math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 7.7s

Alternatives: 15

Speedup: 1.0×

• 49.9% 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: 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 3: 75.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[re_, im_] := N[(N[Cos[re], $MachinePrecision] * N[(0.5 * N[Exp[im], $MachinePrecision] + 0.5), $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. Taylor expanded in im around 0 77.5%

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

5. Final simplification 77.5%

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

Alternative 4: 87.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \left(\cos re \cdot 0.5\right) \cdot \left(2 + \left(im \cdot im + {im}^{4} \cdot 0.08333333333333333\right)\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (*
  (* (cos re) 0.5)
  (+ 2.0 (+ (* im im) (* (pow im 4.0) 0.08333333333333333)))))

C

double code(double re, double im) {
	return (cos(re) * 0.5) * (2.0 + ((im * im) + (pow(im, 4.0) * 0.08333333333333333)));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (cos(re) * 0.5d0) * (2.0d0 + ((im * im) + ((im ** 4.0d0) * 0.08333333333333333d0)))
end function

Java

public static double code(double re, double im) {
	return (Math.cos(re) * 0.5) * (2.0 + ((im * im) + (Math.pow(im, 4.0) * 0.08333333333333333)));
}

Python

def code(re, im):
	return (math.cos(re) * 0.5) * (2.0 + ((im * im) + (math.pow(im, 4.0) * 0.08333333333333333)))

Julia

function code(re, im)
	return Float64(Float64(cos(re) * 0.5) * Float64(2.0 + Float64(Float64(im * im) + Float64((im ^ 4.0) * 0.08333333333333333))))
end

MATLAB

function tmp = code(re, im)
	tmp = (cos(re) * 0.5) * (2.0 + ((im * im) + ((im ^ 4.0) * 0.08333333333333333)));
end

Wolfram

code[re_, im_] := 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]

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 94.4%

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

   1. unpow2 94.4%

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

   2. *-commutative 94.4%

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

4. Simplified 94.4%

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

5. Final simplification 94.4%

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

Alternative 5: 81.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.7:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{4} \cdot \left(\cos re \cdot 0.041666666666666664\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.7)
   (* (* (cos re) 0.5) (+ 2.0 (* im im)))
   (* (pow im 4.0) (* (cos re) 0.041666666666666664))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.7) {
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	} else {
		tmp = pow(im, 4.0) * (cos(re) * 0.041666666666666664);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 3.7d0) then
        tmp = (cos(re) * 0.5d0) * (2.0d0 + (im * im))
    else
        tmp = (im ** 4.0d0) * (cos(re) * 0.041666666666666664d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3.7) {
		tmp = (Math.cos(re) * 0.5) * (2.0 + (im * im));
	} else {
		tmp = Math.pow(im, 4.0) * (Math.cos(re) * 0.041666666666666664);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3.7:
		tmp = (math.cos(re) * 0.5) * (2.0 + (im * im))
	else:
		tmp = math.pow(im, 4.0) * (math.cos(re) * 0.041666666666666664)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.7)
		tmp = Float64(Float64(cos(re) * 0.5) * Float64(2.0 + Float64(im * im)));
	else
		tmp = Float64((im ^ 4.0) * Float64(cos(re) * 0.041666666666666664));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3.7)
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	else
		tmp = (im ^ 4.0) * (cos(re) * 0.041666666666666664);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.7], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[im, 4.0], $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 3.7000000000000002

   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%

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

      1. unpow2 87.0%

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

   4. Simplified 87.0%

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

   if 3.7000000000000002 < 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 91.8%

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

      1. unpow2 91.8%

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

      2. *-commutative 91.8%

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

   4. Simplified 91.8%

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

   5. Taylor expanded in im around inf 91.8%

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

      1. associate-*r* 91.8%

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

   7. Simplified 91.8%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.7:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{4} \cdot \left(\cos re \cdot 0.041666666666666664\right)\\ \end{array} \]

Alternative 6: 81.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 490:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;{im}^{4} \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.020833333333333332\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 490.0)
   (* (* (cos re) 0.5) (+ 2.0 (* im im)))
   (if (<= im 1.35e+154)
     (*
      (pow im 4.0)
      (+ 0.041666666666666664 (* (* re re) -0.020833333333333332)))
     (* 0.5 (* (cos re) (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 490.0) {
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	} else if (im <= 1.35e+154) {
		tmp = pow(im, 4.0) * (0.041666666666666664 + ((re * re) * -0.020833333333333332));
	} else {
		tmp = 0.5 * (cos(re) * (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 (im <= 490.0d0) then
        tmp = (cos(re) * 0.5d0) * (2.0d0 + (im * im))
    else if (im <= 1.35d+154) then
        tmp = (im ** 4.0d0) * (0.041666666666666664d0 + ((re * re) * (-0.020833333333333332d0)))
    else
        tmp = 0.5d0 * (cos(re) * (im * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 490.0) {
		tmp = (Math.cos(re) * 0.5) * (2.0 + (im * im));
	} else if (im <= 1.35e+154) {
		tmp = Math.pow(im, 4.0) * (0.041666666666666664 + ((re * re) * -0.020833333333333332));
	} else {
		tmp = 0.5 * (Math.cos(re) * (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 490.0:
		tmp = (math.cos(re) * 0.5) * (2.0 + (im * im))
	elif im <= 1.35e+154:
		tmp = math.pow(im, 4.0) * (0.041666666666666664 + ((re * re) * -0.020833333333333332))
	else:
		tmp = 0.5 * (math.cos(re) * (im * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 490.0)
		tmp = Float64(Float64(cos(re) * 0.5) * Float64(2.0 + Float64(im * im)));
	elseif (im <= 1.35e+154)
		tmp = Float64((im ^ 4.0) * Float64(0.041666666666666664 + Float64(Float64(re * re) * -0.020833333333333332)));
	else
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 490.0)
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	elseif (im <= 1.35e+154)
		tmp = (im ^ 4.0) * (0.041666666666666664 + ((re * re) * -0.020833333333333332));
	else
		tmp = 0.5 * (cos(re) * (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 490.0], 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[Power[im, 4.0], $MachinePrecision] * N[(0.041666666666666664 + N[(N[(re * re), $MachinePrecision] * -0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 490

   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%

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

      1. unpow2 87.0%

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

   4. Simplified 87.0%

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

   if 490 < im < 1.35000000000000003e154

   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 72.1%

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

      1. unpow2 72.1%

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

      2. *-commutative 72.1%

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

   4. Simplified 72.1%

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

      1. +-commutative 72.1%

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

      2. metadata-eval 72.1%

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

      3. pow-prod-up 72.1%

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

      4. pow-prod-down 72.1%

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

      5. pow2 72.1%

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

      6. associate-*l* 72.1%

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

      7. fma-def 72.1%

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

   6. Applied egg-rr 72.1%

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

   7. Taylor expanded in im around inf 72.1%

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

      1. *-commutative 72.1%

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

      2. *-commutative 72.1%

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

      3. associate-*l* 72.1%

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

   9. Simplified 72.1%

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

   10. Taylor expanded in re around 0 72.1%

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

       1. *-commutative 72.1%

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

       2. unpow2 72.1%

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

   12. Simplified 72.1%

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

   if 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%

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

   7. Simplified 100.0%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 490:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;{im}^{4} \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.020833333333333332\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 7: 67.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.1 \cdot 10^{+37}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;1 + {im}^{4} \cdot 0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.1e+37)
   (cos re)
   (if (<= im 1.35e+154)
     (+ 1.0 (* (pow im 4.0) 0.041666666666666664))
     (* 0.5 (* (cos re) (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.1e+37) {
		tmp = cos(re);
	} else if (im <= 1.35e+154) {
		tmp = 1.0 + (pow(im, 4.0) * 0.041666666666666664);
	} else {
		tmp = 0.5 * (cos(re) * (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 (im <= 1.1d+37) then
        tmp = cos(re)
    else if (im <= 1.35d+154) then
        tmp = 1.0d0 + ((im ** 4.0d0) * 0.041666666666666664d0)
    else
        tmp = 0.5d0 * (cos(re) * (im * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.1e+37) {
		tmp = Math.cos(re);
	} else if (im <= 1.35e+154) {
		tmp = 1.0 + (Math.pow(im, 4.0) * 0.041666666666666664);
	} else {
		tmp = 0.5 * (Math.cos(re) * (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.1e+37:
		tmp = math.cos(re)
	elif im <= 1.35e+154:
		tmp = 1.0 + (math.pow(im, 4.0) * 0.041666666666666664)
	else:
		tmp = 0.5 * (math.cos(re) * (im * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.1e+37)
		tmp = cos(re);
	elseif (im <= 1.35e+154)
		tmp = Float64(1.0 + Float64((im ^ 4.0) * 0.041666666666666664));
	else
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.1e+37)
		tmp = cos(re);
	elseif (im <= 1.35e+154)
		tmp = 1.0 + ((im ^ 4.0) * 0.041666666666666664);
	else
		tmp = 0.5 * (cos(re) * (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.1e+37], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(1.0 + N[(N[Power[im, 4.0], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.1e37

   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 86.1%

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

      1. unpow2 86.1%

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

   4. Simplified 86.1%

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

   5. Taylor expanded in im around 0 71.1%

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

   if 1.1e37 < im < 1.35000000000000003e154

   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 81.2%

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

      1. unpow2 81.2%

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

      2. *-commutative 81.2%

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

   4. Simplified 81.2%

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

   5. Taylor expanded in re around 0 54.5%

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

      1. distribute-lft-in 54.5%

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

      2. metadata-eval 54.5%

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

      3. unpow2 54.5%

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

      4. +-commutative 54.5%

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

      5. *-commutative 54.5%

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

      6. fma-udef 54.5%

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

      7. *-commutative 54.5%

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

   7. Simplified 54.5%

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

   8. Taylor expanded in im around inf 54.5%

      \[\leadsto 1 + \color{blue}{0.041666666666666664 \cdot {im}^{4}} \]

   if 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%

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

   7. Simplified 100.0%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.1 \cdot 10^{+37}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;1 + {im}^{4} \cdot 0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 8: 68.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 200:\\ \;\;\;\;1 + {im}^{4} \cdot 0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 200.0)
   (+ 1.0 (* (pow im 4.0) 0.041666666666666664))
   (* (* (cos re) 0.5) (+ 2.0 (* im im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 200.0) {
		tmp = 1.0 + (pow(im, 4.0) * 0.041666666666666664);
	} else {
		tmp = (cos(re) * 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 <= 200.0d0) then
        tmp = 1.0d0 + ((im ** 4.0d0) * 0.041666666666666664d0)
    else
        tmp = (cos(re) * 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 <= 200.0) {
		tmp = 1.0 + (Math.pow(im, 4.0) * 0.041666666666666664);
	} else {
		tmp = (Math.cos(re) * 0.5) * (2.0 + (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 200.0:
		tmp = 1.0 + (math.pow(im, 4.0) * 0.041666666666666664)
	else:
		tmp = (math.cos(re) * 0.5) * (2.0 + (im * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 200.0)
		tmp = Float64(1.0 + Float64((im ^ 4.0) * 0.041666666666666664));
	else
		tmp = Float64(Float64(cos(re) * 0.5) * Float64(2.0 + Float64(im * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 200.0)
		tmp = 1.0 + ((im ^ 4.0) * 0.041666666666666664);
	else
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 200.0], N[(1.0 + N[(N[Power[im, 4.0], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 200

   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 92.8%

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

      1. unpow2 92.8%

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

      2. *-commutative 92.8%

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

   4. Simplified 92.8%

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

   5. Taylor expanded in re around 0 69.8%

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

      1. distribute-lft-in 69.8%

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

      2. metadata-eval 69.8%

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

      3. unpow2 69.8%

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

      4. +-commutative 69.8%

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

      5. *-commutative 69.8%

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

      6. fma-udef 69.8%

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

      7. *-commutative 69.8%

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

   7. Simplified 69.8%

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

   8. Taylor expanded in im around inf 69.2%

      \[\leadsto 1 + \color{blue}{0.041666666666666664 \cdot {im}^{4}} \]

   if 200 < 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 95.1%

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

      1. unpow2 95.1%

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

   4. Simplified 95.1%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 200:\\ \;\;\;\;1 + {im}^{4} \cdot 0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \end{array} \]

Alternative 9: 64.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.15 \cdot 10^{+37}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;1 + {im}^{4} \cdot 0.041666666666666664\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.15e+37) (cos re) (+ 1.0 (* (pow im 4.0) 0.041666666666666664))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.15e+37) {
		tmp = cos(re);
	} else {
		tmp = 1.0 + (pow(im, 4.0) * 0.041666666666666664);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.15d+37) then
        tmp = cos(re)
    else
        tmp = 1.0d0 + ((im ** 4.0d0) * 0.041666666666666664d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.15e+37) {
		tmp = Math.cos(re);
	} else {
		tmp = 1.0 + (Math.pow(im, 4.0) * 0.041666666666666664);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.15e+37:
		tmp = math.cos(re)
	else:
		tmp = 1.0 + (math.pow(im, 4.0) * 0.041666666666666664)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.15e+37)
		tmp = cos(re);
	else
		tmp = Float64(1.0 + Float64((im ^ 4.0) * 0.041666666666666664));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.15e+37)
		tmp = cos(re);
	else
		tmp = 1.0 + ((im ^ 4.0) * 0.041666666666666664);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.15e+37], N[Cos[re], $MachinePrecision], N[(1.0 + N[(N[Power[im, 4.0], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.15000000000000001e37

   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 86.1%

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

      1. unpow2 86.1%

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

   4. Simplified 86.1%

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

   5. Taylor expanded in im around 0 71.1%

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

   if 1.15000000000000001e37 < 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 95.0%

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

      1. unpow2 95.0%

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

      2. *-commutative 95.0%

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

   4. Simplified 95.0%

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

   5. Taylor expanded in re around 0 70.0%

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

      1. distribute-lft-in 70.0%

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

      2. metadata-eval 70.0%

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

      3. unpow2 70.0%

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

      4. +-commutative 70.0%

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

      5. *-commutative 70.0%

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

      6. fma-udef 70.0%

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

      7. *-commutative 70.0%

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

   7. Simplified 70.0%

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

   8. Taylor expanded in im around inf 70.0%

      \[\leadsto 1 + \color{blue}{0.041666666666666664 \cdot {im}^{4}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.15 \cdot 10^{+37}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;1 + {im}^{4} \cdot 0.041666666666666664\\ \end{array} \]

Alternative 10: 64.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.1 \cdot 10^{+37}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;{im}^{4} \cdot 0.041666666666666664\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.1e+37) (cos re) (* (pow im 4.0) 0.041666666666666664)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.1e+37) {
		tmp = cos(re);
	} else {
		tmp = pow(im, 4.0) * 0.041666666666666664;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.1d+37) then
        tmp = cos(re)
    else
        tmp = (im ** 4.0d0) * 0.041666666666666664d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.1e+37) {
		tmp = Math.cos(re);
	} else {
		tmp = Math.pow(im, 4.0) * 0.041666666666666664;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.1e+37:
		tmp = math.cos(re)
	else:
		tmp = math.pow(im, 4.0) * 0.041666666666666664
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.1e+37)
		tmp = cos(re);
	else
		tmp = Float64((im ^ 4.0) * 0.041666666666666664);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.1e+37)
		tmp = cos(re);
	else
		tmp = (im ^ 4.0) * 0.041666666666666664;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.1e+37], N[Cos[re], $MachinePrecision], N[(N[Power[im, 4.0], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.1e37

   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 86.1%

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

      1. unpow2 86.1%

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

   4. Simplified 86.1%

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

   5. Taylor expanded in im around 0 71.1%

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

   if 1.1e37 < 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 95.0%

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

      1. unpow2 95.0%

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

      2. *-commutative 95.0%

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

   4. Simplified 95.0%

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

      1. +-commutative 95.0%

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

      2. metadata-eval 95.0%

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

      3. pow-prod-up 95.0%

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

      4. pow-prod-down 95.0%

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

      5. pow2 95.0%

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

      6. associate-*l* 95.0%

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

      7. fma-def 95.0%

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

   6. Applied egg-rr 95.0%

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

   7. Taylor expanded in im around inf 95.0%

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

      1. *-commutative 95.0%

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

      2. *-commutative 95.0%

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

      3. associate-*l* 95.0%

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

   9. Simplified 95.0%

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

   10. Taylor expanded in re around 0 70.0%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.1 \cdot 10^{+37}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;{im}^{4} \cdot 0.041666666666666664\\ \end{array} \]

Alternative 11: 62.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 420:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 5 \cdot 10^{+172}:\\ \;\;\;\;0.5 \cdot \left(\left(im \cdot im\right) \cdot \left(1 + re \cdot \left(re \cdot -0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 420.0)
   (cos re)
   (if (<= im 5e+172)
     (* 0.5 (* (* im im) (+ 1.0 (* re (* re -0.5)))))
     (+ 1.0 (* 0.5 (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 420.0) {
		tmp = cos(re);
	} else if (im <= 5e+172) {
		tmp = 0.5 * ((im * im) * (1.0 + (re * (re * -0.5))));
	} else {
		tmp = 1.0 + (0.5 * (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 (im <= 420.0d0) then
        tmp = cos(re)
    else if (im <= 5d+172) then
        tmp = 0.5d0 * ((im * im) * (1.0d0 + (re * (re * (-0.5d0)))))
    else
        tmp = 1.0d0 + (0.5d0 * (im * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 420.0) {
		tmp = Math.cos(re);
	} else if (im <= 5e+172) {
		tmp = 0.5 * ((im * im) * (1.0 + (re * (re * -0.5))));
	} else {
		tmp = 1.0 + (0.5 * (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 420.0:
		tmp = math.cos(re)
	elif im <= 5e+172:
		tmp = 0.5 * ((im * im) * (1.0 + (re * (re * -0.5))))
	else:
		tmp = 1.0 + (0.5 * (im * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 420.0)
		tmp = cos(re);
	elseif (im <= 5e+172)
		tmp = Float64(0.5 * Float64(Float64(im * im) * Float64(1.0 + Float64(re * Float64(re * -0.5)))));
	else
		tmp = Float64(1.0 + Float64(0.5 * Float64(im * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 420.0)
		tmp = cos(re);
	elseif (im <= 5e+172)
		tmp = 0.5 * ((im * im) * (1.0 + (re * (re * -0.5))));
	else
		tmp = 1.0 + (0.5 * (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 420.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 5e+172], N[(0.5 * N[(N[(im * im), $MachinePrecision] * N[(1.0 + N[(re * N[(re * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 420

   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%

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

      1. unpow2 87.0%

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

   4. Simplified 87.0%

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

   5. Taylor expanded in im around 0 71.8%

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

   if 420 < im < 5.0000000000000001e172

   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 11.1%

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

      1. unpow2 11.1%

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

   4. Simplified 11.1%

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

   5. Taylor expanded in im around inf 11.1%

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

      1. unpow2 11.1%

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

   7. Simplified 11.1%

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

   8. Taylor expanded in re around 0 26.3%

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

      1. unpow2 26.3%

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

      2. unpow2 26.3%

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

      3. unpow2 26.3%

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

      4. associate-*r* 26.3%

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

      5. distribute-rgt1-in 31.9%

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

      6. +-commutative 31.9%

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

      7. associate-*r* 31.9%

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

   10. Simplified 31.9%

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

   if 5.0000000000000001e172 < 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 re around 0 75.0%

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

      1. distribute-lft-in 75.0%

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

      2. metadata-eval 75.0%

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

      3. unpow2 75.0%

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

      4. +-commutative 75.0%

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

      5. *-commutative 75.0%

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

      6. fma-udef 75.0%

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

      7. *-commutative 75.0%

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

   7. Simplified 75.0%

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

   8. Taylor expanded in im around 0 75.0%

      \[\leadsto 1 + \color{blue}{0.5 \cdot {im}^{2}} \]
   9. Step-by-step derivation

      1. unpow2 75.0%

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

   10. Simplified 75.0%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 420:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 5 \cdot 10^{+172}:\\ \;\;\;\;0.5 \cdot \left(\left(im \cdot im\right) \cdot \left(1 + re \cdot \left(re \cdot -0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \end{array} \]

Alternative 12: 49.1% accurate, 18.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 880000000000 \lor \neg \left(im \leq 10^{+172}\right):\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(im \cdot im\right) \cdot \left(1 + re \cdot \left(re \cdot -0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 880000000000.0) (not (<= im 1e+172)))
   (+ 1.0 (* 0.5 (* im im)))
   (* 0.5 (* (* im im) (+ 1.0 (* re (* re -0.5)))))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 880000000000.0) || !(im <= 1e+172)) {
		tmp = 1.0 + (0.5 * (im * im));
	} else {
		tmp = 0.5 * ((im * im) * (1.0 + (re * (re * -0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= 880000000000.0d0) .or. (.not. (im <= 1d+172))) then
        tmp = 1.0d0 + (0.5d0 * (im * im))
    else
        tmp = 0.5d0 * ((im * im) * (1.0d0 + (re * (re * (-0.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= 880000000000.0) || !(im <= 1e+172)) {
		tmp = 1.0 + (0.5 * (im * im));
	} else {
		tmp = 0.5 * ((im * im) * (1.0 + (re * (re * -0.5))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= 880000000000.0) or not (im <= 1e+172):
		tmp = 1.0 + (0.5 * (im * im))
	else:
		tmp = 0.5 * ((im * im) * (1.0 + (re * (re * -0.5))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 880000000000.0) || !(im <= 1e+172))
		tmp = Float64(1.0 + Float64(0.5 * Float64(im * im)));
	else
		tmp = Float64(0.5 * Float64(Float64(im * im) * Float64(1.0 + Float64(re * Float64(re * -0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= 880000000000.0) || ~((im <= 1e+172)))
		tmp = 1.0 + (0.5 * (im * im));
	else
		tmp = 0.5 * ((im * im) * (1.0 + (re * (re * -0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 880000000000.0], N[Not[LessEqual[im, 1e+172]], $MachinePrecision]], N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(im * im), $MachinePrecision] * N[(1.0 + N[(re * N[(re * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 8.8e11 or 1.0000000000000001e172 < 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 95.6%

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

      1. unpow2 95.6%

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

      2. *-commutative 95.6%

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

   4. Simplified 95.6%

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

   5. Taylor expanded in re around 0 59.5%

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

      1. distribute-lft-in 59.5%

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

      2. metadata-eval 59.5%

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

      3. unpow2 59.5%

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

      4. +-commutative 59.5%

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

      5. *-commutative 59.5%

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

      6. fma-udef 59.5%

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

      7. *-commutative 59.5%

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

   7. Simplified 59.5%

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

   8. Taylor expanded in im around 0 54.3%

      \[\leadsto 1 + \color{blue}{0.5 \cdot {im}^{2}} \]
   9. Step-by-step derivation

      1. unpow2 54.3%

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

   10. Simplified 54.3%

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

   if 8.8e11 < im < 1.0000000000000001e172

   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 11.5%

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

      1. unpow2 11.5%

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

   4. Simplified 11.5%

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

   5. Taylor expanded in im around inf 11.5%

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

      1. unpow2 11.5%

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

   7. Simplified 11.5%

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

   8. Taylor expanded in re around 0 27.7%

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

      1. unpow2 27.7%

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

      2. unpow2 27.7%

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

      3. unpow2 27.7%

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

      4. associate-*r* 27.7%

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

      5. distribute-rgt1-in 33.6%

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

      6. +-commutative 33.6%

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

      7. associate-*r* 33.6%

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

   10. Simplified 33.6%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 880000000000 \lor \neg \left(im \leq 10^{+172}\right):\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(im \cdot im\right) \cdot \left(1 + re \cdot \left(re \cdot -0.5\right)\right)\right)\\ \end{array} \]

Alternative 13: 48.4% accurate, 23.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 4500000000000 \lor \neg \left(im \leq 1.55 \cdot 10^{+138}\right):\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(im \cdot \left(\left(re \cdot re\right) \cdot -0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 4500000000000.0) (not (<= im 1.55e+138)))
   (+ 1.0 (* 0.5 (* im im)))
   (* im (* im (* (* re re) -0.25)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 4500000000000.0) || !(im <= 1.55e+138)) {
		tmp = 1.0 + (0.5 * (im * im));
	} else {
		tmp = im * (im * ((re * re) * -0.25));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= 4500000000000.0d0) .or. (.not. (im <= 1.55d+138))) then
        tmp = 1.0d0 + (0.5d0 * (im * im))
    else
        tmp = im * (im * ((re * re) * (-0.25d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= 4500000000000.0) || !(im <= 1.55e+138)) {
		tmp = 1.0 + (0.5 * (im * im));
	} else {
		tmp = im * (im * ((re * re) * -0.25));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= 4500000000000.0) or not (im <= 1.55e+138):
		tmp = 1.0 + (0.5 * (im * im))
	else:
		tmp = im * (im * ((re * re) * -0.25))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 4500000000000.0) || !(im <= 1.55e+138))
		tmp = Float64(1.0 + Float64(0.5 * Float64(im * im)));
	else
		tmp = Float64(im * Float64(im * Float64(Float64(re * re) * -0.25)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= 4500000000000.0) || ~((im <= 1.55e+138)))
		tmp = 1.0 + (0.5 * (im * im));
	else
		tmp = im * (im * ((re * re) * -0.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 4500000000000.0], N[Not[LessEqual[im, 1.55e+138]], $MachinePrecision]], N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(im * N[(N[(re * re), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 4.5e12 or 1.5499999999999999e138 < 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 95.6%

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

      1. unpow2 95.6%

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

      2. *-commutative 95.6%

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

   4. Simplified 95.6%

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

   5. Taylor expanded in re around 0 60.0%

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

      1. distribute-lft-in 60.0%

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

      2. metadata-eval 60.0%

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

      3. unpow2 60.0%

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

      4. +-commutative 60.0%

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

      5. *-commutative 60.0%

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

      6. fma-udef 60.0%

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

      7. *-commutative 60.0%

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

   7. Simplified 60.0%

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

   8. Taylor expanded in im around 0 54.1%

      \[\leadsto 1 + \color{blue}{0.5 \cdot {im}^{2}} \]
   9. Step-by-step derivation

      1. unpow2 54.1%

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

   10. Simplified 54.1%

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

   if 4.5e12 < im < 1.5499999999999999e138

   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 5.4%

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

      1. unpow2 5.4%

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

   4. Simplified 5.4%

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

   5. Taylor expanded in re around 0 32.2%

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

      1. *-commutative 32.2%

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

      2. unpow2 32.2%

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

      3. +-commutative 32.2%

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

      4. fma-udef 32.2%

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

      5. *-commutative 32.2%

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

      6. unpow2 32.2%

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

      7. +-commutative 32.2%

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

      8. fma-udef 32.2%

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

      9. associate-*l* 32.2%

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

      10. distribute-lft-out 32.2%

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

      11. unpow2 32.2%

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

   7. Simplified 32.2%

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

   8. Taylor expanded in im around inf 32.2%

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

      1. unpow2 32.2%

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

      2. *-commutative 32.2%

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

      3. +-commutative 32.2%

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

      4. unpow2 32.2%

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

      5. *-commutative 32.2%

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

      6. associate-*r* 32.2%

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

      7. fma-udef 32.2%

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

      8. associate-*l* 32.2%

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

      9. fma-udef 32.2%

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

      10. associate-*r* 32.2%

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

      11. *-commutative 32.2%

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

      12. fma-def 32.2%

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

   10. Simplified 32.2%

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

   11. Taylor expanded in re around inf 29.5%

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

       1. *-commutative 29.5%

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

       2. *-commutative 29.5%

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

       3. associate-*r* 29.5%

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

       4. unpow2 29.5%

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

   13. Simplified 29.5%

       \[\leadsto im \cdot \color{blue}{\left(im \cdot \left(\left(re \cdot re\right) \cdot -0.25\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4500000000000 \lor \neg \left(im \leq 1.55 \cdot 10^{+138}\right):\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(im \cdot \left(\left(re \cdot re\right) \cdot -0.25\right)\right)\\ \end{array} \]

Alternative 14: 47.0% accurate, 44.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(1.0 + N[(0.5 * 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 94.4%

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

   1. unpow2 94.4%

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

   2. *-commutative 94.4%

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

4. Simplified 94.4%

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

5. Taylor expanded in re around 0 59.2%

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

   1. distribute-lft-in 59.2%

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

   2. metadata-eval 59.2%

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

   3. unpow2 59.2%

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

   4. +-commutative 59.2%

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

   5. *-commutative 59.2%

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

   6. fma-udef 59.2%

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

   7. *-commutative 59.2%

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

7. Simplified 59.2%

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

8. Taylor expanded in im around 0 51.3%

   \[\leadsto 1 + \color{blue}{0.5 \cdot {im}^{2}} \]
9. Step-by-step derivation

   1. unpow2 51.3%

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

10. Simplified 51.3%

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

11. Final simplification 51.3%

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

Alternative 15: 21.9% accurate, 61.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(im * N[(0.5 * im), $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 83.7%

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

   1. unpow2 83.7%

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

4. Simplified 83.7%

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

5. Taylor expanded in re around 0 32.6%

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

   1. *-commutative 32.6%

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

   2. unpow2 32.6%

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

   3. +-commutative 32.6%

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

   4. fma-udef 32.6%

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

   5. *-commutative 32.6%

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

   6. unpow2 32.6%

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

   7. +-commutative 32.6%

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

   8. fma-udef 32.6%

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

   9. associate-*l* 32.6%

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

   10. distribute-lft-out 48.2%

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

   11. unpow2 48.2%

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

7. Simplified 48.2%

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

8. Taylor expanded in im around inf 21.3%

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

   1. unpow2 21.3%

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

   2. *-commutative 21.3%

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

   3. +-commutative 21.3%

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

   4. unpow2 21.3%

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

   5. *-commutative 21.3%

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

   6. associate-*r* 21.3%

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

   7. fma-udef 21.3%

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

   8. associate-*l* 21.4%

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

   9. fma-udef 21.4%

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

   10. associate-*r* 21.4%

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

   11. *-commutative 21.4%

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

   12. fma-def 21.4%

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

10. Simplified 21.4%

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

11. Taylor expanded in re around 0 22.8%

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

12. Final simplification 22.8%

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

Reproduce

herbie shell --seed 2023238 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))