       \[\leadsto \color{blue}{\left(im \cdot im\right) \cdot \left(-0.25 \cdot \left(re \cdot re\right) + 0.5\right)} \]

   if -510 < im < 2.2e14

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 97.7%

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

      1. unpow2 97.7%

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

   4. Simplified 97.7%

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

   5. Taylor expanded in im around 0 97.5%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -510 \lor \neg \left(im \leq 2.2 \cdot 10^{+14}\right):\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re\\ \end{array} \]

Alternative 16: 47.3% accurate, 23.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -5.7 \cdot 10^{+200}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 + im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -5.7e+200)
   (* (* im im) (+ 0.5 (* (* re re) -0.25)))
   (* 0.5 (+ 2.0 (* im im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -5.7e+200) {
		tmp = (im * im) * (0.5 + ((re * re) * -0.25));
	} else {
		tmp = 0.5 * (2.0 + (im * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-5.7d+200)) then
        tmp = (im * im) * (0.5d0 + ((re * re) * (-0.25d0)))
    else
        tmp = 0.5d0 * (2.0d0 + (im * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -5.7e+200) {
		tmp = (im * im) * (0.5 + ((re * re) * -0.25));
	} else {
		tmp = 0.5 * (2.0 + (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -5.7e+200:
		tmp = (im * im) * (0.5 + ((re * re) * -0.25))
	else:
		tmp = 0.5 * (2.0 + (im * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -5.7e+200)
		tmp = Float64(Float64(im * im) * Float64(0.5 + Float64(Float64(re * re) * -0.25)));
	else
		tmp = Float64(0.5 * Float64(2.0 + Float64(im * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -5.7e+200)
		tmp = (im * im) * (0.5 + ((re * re) * -0.25));
	else
		tmp = 0.5 * (2.0 + (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -5.7e+200], N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(N[(re * re), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -5.70000000000000007e200

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 77.4%

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

      1. unpow2 77.4%

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

   4. Simplified 77.4%

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

   5. Taylor expanded in im around inf 31.6%

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

      1. unpow2 31.6%

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

      2. associate-*r* 31.6%

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

      3. *-commutative 31.6%

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

      4. associate-*r* 31.6%

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

   7. Simplified 31.6%

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

   8. Taylor expanded in re around 0 9.7%

      \[\leadsto \color{blue}{0.5 \cdot {im}^{2} + -0.25 \cdot \left({re}^{2} \cdot {im}^{2}\right)} \]
   9. Step-by-step derivation

      1. +-commutative 9.7%

         \[\leadsto \color{blue}{-0.25 \cdot \left({re}^{2} \cdot {im}^{2}\right) + 0.5 \cdot {im}^{2}} \]

      2. unpow2 9.7%

         \[\leadsto -0.25 \cdot \left({re}^{2} \cdot \color{blue}{\left(im \cdot im\right)}\right) + 0.5 \cdot {im}^{2} \]

      3. associate-*r* 9.7%

         \[\leadsto \color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot \left(im \cdot im\right)} + 0.5 \cdot {im}^{2} \]

      4. unpow2 9.7%

         \[\leadsto \left(-0.25 \cdot {re}^{2}\right) \cdot \left(im \cdot im\right) + 0.5 \cdot \color{blue}{\left(im \cdot im\right)} \]

      5. distribute-rgt-out 33.5%

         \[\leadsto \color{blue}{\left(im \cdot im\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right)} \]

      6. unpow2 33.5%

         \[\leadsto \left(im \cdot im\right) \cdot \left(-0.25 \cdot \color{blue}{\left(re \cdot re\right)} + 0.5\right) \]

   10. Simplified 33.5%

       \[\leadsto \color{blue}{\left(im \cdot im\right) \cdot \left(-0.25 \cdot \left(re \cdot re\right) + 0.5\right)} \]

   if -5.70000000000000007e200 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 85.2%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]
   3. Step-by-step derivation

      1. unpow2 85.2%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(\color{blue}{im \cdot im} + 0.08333333333333333 \cdot {im}^{4}\right)\right) \]

      2. *-commutative 85.2%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(im \cdot im + \color{blue}{{im}^{4} \cdot 0.08333333333333333}\right)\right) \]

   4. Simplified 85.2%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left(im \cdot im + {im}^{4} \cdot 0.08333333333333333\right)\right)} \]

   5. Taylor expanded in re around 0 64.2%

      \[\leadsto \color{blue}{0.5 \cdot \left(2 + \left(0.08333333333333333 \cdot {im}^{4} + {im}^{2}\right)\right)} \]

   6. Taylor expanded in im around 0 52.4%

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

      1. unpow2 52.4%

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

   8. Simplified 52.4%

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

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5.7 \cdot 10^{+200}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 + im \cdot im\right)\\ \end{array} \]

Alternative 17: 46.3% accurate, 33.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1.42 \lor \neg \left(im \leq 1.4\right):\\ \;\;\;\;0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -1.42) (not (<= im 1.4))) (* 0.5 (* im im)) 1.0))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -1.42) || !(im <= 1.4)) {
		tmp = 0.5 * (im * im);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= (-1.42d0)) .or. (.not. (im <= 1.4d0))) then
        tmp = 0.5d0 * (im * im)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -1.42) || !(im <= 1.4)) {
		tmp = 0.5 * (im * im);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -1.42) or not (im <= 1.4):
		tmp = 0.5 * (im * im)
	else:
		tmp = 1.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -1.42) || !(im <= 1.4))
		tmp = Float64(0.5 * Float64(im * im));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -1.42) || ~((im <= 1.4)))
		tmp = 0.5 * (im * im);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -1.42], N[Not[LessEqual[im, 1.4]], $MachinePrecision]], N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if im < -1.4199999999999999 or 1.3999999999999999 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 71.9%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]
   3. Step-by-step derivation

      1. unpow2 71.9%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(\color{blue}{im \cdot im} + 0.08333333333333333 \cdot {im}^{4}\right)\right) \]

      2. *-commutative 71.9%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(im \cdot im + \color{blue}{{im}^{4} \cdot 0.08333333333333333}\right)\right) \]

   4. Simplified 71.9%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left(im \cdot im + {im}^{4} \cdot 0.08333333333333333\right)\right)} \]

   5. Taylor expanded in re around 0 56.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(2 + \left(0.08333333333333333 \cdot {im}^{4} + {im}^{2}\right)\right)} \]

   6. Taylor expanded in im around 0 34.8%

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

      1. unpow2 34.8%

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

   8. Simplified 34.8%

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

   9. Taylor expanded in im around inf 34.8%

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

       1. unpow2 34.8%

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

   11. Simplified 34.8%

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

   if -1.4199999999999999 < im < 1.3999999999999999

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]
   3. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(\color{blue}{im \cdot im} + 0.08333333333333333 \cdot {im}^{4}\right)\right) \]

      2. *-commutative 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(im \cdot im + \color{blue}{{im}^{4} \cdot 0.08333333333333333}\right)\right) \]

   4. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left(im \cdot im + {im}^{4} \cdot 0.08333333333333333\right)\right)} \]

   5. Taylor expanded in re around 0 64.6%

      \[\leadsto \color{blue}{0.5 \cdot \left(2 + \left(0.08333333333333333 \cdot {im}^{4} + {im}^{2}\right)\right)} \]

   6. Taylor expanded in im around 0 64.6%

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

      1. unpow2 64.6%

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

   8. Simplified 64.6%

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

   9. Taylor expanded in im around 0 64.4%

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

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.42 \lor \neg \left(im \leq 1.4\right):\\ \;\;\;\;0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 18: 46.4% accurate, 44.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(2 + im \cdot im\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* 0.5 (+ 2.0 (* im im))))

C

double code(double re, double im) {
	return 0.5 * (2.0 + (im * im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * (2.0d0 + (im * im))
end function

Java

public static double code(double re, double im) {
	return 0.5 * (2.0 + (im * im));
}

Python

def code(re, im):
	return 0.5 * (2.0 + (im * im))

Julia

function code(re, im)
	return Float64(0.5 * Float64(2.0 + Float64(im * im)))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * (2.0 + (im * im));
end

Wolfram

code[re_, im_] := N[(0.5 * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in im around 0 85.3%

   \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]
3. Step-by-step derivation

   1. unpow2 85.3%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(\color{blue}{im \cdot im} + 0.08333333333333333 \cdot {im}^{4}\right)\right) \]

   2. *-commutative 85.3%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(im \cdot im + \color{blue}{{im}^{4} \cdot 0.08333333333333333}\right)\right) \]

4. Simplified 85.3%

   \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left(im \cdot im + {im}^{4} \cdot 0.08333333333333333\right)\right)} \]

5. Taylor expanded in re around 0 60.1%

   \[\leadsto \color{blue}{0.5 \cdot \left(2 + \left(0.08333333333333333 \cdot {im}^{4} + {im}^{2}\right)\right)} \]

6. Taylor expanded in im around 0 49.0%

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

   1. unpow2 49.0%

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

8. Simplified 49.0%

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

9. Final simplification 49.0%

   \[\leadsto 0.5 \cdot \left(2 + im \cdot im\right) \]

Alternative 19: 8.6% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ 0.501953125 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.501953125)

C

double code(double re, double im) {
	return 0.501953125;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.501953125d0
end function

Java

public static double code(double re, double im) {
	return 0.501953125;
}

Python

def code(re, im):
	return 0.501953125

Julia

function code(re, im)
	return 0.501953125
end

MATLAB

function tmp = code(re, im)
	tmp = 0.501953125;
end

Wolfram

code[re_, im_] := 0.501953125

Derivation

1. Initial program 100.0%

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

   1. *-commutative 100.0%

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

   2. associate-*l* 100.0%

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

   3. +-commutative 100.0%

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

   4. distribute-lft-in 100.0%

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

   5. distribute-lft-in 100.0%

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

   6. distribute-rgt-in 100.0%

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

   7. *-commutative 100.0%

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

   8. fma-def 100.0%

      \[\leadsto \cos re \cdot \color{blue}{\mathsf{fma}\left(0.5, e^{im}, e^{-im} \cdot 0.5\right)} \]

   9. exp-neg 100.0%

      \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

   10. associate-*l/ 100.0%

       \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

   11. metadata-eval 100.0%

       \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{\color{blue}{0.5}}{e^{im}}\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]

5. Applied egg-rr 42.0%

   \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{0.001953125}\right) \]

6. Taylor expanded in re around 0 34.9%

   \[\leadsto \color{blue}{0.001953125 + 0.5 \cdot e^{im}} \]

7. Taylor expanded in im around 0 9.2%

   \[\leadsto \color{blue}{0.501953125} \]

8. Final simplification 9.2%

   \[\leadsto 0.501953125 \]

Alternative 20: 27.5% accurate, 308.0× 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%

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

2. Taylor expanded in im around 0 85.3%

   \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]
3. Step-by-step derivation

   1. unpow2 85.3%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(\color{blue}{im \cdot im} + 0.08333333333333333 \cdot {im}^{4}\right)\right) \]

   2. *-commutative 85.3%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(2 + \left(im \cdot im + \color{blue}{{im}^{4} \cdot 0.08333333333333333}\right)\right) \]

4. Simplified 85.3%

   \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left(im \cdot im + {im}^{4} \cdot 0.08333333333333333\right)\right)} \]

5. Taylor expanded in re around 0 60.1%

   \[\leadsto \color{blue}{0.5 \cdot \left(2 + \left(0.08333333333333333 \cdot {im}^{4} + {im}^{2}\right)\right)} \]

6. Taylor expanded in im around 0 49.0%

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

   1. unpow2 49.0%

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

8. Simplified 49.0%

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

9. Taylor expanded in im around 0 32.1%

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

10. Final simplification 32.1%

    \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023194 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))