math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 8.4s

Alternatives: 14

Speedup: 1.0×

• Unsound rule application detected in e-graph. Results may not be sound.

• 50.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 92.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(\cos re \cdot \left(0.5 \cdot im\right)\right)\\ t_1 := 0.5 \cdot \left(e^{im} + e^{-im}\right)\\ \mathbf{if}\;im \leq -1.4 \cdot 10^{+159}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -0.057:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 0.0145:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 8 \cdot 10^{+151}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (* (cos re) (* 0.5 im))))
        (t_1 (* 0.5 (+ (exp im) (exp (- im))))))
   (if (<= im -1.4e+159)
     t_0
     (if (<= im -0.057)
       t_1
       (if (<= im 0.0145)
         (* (* (cos re) 0.5) (+ 2.0 (* im im)))
         (if (<= im 8e+151) t_1 t_0))))))

C

double code(double re, double im) {
	double t_0 = im * (cos(re) * (0.5 * im));
	double t_1 = 0.5 * (exp(im) + exp(-im));
	double tmp;
	if (im <= -1.4e+159) {
		tmp = t_0;
	} else if (im <= -0.057) {
		tmp = t_1;
	} else if (im <= 0.0145) {
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	} else if (im <= 8e+151) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = im * (cos(re) * (0.5d0 * im))
    t_1 = 0.5d0 * (exp(im) + exp(-im))
    if (im <= (-1.4d+159)) then
        tmp = t_0
    else if (im <= (-0.057d0)) then
        tmp = t_1
    else if (im <= 0.0145d0) then
        tmp = (cos(re) * 0.5d0) * (2.0d0 + (im * im))
    else if (im <= 8d+151) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * (Math.cos(re) * (0.5 * im));
	double t_1 = 0.5 * (Math.exp(im) + Math.exp(-im));
	double tmp;
	if (im <= -1.4e+159) {
		tmp = t_0;
	} else if (im <= -0.057) {
		tmp = t_1;
	} else if (im <= 0.0145) {
		tmp = (Math.cos(re) * 0.5) * (2.0 + (im * im));
	} else if (im <= 8e+151) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (math.cos(re) * (0.5 * im))
	t_1 = 0.5 * (math.exp(im) + math.exp(-im))
	tmp = 0
	if im <= -1.4e+159:
		tmp = t_0
	elif im <= -0.057:
		tmp = t_1
	elif im <= 0.0145:
		tmp = (math.cos(re) * 0.5) * (2.0 + (im * im))
	elif im <= 8e+151:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(cos(re) * Float64(0.5 * im)))
	t_1 = Float64(0.5 * Float64(exp(im) + exp(Float64(-im))))
	tmp = 0.0
	if (im <= -1.4e+159)
		tmp = t_0;
	elseif (im <= -0.057)
		tmp = t_1;
	elseif (im <= 0.0145)
		tmp = Float64(Float64(cos(re) * 0.5) * Float64(2.0 + Float64(im * im)));
	elseif (im <= 8e+151)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (cos(re) * (0.5 * im));
	t_1 = 0.5 * (exp(im) + exp(-im));
	tmp = 0.0;
	if (im <= -1.4e+159)
		tmp = t_0;
	elseif (im <= -0.057)
		tmp = t_1;
	elseif (im <= 0.0145)
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	elseif (im <= 8e+151)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(N[Cos[re], $MachinePrecision] * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.4e+159], t$95$0, If[LessEqual[im, -0.057], t$95$1, If[LessEqual[im, 0.0145], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 8e+151], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.4000000000000001e159 or 8.00000000000000014e151 < 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 98.6%

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

      1. unpow2 98.6%

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

   4. Simplified 98.6%

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

   5. Taylor expanded in im around inf 98.6%

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

      1. *-commutative 98.6%

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

      2. associate-*r* 98.6%

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

      3. *-commutative 98.6%

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

      4. unpow2 98.6%

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

      5. associate-*l* 98.6%

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

      6. associate-*l* 98.6%

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

   7. Simplified 98.6%

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

   if -1.4000000000000001e159 < im < -0.0570000000000000021 or 0.0145000000000000007 < im < 8.00000000000000014e151

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 82.6%

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

   if -0.0570000000000000021 < im < 0.0145000000000000007

   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 99.6%

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

      1. unpow2 99.6%

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

   4. Simplified 99.6%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.4 \cdot 10^{+159}:\\ \;\;\;\;im \cdot \left(\cos re \cdot \left(0.5 \cdot im\right)\right)\\ \mathbf{elif}\;im \leq -0.057:\\ \;\;\;\;0.5 \cdot \left(e^{im} + e^{-im}\right)\\ \mathbf{elif}\;im \leq 0.0145:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 8 \cdot 10^{+151}:\\ \;\;\;\;0.5 \cdot \left(e^{im} + e^{-im}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\cos re \cdot \left(0.5 \cdot im\right)\right)\\ \end{array} \]

Alternative 4: 95.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(e^{im} + e^{-im}\right)\\ t_1 := {im}^{4} \cdot \left(\cos re \cdot 0.041666666666666664\right)\\ \mathbf{if}\;im \leq -5.1 \cdot 10^{+82}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -0.088:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 0.0145:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+73}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ (exp im) (exp (- im)))))
        (t_1 (* (pow im 4.0) (* (cos re) 0.041666666666666664))))
   (if (<= im -5.1e+82)
     t_1
     (if (<= im -0.088)
       t_0
       (if (<= im 0.0145)
         (* (* (cos re) 0.5) (+ 2.0 (* im im)))
         (if (<= im 1.15e+73) t_0 t_1))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (exp(im) + exp(-im));
	double t_1 = pow(im, 4.0) * (cos(re) * 0.041666666666666664);
	double tmp;
	if (im <= -5.1e+82) {
		tmp = t_1;
	} else if (im <= -0.088) {
		tmp = t_0;
	} else if (im <= 0.0145) {
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	} else if (im <= 1.15e+73) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * (exp(im) + exp(-im))
    t_1 = (im ** 4.0d0) * (cos(re) * 0.041666666666666664d0)
    if (im <= (-5.1d+82)) then
        tmp = t_1
    else if (im <= (-0.088d0)) then
        tmp = t_0
    else if (im <= 0.0145d0) then
        tmp = (cos(re) * 0.5d0) * (2.0d0 + (im * im))
    else if (im <= 1.15d+73) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * (Math.exp(im) + Math.exp(-im));
	double t_1 = Math.pow(im, 4.0) * (Math.cos(re) * 0.041666666666666664);
	double tmp;
	if (im <= -5.1e+82) {
		tmp = t_1;
	} else if (im <= -0.088) {
		tmp = t_0;
	} else if (im <= 0.0145) {
		tmp = (Math.cos(re) * 0.5) * (2.0 + (im * im));
	} else if (im <= 1.15e+73) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * (math.exp(im) + math.exp(-im))
	t_1 = math.pow(im, 4.0) * (math.cos(re) * 0.041666666666666664)
	tmp = 0
	if im <= -5.1e+82:
		tmp = t_1
	elif im <= -0.088:
		tmp = t_0
	elif im <= 0.0145:
		tmp = (math.cos(re) * 0.5) * (2.0 + (im * im))
	elif im <= 1.15e+73:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(exp(im) + exp(Float64(-im))))
	t_1 = Float64((im ^ 4.0) * Float64(cos(re) * 0.041666666666666664))
	tmp = 0.0
	if (im <= -5.1e+82)
		tmp = t_1;
	elseif (im <= -0.088)
		tmp = t_0;
	elseif (im <= 0.0145)
		tmp = Float64(Float64(cos(re) * 0.5) * Float64(2.0 + Float64(im * im)));
	elseif (im <= 1.15e+73)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (exp(im) + exp(-im));
	t_1 = (im ^ 4.0) * (cos(re) * 0.041666666666666664);
	tmp = 0.0;
	if (im <= -5.1e+82)
		tmp = t_1;
	elseif (im <= -0.088)
		tmp = t_0;
	elseif (im <= 0.0145)
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	elseif (im <= 1.15e+73)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[im, 4.0], $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -5.1e+82], t$95$1, If[LessEqual[im, -0.088], t$95$0, If[LessEqual[im, 0.0145], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.15e+73], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -5.1000000000000003e82 or 1.15e73 < 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 99.1%

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

      1. unpow2 99.1%

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

      2. *-commutative 99.1%

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

   4. Simplified 99.1%

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

   5. Taylor expanded in im around inf 99.1%

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

      1. *-commutative 99.1%

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

      2. *-commutative 99.1%

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

      3. associate-*l* 99.1%

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

   7. Simplified 99.1%

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

   if -5.1000000000000003e82 < im < -0.087999999999999995 or 0.0145000000000000007 < im < 1.15e73

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 78.7%

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

   if -0.087999999999999995 < im < 0.0145000000000000007

   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 99.6%

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

      1. unpow2 99.6%

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

   4. Simplified 99.6%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -5.1 \cdot 10^{+82}:\\ \;\;\;\;{im}^{4} \cdot \left(\cos re \cdot 0.041666666666666664\right)\\ \mathbf{elif}\;im \leq -0.088:\\ \;\;\;\;0.5 \cdot \left(e^{im} + e^{-im}\right)\\ \mathbf{elif}\;im \leq 0.0145:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+73}:\\ \;\;\;\;0.5 \cdot \left(e^{im} + e^{-im}\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{4} \cdot \left(\cos re \cdot 0.041666666666666664\right)\\ \end{array} \]

Alternative 5: 97.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -5.1 \cdot 10^{+82}:\\ \;\;\;\;{im}^{4} \cdot \left(\cos re \cdot 0.041666666666666664\right)\\ \mathbf{elif}\;im \leq -9 \cdot 10^{-10}:\\ \;\;\;\;0.5 \cdot \left(e^{im} + e^{-im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(0.5 + 0.5 \cdot e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -5.1e+82)
   (* (pow im 4.0) (* (cos re) 0.041666666666666664))
   (if (<= im -9e-10)
     (* 0.5 (+ (exp im) (exp (- im))))
     (* (cos re) (+ 0.5 (* 0.5 (exp im)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -5.1e+82) {
		tmp = pow(im, 4.0) * (cos(re) * 0.041666666666666664);
	} else if (im <= -9e-10) {
		tmp = 0.5 * (exp(im) + exp(-im));
	} else {
		tmp = cos(re) * (0.5 + (0.5 * exp(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 <= (-5.1d+82)) then
        tmp = (im ** 4.0d0) * (cos(re) * 0.041666666666666664d0)
    else if (im <= (-9d-10)) then
        tmp = 0.5d0 * (exp(im) + exp(-im))
    else
        tmp = cos(re) * (0.5d0 + (0.5d0 * exp(im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -5.1e+82) {
		tmp = Math.pow(im, 4.0) * (Math.cos(re) * 0.041666666666666664);
	} else if (im <= -9e-10) {
		tmp = 0.5 * (Math.exp(im) + Math.exp(-im));
	} else {
		tmp = Math.cos(re) * (0.5 + (0.5 * Math.exp(im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -5.1e+82:
		tmp = math.pow(im, 4.0) * (math.cos(re) * 0.041666666666666664)
	elif im <= -9e-10:
		tmp = 0.5 * (math.exp(im) + math.exp(-im))
	else:
		tmp = math.cos(re) * (0.5 + (0.5 * math.exp(im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -5.1e+82)
		tmp = Float64((im ^ 4.0) * Float64(cos(re) * 0.041666666666666664));
	elseif (im <= -9e-10)
		tmp = Float64(0.5 * Float64(exp(im) + exp(Float64(-im))));
	else
		tmp = Float64(cos(re) * Float64(0.5 + Float64(0.5 * exp(im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -5.1e+82)
		tmp = (im ^ 4.0) * (cos(re) * 0.041666666666666664);
	elseif (im <= -9e-10)
		tmp = 0.5 * (exp(im) + exp(-im));
	else
		tmp = cos(re) * (0.5 + (0.5 * exp(im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -5.1e+82], N[(N[Power[im, 4.0], $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, -9e-10], N[(0.5 * N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(0.5 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < -5.1000000000000003e82

   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%

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

      1. *-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

   7. Simplified 100.0%

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

   if -5.1000000000000003e82 < im < -8.9999999999999999e-10

   1. Initial program 99.9%

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

   2. Taylor expanded in re around 0 67.5%

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

   if -8.9999999999999999e-10 < 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%

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

   4. Taylor expanded in im around 0 99.5%

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

      1. fma-udef 99.5%

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

   6. Applied egg-rr 99.5%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -5.1 \cdot 10^{+82}:\\ \;\;\;\;{im}^{4} \cdot \left(\cos re \cdot 0.041666666666666664\right)\\ \mathbf{elif}\;im \leq -9 \cdot 10^{-10}:\\ \;\;\;\;0.5 \cdot \left(e^{im} + e^{-im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(0.5 + 0.5 \cdot e^{im}\right)\\ \end{array} \]

Alternative 6: 83.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(\cos re \cdot \left(0.5 \cdot im\right)\right)\\ \mathbf{if}\;im \leq -1.4:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 1.75:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 8 \cdot 10^{+151}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{im}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (* (cos re) (* 0.5 im)))))
   (if (<= im -1.4)
     t_0
     (if (<= im 1.75)
       (cos re)
       (if (<= im 8e+151) (+ 0.5 (* 0.5 (exp im))) t_0)))))

C

double code(double re, double im) {
	double t_0 = im * (cos(re) * (0.5 * im));
	double tmp;
	if (im <= -1.4) {
		tmp = t_0;
	} else if (im <= 1.75) {
		tmp = cos(re);
	} else if (im <= 8e+151) {
		tmp = 0.5 + (0.5 * exp(im));
	} 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 = im * (cos(re) * (0.5d0 * im))
    if (im <= (-1.4d0)) then
        tmp = t_0
    else if (im <= 1.75d0) then
        tmp = cos(re)
    else if (im <= 8d+151) then
        tmp = 0.5d0 + (0.5d0 * exp(im))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * (Math.cos(re) * (0.5 * im));
	double tmp;
	if (im <= -1.4) {
		tmp = t_0;
	} else if (im <= 1.75) {
		tmp = Math.cos(re);
	} else if (im <= 8e+151) {
		tmp = 0.5 + (0.5 * Math.exp(im));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (math.cos(re) * (0.5 * im))
	tmp = 0
	if im <= -1.4:
		tmp = t_0
	elif im <= 1.75:
		tmp = math.cos(re)
	elif im <= 8e+151:
		tmp = 0.5 + (0.5 * math.exp(im))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(cos(re) * Float64(0.5 * im)))
	tmp = 0.0
	if (im <= -1.4)
		tmp = t_0;
	elseif (im <= 1.75)
		tmp = cos(re);
	elseif (im <= 8e+151)
		tmp = Float64(0.5 + Float64(0.5 * exp(im)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (cos(re) * (0.5 * im));
	tmp = 0.0;
	if (im <= -1.4)
		tmp = t_0;
	elseif (im <= 1.75)
		tmp = cos(re);
	elseif (im <= 8e+151)
		tmp = 0.5 + (0.5 * exp(im));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(N[Cos[re], $MachinePrecision] * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.4], t$95$0, If[LessEqual[im, 1.75], N[Cos[re], $MachinePrecision], If[LessEqual[im, 8e+151], N[(0.5 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.3999999999999999 or 8.00000000000000014e151 < 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 66.0%

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

      1. unpow2 66.0%

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

   4. Simplified 66.0%

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

   5. Taylor expanded in im around inf 66.0%

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

      1. *-commutative 66.0%

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

      2. associate-*r* 66.0%

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

      3. *-commutative 66.0%

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

      4. unpow2 66.0%

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

      5. associate-*l* 66.0%

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

      6. associate-*l* 66.0%

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

   7. Simplified 66.0%

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

   if -1.3999999999999999 < im < 1.75

   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 99.5%

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

   if 1.75 < im < 8.00000000000000014e151

   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 0 100.0%

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

   5. Taylor expanded in re around 0 84.4%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.4:\\ \;\;\;\;im \cdot \left(\cos re \cdot \left(0.5 \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 1.75:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 8 \cdot 10^{+151}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{im}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\cos re \cdot \left(0.5 \cdot im\right)\right)\\ \end{array} \]

Alternative 7: 84.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.5:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 8 \cdot 10^{+151}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{im}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\cos re \cdot \left(0.5 \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.5)
   (* (* (cos re) 0.5) (+ 2.0 (* im im)))
   (if (<= im 8e+151)
     (+ 0.5 (* 0.5 (exp im)))
     (* im (* (cos re) (* 0.5 im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.5) {
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	} else if (im <= 8e+151) {
		tmp = 0.5 + (0.5 * exp(im));
	} else {
		tmp = im * (cos(re) * (0.5 * 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 <= 3.5d0) then
        tmp = (cos(re) * 0.5d0) * (2.0d0 + (im * im))
    else if (im <= 8d+151) then
        tmp = 0.5d0 + (0.5d0 * exp(im))
    else
        tmp = im * (cos(re) * (0.5d0 * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3.5) {
		tmp = (Math.cos(re) * 0.5) * (2.0 + (im * im));
	} else if (im <= 8e+151) {
		tmp = 0.5 + (0.5 * Math.exp(im));
	} else {
		tmp = im * (Math.cos(re) * (0.5 * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3.5:
		tmp = (math.cos(re) * 0.5) * (2.0 + (im * im))
	elif im <= 8e+151:
		tmp = 0.5 + (0.5 * math.exp(im))
	else:
		tmp = im * (math.cos(re) * (0.5 * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.5)
		tmp = Float64(Float64(cos(re) * 0.5) * Float64(2.0 + Float64(im * im)));
	elseif (im <= 8e+151)
		tmp = Float64(0.5 + Float64(0.5 * exp(im)));
	else
		tmp = Float64(im * Float64(cos(re) * Float64(0.5 * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3.5)
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	elseif (im <= 8e+151)
		tmp = 0.5 + (0.5 * exp(im));
	else
		tmp = im * (cos(re) * (0.5 * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.5], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 8e+151], N[(0.5 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(N[Cos[re], $MachinePrecision] * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 3.5

   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%

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

   if 3.5 < im < 8.00000000000000014e151

   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 0 100.0%

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

   5. Taylor expanded in re around 0 84.4%

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

   if 8.00000000000000014e151 < 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 96.3%

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

      1. unpow2 96.3%

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

   4. Simplified 96.3%

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

   5. Taylor expanded in im around inf 96.3%

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

      1. *-commutative 96.3%

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

      2. associate-*r* 96.3%

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

      3. *-commutative 96.3%

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

      4. unpow2 96.3%

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

      5. associate-*l* 96.3%

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

      6. associate-*l* 96.3%

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

   7. Simplified 96.3%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.5:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 8 \cdot 10^{+151}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{im}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\cos re \cdot \left(0.5 \cdot im\right)\right)\\ \end{array} \]

Alternative 8: 77.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1.2 \cdot 10^{+47}:\\ \;\;\;\;im \cdot \left(0.5 \cdot im\right)\\ \mathbf{elif}\;im \leq 2:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -1.2e+47)
   (* im (* 0.5 im))
   (if (<= im 2.0) (cos re) (+ 0.5 (* 0.5 (exp im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -1.2e+47) {
		tmp = im * (0.5 * im);
	} else if (im <= 2.0) {
		tmp = cos(re);
	} else {
		tmp = 0.5 + (0.5 * exp(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 <= (-1.2d+47)) then
        tmp = im * (0.5d0 * im)
    else if (im <= 2.0d0) then
        tmp = cos(re)
    else
        tmp = 0.5d0 + (0.5d0 * exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -1.2e+47) {
		tmp = im * (0.5 * im);
	} else if (im <= 2.0) {
		tmp = Math.cos(re);
	} else {
		tmp = 0.5 + (0.5 * Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -1.2e+47:
		tmp = im * (0.5 * im)
	elif im <= 2.0:
		tmp = math.cos(re)
	else:
		tmp = 0.5 + (0.5 * math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -1.2e+47)
		tmp = Float64(im * Float64(0.5 * im));
	elseif (im <= 2.0)
		tmp = cos(re);
	else
		tmp = Float64(0.5 + Float64(0.5 * exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -1.2e+47)
		tmp = im * (0.5 * im);
	elseif (im <= 2.0)
		tmp = cos(re);
	else
		tmp = 0.5 + (0.5 * exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -1.2e+47], N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.0], N[Cos[re], $MachinePrecision], N[(0.5 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.20000000000000009e47

   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 69.3%

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

      1. unpow2 69.3%

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

   4. Simplified 69.3%

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

   5. Taylor expanded in re around 0 2.7%

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

      1. *-commutative 2.7%

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

      2. *-commutative 2.7%

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

      3. associate-*l* 2.7%

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

      4. distribute-lft-out 49.2%

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

      5. +-commutative 49.2%

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

      6. unpow2 49.2%

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

      7. fma-def 49.2%

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

      8. *-commutative 49.2%

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

      9. unpow2 49.2%

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

   7. Simplified 49.2%

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

   8. Taylor expanded in im around inf 49.2%

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

      1. *-commutative 49.2%

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

      2. unpow2 49.2%

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

      3. unpow2 49.2%

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

   10. Simplified 49.2%

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

   11. Taylor expanded in re around 0 55.4%

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

       1. unpow2 55.4%

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

       2. associate-*r* 55.4%

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

   13. Simplified 55.4%

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

   if -1.20000000000000009e47 < im < 2

   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 89.4%

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

   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%

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

   4. Taylor expanded in im around 0 100.0%

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

   5. Taylor expanded in re around 0 78.6%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.2 \cdot 10^{+47}:\\ \;\;\;\;im \cdot \left(0.5 \cdot im\right)\\ \mathbf{elif}\;im \leq 2:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{im}\\ \end{array} \]

Alternative 9: 69.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(im \cdot im\right)\\ t_1 := im \cdot \left(0.5 \cdot im\right)\\ t_2 := im \cdot \left(im \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right)\\ \mathbf{if}\;im \leq -4 \cdot 10^{+47}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 5.8 \cdot 10^{+86}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2.3 \cdot 10^{+139}:\\ \;\;\;\;\frac{t_0 \cdot t_0 - t_2 \cdot t_2}{t_0 - t_2}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (* im im)))
        (t_1 (* im (* 0.5 im)))
        (t_2 (* im (* im (* re (* re -0.25))))))
   (if (<= im -4e+47)
     t_1
     (if (<= im 5.8e+86)
       (cos re)
       (if (<= im 2.3e+139)
         (/ (- (* t_0 t_0) (* t_2 t_2)) (- t_0 t_2))
         t_1)))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (im * im);
	double t_1 = im * (0.5 * im);
	double t_2 = im * (im * (re * (re * -0.25)));
	double tmp;
	if (im <= -4e+47) {
		tmp = t_1;
	} else if (im <= 5.8e+86) {
		tmp = cos(re);
	} else if (im <= 2.3e+139) {
		tmp = ((t_0 * t_0) - (t_2 * t_2)) / (t_0 - t_2);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 0.5d0 * (im * im)
    t_1 = im * (0.5d0 * im)
    t_2 = im * (im * (re * (re * (-0.25d0))))
    if (im <= (-4d+47)) then
        tmp = t_1
    else if (im <= 5.8d+86) then
        tmp = cos(re)
    else if (im <= 2.3d+139) then
        tmp = ((t_0 * t_0) - (t_2 * t_2)) / (t_0 - t_2)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * (im * im);
	double t_1 = im * (0.5 * im);
	double t_2 = im * (im * (re * (re * -0.25)));
	double tmp;
	if (im <= -4e+47) {
		tmp = t_1;
	} else if (im <= 5.8e+86) {
		tmp = Math.cos(re);
	} else if (im <= 2.3e+139) {
		tmp = ((t_0 * t_0) - (t_2 * t_2)) / (t_0 - t_2);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * (im * im)
	t_1 = im * (0.5 * im)
	t_2 = im * (im * (re * (re * -0.25)))
	tmp = 0
	if im <= -4e+47:
		tmp = t_1
	elif im <= 5.8e+86:
		tmp = math.cos(re)
	elif im <= 2.3e+139:
		tmp = ((t_0 * t_0) - (t_2 * t_2)) / (t_0 - t_2)
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(im * im))
	t_1 = Float64(im * Float64(0.5 * im))
	t_2 = Float64(im * Float64(im * Float64(re * Float64(re * -0.25))))
	tmp = 0.0
	if (im <= -4e+47)
		tmp = t_1;
	elseif (im <= 5.8e+86)
		tmp = cos(re);
	elseif (im <= 2.3e+139)
		tmp = Float64(Float64(Float64(t_0 * t_0) - Float64(t_2 * t_2)) / Float64(t_0 - t_2));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (im * im);
	t_1 = im * (0.5 * im);
	t_2 = im * (im * (re * (re * -0.25)));
	tmp = 0.0;
	if (im <= -4e+47)
		tmp = t_1;
	elseif (im <= 5.8e+86)
		tmp = cos(re);
	elseif (im <= 2.3e+139)
		tmp = ((t_0 * t_0) - (t_2 * t_2)) / (t_0 - t_2);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(im * N[(im * N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -4e+47], t$95$1, If[LessEqual[im, 5.8e+86], N[Cos[re], $MachinePrecision], If[LessEqual[im, 2.3e+139], N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - t$95$2), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -4.0000000000000002e47 or 2.3e139 < 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 76.4%

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

      1. unpow2 76.4%

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

   4. Simplified 76.4%

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

   5. Taylor expanded in re around 0 3.1%

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

      1. *-commutative 3.1%

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

      2. *-commutative 3.1%

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

      3. associate-*l* 3.1%

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

      4. distribute-lft-out 53.7%

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

      5. +-commutative 53.7%

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

      6. unpow2 53.7%

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

      7. fma-def 53.7%

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

      8. *-commutative 53.7%

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

      9. unpow2 53.7%

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

   7. Simplified 53.7%

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

   8. Taylor expanded in im around inf 53.7%

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

      1. *-commutative 53.7%

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

      2. unpow2 53.7%

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

      3. unpow2 53.7%

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

   10. Simplified 53.7%

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

   11. Taylor expanded in re around 0 59.3%

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

       1. unpow2 59.3%

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

       2. associate-*r* 59.3%

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

   13. Simplified 59.3%

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

   if -4.0000000000000002e47 < im < 5.79999999999999981e86

   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 81.2%

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

   if 5.79999999999999981e86 < im < 2.3e139

   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 6.6%

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

      1. unpow2 6.6%

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

   4. Simplified 6.6%

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

   5. Taylor expanded in re around 0 11.1%

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

      1. *-commutative 11.1%

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

      2. *-commutative 11.1%

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

      3. associate-*l* 11.1%

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

      4. distribute-lft-out 11.1%

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

      5. +-commutative 11.1%

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

      6. unpow2 11.1%

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

      7. fma-def 11.1%

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

      8. *-commutative 11.1%

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

      9. unpow2 11.1%

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

   7. Simplified 11.1%

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

   8. Taylor expanded in im around inf 11.1%

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

      1. *-commutative 11.1%

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

      2. unpow2 11.1%

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

      3. unpow2 11.1%

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

   10. Simplified 11.1%

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

       1. distribute-lft-in 11.1%

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

       2. flip-+ 56.3%

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

       3. *-commutative 56.3%

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

       4. *-commutative 56.3%

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

       5. associate-*l* 56.3%

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

       6. *-commutative 56.3%

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

       7. associate-*l* 56.3%

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

       8. associate-*l* 56.3%

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

       9. *-commutative 56.3%

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

       10. associate-*l* 56.3%

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

       11. *-commutative 56.3%

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

       12. associate-*l* 56.3%

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

   12. Applied egg-rr 56.3%

       \[\leadsto \color{blue}{\frac{\left(0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(0.5 \cdot \left(im \cdot im\right)\right) - \left(im \cdot \left(im \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right)\right) \cdot \left(im \cdot \left(im \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right)\right)}{0.5 \cdot \left(im \cdot im\right) - im \cdot \left(im \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -4 \cdot 10^{+47}:\\ \;\;\;\;im \cdot \left(0.5 \cdot im\right)\\ \mathbf{elif}\;im \leq 5.8 \cdot 10^{+86}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2.3 \cdot 10^{+139}:\\ \;\;\;\;\frac{\left(0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(0.5 \cdot \left(im \cdot im\right)\right) - \left(im \cdot \left(im \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right)\right) \cdot \left(im \cdot \left(im \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right)\right)}{0.5 \cdot \left(im \cdot im\right) - im \cdot \left(im \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot im\right)\\ \end{array} \]

Alternative 10: 47.9% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(im \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right)\\ t_1 := 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{if}\;im \leq 5.8 \cdot 10^{+86}:\\ \;\;\;\;t_1 + 1\\ \mathbf{elif}\;im \leq 2.3 \cdot 10^{+139}:\\ \;\;\;\;\frac{t_1 \cdot t_1 - t_0 \cdot t_0}{t_1 - t_0}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (* im (* re (* re -0.25))))) (t_1 (* 0.5 (* im im))))
   (if (<= im 5.8e+86)
     (+ t_1 1.0)
     (if (<= im 2.3e+139)
       (/ (- (* t_1 t_1) (* t_0 t_0)) (- t_1 t_0))
       (* im (* 0.5 im))))))

C

double code(double re, double im) {
	double t_0 = im * (im * (re * (re * -0.25)));
	double t_1 = 0.5 * (im * im);
	double tmp;
	if (im <= 5.8e+86) {
		tmp = t_1 + 1.0;
	} else if (im <= 2.3e+139) {
		tmp = ((t_1 * t_1) - (t_0 * t_0)) / (t_1 - t_0);
	} else {
		tmp = im * (0.5 * im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = im * (im * (re * (re * (-0.25d0))))
    t_1 = 0.5d0 * (im * im)
    if (im <= 5.8d+86) then
        tmp = t_1 + 1.0d0
    else if (im <= 2.3d+139) then
        tmp = ((t_1 * t_1) - (t_0 * t_0)) / (t_1 - t_0)
    else
        tmp = im * (0.5d0 * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * (im * (re * (re * -0.25)));
	double t_1 = 0.5 * (im * im);
	double tmp;
	if (im <= 5.8e+86) {
		tmp = t_1 + 1.0;
	} else if (im <= 2.3e+139) {
		tmp = ((t_1 * t_1) - (t_0 * t_0)) / (t_1 - t_0);
	} else {
		tmp = im * (0.5 * im);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (im * (re * (re * -0.25)))
	t_1 = 0.5 * (im * im)
	tmp = 0
	if im <= 5.8e+86:
		tmp = t_1 + 1.0
	elif im <= 2.3e+139:
		tmp = ((t_1 * t_1) - (t_0 * t_0)) / (t_1 - t_0)
	else:
		tmp = im * (0.5 * im)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(im * Float64(re * Float64(re * -0.25))))
	t_1 = Float64(0.5 * Float64(im * im))
	tmp = 0.0
	if (im <= 5.8e+86)
		tmp = Float64(t_1 + 1.0);
	elseif (im <= 2.3e+139)
		tmp = Float64(Float64(Float64(t_1 * t_1) - Float64(t_0 * t_0)) / Float64(t_1 - t_0));
	else
		tmp = Float64(im * Float64(0.5 * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (im * (re * (re * -0.25)));
	t_1 = 0.5 * (im * im);
	tmp = 0.0;
	if (im <= 5.8e+86)
		tmp = t_1 + 1.0;
	elseif (im <= 2.3e+139)
		tmp = ((t_1 * t_1) - (t_0 * t_0)) / (t_1 - t_0);
	else
		tmp = im * (0.5 * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(im * N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 5.8e+86], N[(t$95$1 + 1.0), $MachinePrecision], If[LessEqual[im, 2.3e+139], N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 - t$95$0), $MachinePrecision]), $MachinePrecision], N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if im < 5.79999999999999981e86

   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 78.2%

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

      1. unpow2 78.2%

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

   4. Simplified 78.2%

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

   5. Taylor expanded in re around 0 45.6%

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

      1. distribute-lft-in 45.6%

         \[\leadsto \color{blue}{0.5 \cdot 2 + 0.5 \cdot {im}^{2}} \]

      2. metadata-eval 45.6%

         \[\leadsto \color{blue}{1} + 0.5 \cdot {im}^{2} \]

      3. unpow2 45.6%

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

   7. Simplified 45.6%

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

   if 5.79999999999999981e86 < im < 2.3e139

   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 6.6%

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

      1. unpow2 6.6%

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

   4. Simplified 6.6%

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

   5. Taylor expanded in re around 0 11.1%

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

      1. *-commutative 11.1%

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

      2. *-commutative 11.1%

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

      3. associate-*l* 11.1%

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

      4. distribute-lft-out 11.1%

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

      5. +-commutative 11.1%

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

      6. unpow2 11.1%

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

      7. fma-def 11.1%

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

      8. *-commutative 11.1%

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

      9. unpow2 11.1%

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

   7. Simplified 11.1%

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

   8. Taylor expanded in im around inf 11.1%

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

      1. *-commutative 11.1%

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

      2. unpow2 11.1%

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

      3. unpow2 11.1%

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

   10. Simplified 11.1%

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

       1. distribute-lft-in 11.1%

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

       2. flip-+ 56.3%

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

       3. *-commutative 56.3%

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

       4. *-commutative 56.3%

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

       5. associate-*l* 56.3%

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

       6. *-commutative 56.3%

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

       7. associate-*l* 56.3%

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

       8. associate-*l* 56.3%

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

       9. *-commutative 56.3%

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

       10. associate-*l* 56.3%

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

       11. *-commutative 56.3%

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

       12. associate-*l* 56.3%

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

   12. Applied egg-rr 56.3%

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

   if 2.3e139 < 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 92.8%

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

      1. unpow2 92.8%

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

   4. Simplified 92.8%

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

   5. Taylor expanded in re around 0 4.0%

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

      1. *-commutative 4.0%

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

      2. *-commutative 4.0%

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

      3. associate-*l* 4.0%

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

      4. distribute-lft-out 64.0%

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

      5. +-commutative 64.0%

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

      6. unpow2 64.0%

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

      7. fma-def 64.0%

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

      8. *-commutative 64.0%

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

      9. unpow2 64.0%

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

   7. Simplified 64.0%

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

   8. Taylor expanded in im around inf 64.0%

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

      1. *-commutative 64.0%

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

      2. unpow2 64.0%

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

      3. unpow2 64.0%

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

   10. Simplified 64.0%

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

   11. Taylor expanded in re around 0 68.3%

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

       1. unpow2 68.3%

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

       2. associate-*r* 68.3%

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

   13. Simplified 68.3%

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

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 5.8 \cdot 10^{+86}:\\ \;\;\;\;0.5 \cdot \left(im \cdot im\right) + 1\\ \mathbf{elif}\;im \leq 2.3 \cdot 10^{+139}:\\ \;\;\;\;\frac{\left(0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(0.5 \cdot \left(im \cdot im\right)\right) - \left(im \cdot \left(im \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right)\right) \cdot \left(im \cdot \left(im \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right)\right)}{0.5 \cdot \left(im \cdot im\right) - im \cdot \left(im \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot im\right)\\ \end{array} \]

Alternative 11: 46.0% accurate, 33.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -0.042 \lor \neg \left(im \leq 1.45\right):\\ \;\;\;\;im \cdot \left(0.5 \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -0.042) (not (<= im 1.45))) (* im (* 0.5 im)) 1.0))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -0.042) || !(im <= 1.45)) {
		tmp = im * (0.5 * 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 <= (-0.042d0)) .or. (.not. (im <= 1.45d0))) then
        tmp = im * (0.5d0 * im)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -0.042) || !(im <= 1.45)) {
		tmp = im * (0.5 * im);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -0.042) or not (im <= 1.45):
		tmp = im * (0.5 * im)
	else:
		tmp = 1.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -0.042) || !(im <= 1.45))
		tmp = Float64(im * Float64(0.5 * im));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -0.042) || ~((im <= 1.45)))
		tmp = im * (0.5 * im);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -0.042], N[Not[LessEqual[im, 1.45]], $MachinePrecision]], N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if im < -0.0420000000000000026 or 1.44999999999999996 < 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 51.0%

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

      1. unpow2 51.0%

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

   4. Simplified 51.0%

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

   5. Taylor expanded in re around 0 6.3%

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

      1. *-commutative 6.3%

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

      2. *-commutative 6.3%

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

      3. associate-*l* 6.3%

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

      4. distribute-lft-out 38.6%

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

      5. +-commutative 38.6%

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

      6. unpow2 38.6%

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

      7. fma-def 38.6%

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

      8. *-commutative 38.6%

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

      9. unpow2 38.6%

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

   7. Simplified 38.6%

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

   8. Taylor expanded in im around inf 38.6%

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

      1. *-commutative 38.6%

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

      2. unpow2 38.6%

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

      3. unpow2 38.6%

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

   10. Simplified 38.6%

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

   11. Taylor expanded in re around 0 39.4%

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

       1. unpow2 39.4%

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

       2. associate-*r* 39.4%

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

   13. Simplified 39.4%

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

   if -0.0420000000000000026 < im < 1.44999999999999996

   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%

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

   6. Applied egg-rr 51.5%

      \[\leadsto \color{blue}{\frac{\cos re - 1.9380669946781485 \cdot 10^{-10} \cdot \cos re}{\cos re - 1.9380669946781485 \cdot 10^{-10} \cdot \cos re}} \]
   7. Step-by-step derivation

      1. *-inverses 51.5%

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

   8. Simplified 51.5%

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

4. Final simplification 45.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -0.042 \lor \neg \left(im \leq 1.45\right):\\ \;\;\;\;im \cdot \left(0.5 \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 12: 46.2% accurate, 44.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision] + 1.0), $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 75.1%

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

   1. unpow2 75.1%

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

4. Simplified 75.1%

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

5. Taylor expanded in re around 0 45.3%

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

   1. distribute-lft-in 45.3%

      \[\leadsto \color{blue}{0.5 \cdot 2 + 0.5 \cdot {im}^{2}} \]

   2. metadata-eval 45.3%

      \[\leadsto \color{blue}{1} + 0.5 \cdot {im}^{2} \]

   3. unpow2 45.3%

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

7. Simplified 45.3%

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

8. Final simplification 45.3%

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

Alternative 13: 4.0% 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 75.1%

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

   1. unpow2 75.1%

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

4. Simplified 75.1%

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

6. Applied egg-rr 3.6%

   \[\leadsto \color{blue}{-2 + \cos re} \]
7. Step-by-step derivation

   1. +-commutative 3.6%

      \[\leadsto \color{blue}{\cos re + -2} \]

8. Simplified 3.6%

   \[\leadsto \color{blue}{\cos re + -2} \]

9. Taylor expanded in re around 0 4.2%

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

10. Final simplification 4.2%

    \[\leadsto -1 \]

Alternative 14: 28.2% 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 88.1%

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

   1. unpow2 88.1%

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

   2. *-commutative 88.1%

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

4. Simplified 88.1%

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

6. Applied egg-rr 26.7%

   \[\leadsto \color{blue}{\frac{\cos re - 1.9380669946781485 \cdot 10^{-10} \cdot \cos re}{\cos re - 1.9380669946781485 \cdot 10^{-10} \cdot \cos re}} \]
7. Step-by-step derivation

   1. *-inverses 26.7%

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

8. Simplified 26.7%

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

9. Final simplification 26.7%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023178 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))