math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 5.4s

Alternatives: 12

Speedup: 1.0×

• 49.2% 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, 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]

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%

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

Alternative 2: 85.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 4.2 \cdot 10^{-7} \lor \neg \left(im \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;\mathsf{fma}\left(0.5, \cos re \cdot \left(im \cdot im\right), \cos re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 4.2e-7) (not (<= im 1.35e+154)))
   (fma 0.5 (* (cos re) (* im im)) (cos re))
   (* 0.5 (+ (exp (- im)) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 4.2e-7) || !(im <= 1.35e+154)) {
		tmp = fma(0.5, (cos(re) * (im * im)), cos(re));
	} else {
		tmp = 0.5 * (exp(-im) + exp(im));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 4.2e-7) || !(im <= 1.35e+154))
		tmp = fma(0.5, Float64(cos(re) * Float64(im * im)), cos(re));
	else
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 4.2e-7], N[Not[LessEqual[im, 1.35e+154]], $MachinePrecision]], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] + N[Cos[re], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 4.2e-7 or 1.35000000000000003e154 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 81.0%

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

   4. Simplified 81.0%

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

   if 4.2e-7 < im < 1.35000000000000003e154

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 90.6%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.2 \cdot 10^{-7} \lor \neg \left(im \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;\mathsf{fma}\left(0.5, \cos re \cdot \left(im \cdot im\right), \cos re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \end{array} \]

Alternative 3: 69.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} + e^{im}\\ \mathbf{if}\;im \leq 4.2 \cdot 10^{-7}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 9 \cdot 10^{+127}:\\ \;\;\;\;0.5 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ (exp (- im)) (exp im))))
   (if (<= im 4.2e-7)
     (cos re)
     (if (<= im 9e+127) (* 0.5 t_0) (* t_0 (+ 0.5 (* -0.25 (* re re))))))))

C

double code(double re, double im) {
	double t_0 = exp(-im) + exp(im);
	double tmp;
	if (im <= 4.2e-7) {
		tmp = cos(re);
	} else if (im <= 9e+127) {
		tmp = 0.5 * t_0;
	} else {
		tmp = t_0 * (0.5 + (-0.25 * (re * re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-im) + exp(im)
    if (im <= 4.2d-7) then
        tmp = cos(re)
    else if (im <= 9d+127) then
        tmp = 0.5d0 * t_0
    else
        tmp = t_0 * (0.5d0 + ((-0.25d0) * (re * re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(-im) + Math.exp(im);
	double tmp;
	if (im <= 4.2e-7) {
		tmp = Math.cos(re);
	} else if (im <= 9e+127) {
		tmp = 0.5 * t_0;
	} else {
		tmp = t_0 * (0.5 + (-0.25 * (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) + math.exp(im)
	tmp = 0
	if im <= 4.2e-7:
		tmp = math.cos(re)
	elif im <= 9e+127:
		tmp = 0.5 * t_0
	else:
		tmp = t_0 * (0.5 + (-0.25 * (re * re)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) + exp(im))
	tmp = 0.0
	if (im <= 4.2e-7)
		tmp = cos(re);
	elseif (im <= 9e+127)
		tmp = Float64(0.5 * t_0);
	else
		tmp = Float64(t_0 * Float64(0.5 + Float64(-0.25 * Float64(re * re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) + exp(im);
	tmp = 0.0;
	if (im <= 4.2e-7)
		tmp = cos(re);
	elseif (im <= 9e+127)
		tmp = 0.5 * t_0;
	else
		tmp = t_0 * (0.5 + (-0.25 * (re * re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 4.2e-7], N[Cos[re], $MachinePrecision], If[LessEqual[im, 9e+127], N[(0.5 * t$95$0), $MachinePrecision], N[(t$95$0 * N[(0.5 + N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < 4.2e-7

   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 62.9%

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

   if 4.2e-7 < im < 9.00000000000000068e127

   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 92.9%

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

   if 9.00000000000000068e127 < im

   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 0.0%

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

   4. Simplified 85.0%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.2 \cdot 10^{-7}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 9 \cdot 10^{+127}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]

Alternative 4: 73.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left(re \cdot -0.5\right)\\ \mathbf{if}\;im \leq 1.95 \cdot 10^{+23} \lor \neg \left(im \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;\mathsf{fma}\left(0.5, im \cdot im, \cos re\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{t_0 \cdot t_0 - {im}^{4} \cdot 0.25}{t_0 - 0.5 \cdot \left(im \cdot im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (* re -0.5))))
   (if (or (<= im 1.95e+23) (not (<= im 1.35e+154)))
     (fma 0.5 (* im im) (cos re))
     (+
      1.0
      (/ (- (* t_0 t_0) (* (pow im 4.0) 0.25)) (- t_0 (* 0.5 (* im im))))))))

C

double code(double re, double im) {
	double t_0 = re * (re * -0.5);
	double tmp;
	if ((im <= 1.95e+23) || !(im <= 1.35e+154)) {
		tmp = fma(0.5, (im * im), cos(re));
	} else {
		tmp = 1.0 + (((t_0 * t_0) - (pow(im, 4.0) * 0.25)) / (t_0 - (0.5 * (im * im))));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(re * Float64(re * -0.5))
	tmp = 0.0
	if ((im <= 1.95e+23) || !(im <= 1.35e+154))
		tmp = fma(0.5, Float64(im * im), cos(re));
	else
		tmp = Float64(1.0 + Float64(Float64(Float64(t_0 * t_0) - Float64((im ^ 4.0) * 0.25)) / Float64(t_0 - Float64(0.5 * Float64(im * im)))));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(re * -0.5), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[im, 1.95e+23], N[Not[LessEqual[im, 1.35e+154]], $MachinePrecision]], N[(0.5 * N[(im * im), $MachinePrecision] + N[Cos[re], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(N[Power[im, 4.0], $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if im < 1.95e23 or 1.35000000000000003e154 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 79.7%

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

   4. Simplified 79.7%

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

   5. Taylor expanded in re around 0 71.2%

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

      1. unpow2 71.2%

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

   7. Simplified 71.2%

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

   if 1.95e23 < im < 1.35000000000000003e154

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 5.3%

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

   4. Simplified 5.3%

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

   5. Taylor expanded in re around 0 4.7%

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

      1. unpow2 4.7%

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

   7. Simplified 4.7%

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

   8. Taylor expanded in re around 0 10.6%

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

      1. +-commutative 10.6%

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

      2. *-commutative 10.6%

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

      3. unpow2 10.6%

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

      4. fma-def 10.6%

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

      5. unpow2 10.6%

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

   10. Simplified 10.6%

       \[\leadsto \color{blue}{1 + \mathsf{fma}\left(re \cdot re, -0.5, 0.5 \cdot \left(im \cdot im\right)\right)} \]
   11. Step-by-step derivation

       1. fma-udef 10.6%

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

       2. flip-+ 23.6%

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

       3. associate-*l* 23.6%

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

       4. associate-*l* 23.6%

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

       5. *-commutative 23.6%

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

       6. *-commutative 23.6%

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

       7. swap-sqr 23.6%

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

       8. pow2 23.6%

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

       9. pow2 23.6%

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

       10. pow-prod-up 23.6%

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

       11. metadata-eval 23.6%

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

       12. metadata-eval 23.6%

           \[\leadsto 1 + \frac{\left(re \cdot \left(re \cdot -0.5\right)\right) \cdot \left(re \cdot \left(re \cdot -0.5\right)\right) - {im}^{4} \cdot \color{blue}{0.25}}{\left(re \cdot re\right) \cdot -0.5 - 0.5 \cdot \left(im \cdot im\right)} \]

       13. associate-*l* 23.6%

           \[\leadsto 1 + \frac{\left(re \cdot \left(re \cdot -0.5\right)\right) \cdot \left(re \cdot \left(re \cdot -0.5\right)\right) - {im}^{4} \cdot 0.25}{\color{blue}{re \cdot \left(re \cdot -0.5\right)} - 0.5 \cdot \left(im \cdot im\right)} \]

   12. Applied egg-rr 23.6%

       \[\leadsto 1 + \color{blue}{\frac{\left(re \cdot \left(re \cdot -0.5\right)\right) \cdot \left(re \cdot \left(re \cdot -0.5\right)\right) - {im}^{4} \cdot 0.25}{re \cdot \left(re \cdot -0.5\right) - 0.5 \cdot \left(im \cdot im\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.95 \cdot 10^{+23} \lor \neg \left(im \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;\mathsf{fma}\left(0.5, im \cdot im, \cos re\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\left(re \cdot \left(re \cdot -0.5\right)\right) \cdot \left(re \cdot \left(re \cdot -0.5\right)\right) - {im}^{4} \cdot 0.25}{re \cdot \left(re \cdot -0.5\right) - 0.5 \cdot \left(im \cdot im\right)}\\ \end{array} \]

Alternative 5: 69.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 4.2 \cdot 10^{-7}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 4.2e-7) (cos re) (* 0.5 (+ (exp (- im)) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 4.2e-7) {
		tmp = cos(re);
	} else {
		tmp = 0.5 * (exp(-im) + 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 <= 4.2d-7) then
        tmp = cos(re)
    else
        tmp = 0.5d0 * (exp(-im) + exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 4.2e-7) {
		tmp = Math.cos(re);
	} else {
		tmp = 0.5 * (Math.exp(-im) + Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 4.2e-7:
		tmp = math.cos(re)
	else:
		tmp = 0.5 * (math.exp(-im) + math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 4.2e-7)
		tmp = cos(re);
	else
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 4.2e-7)
		tmp = cos(re);
	else
		tmp = 0.5 * (exp(-im) + exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 4.2e-7], N[Cos[re], $MachinePrecision], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 4.2e-7

   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 62.9%

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

   if 4.2e-7 < im

   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 72.1%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.2 \cdot 10^{-7}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \end{array} \]

Alternative 6: 64.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left(re \cdot -0.5\right)\\ t_1 := 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{if}\;im \leq 2.6 \cdot 10^{+25}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;1 + \frac{t_0 \cdot t_0 - {im}^{4} \cdot 0.25}{t_0 - t_1}\\ \mathbf{else}:\\ \;\;\;\;1 + t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (* re -0.5))) (t_1 (* 0.5 (* im im))))
   (if (<= im 2.6e+25)
     (cos re)
     (if (<= im 1.35e+154)
       (+ 1.0 (/ (- (* t_0 t_0) (* (pow im 4.0) 0.25)) (- t_0 t_1)))
       (+ 1.0 t_1)))))

C

double code(double re, double im) {
	double t_0 = re * (re * -0.5);
	double t_1 = 0.5 * (im * im);
	double tmp;
	if (im <= 2.6e+25) {
		tmp = cos(re);
	} else if (im <= 1.35e+154) {
		tmp = 1.0 + (((t_0 * t_0) - (pow(im, 4.0) * 0.25)) / (t_0 - t_1));
	} else {
		tmp = 1.0 + 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 = re * (re * (-0.5d0))
    t_1 = 0.5d0 * (im * im)
    if (im <= 2.6d+25) then
        tmp = cos(re)
    else if (im <= 1.35d+154) then
        tmp = 1.0d0 + (((t_0 * t_0) - ((im ** 4.0d0) * 0.25d0)) / (t_0 - t_1))
    else
        tmp = 1.0d0 + t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re * (re * -0.5);
	double t_1 = 0.5 * (im * im);
	double tmp;
	if (im <= 2.6e+25) {
		tmp = Math.cos(re);
	} else if (im <= 1.35e+154) {
		tmp = 1.0 + (((t_0 * t_0) - (Math.pow(im, 4.0) * 0.25)) / (t_0 - t_1));
	} else {
		tmp = 1.0 + t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * (re * -0.5)
	t_1 = 0.5 * (im * im)
	tmp = 0
	if im <= 2.6e+25:
		tmp = math.cos(re)
	elif im <= 1.35e+154:
		tmp = 1.0 + (((t_0 * t_0) - (math.pow(im, 4.0) * 0.25)) / (t_0 - t_1))
	else:
		tmp = 1.0 + t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(re * -0.5))
	t_1 = Float64(0.5 * Float64(im * im))
	tmp = 0.0
	if (im <= 2.6e+25)
		tmp = cos(re);
	elseif (im <= 1.35e+154)
		tmp = Float64(1.0 + Float64(Float64(Float64(t_0 * t_0) - Float64((im ^ 4.0) * 0.25)) / Float64(t_0 - t_1)));
	else
		tmp = Float64(1.0 + t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * (re * -0.5);
	t_1 = 0.5 * (im * im);
	tmp = 0.0;
	if (im <= 2.6e+25)
		tmp = cos(re);
	elseif (im <= 1.35e+154)
		tmp = 1.0 + (((t_0 * t_0) - ((im ^ 4.0) * 0.25)) / (t_0 - t_1));
	else
		tmp = 1.0 + t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(re * -0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 2.6e+25], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(1.0 + N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(N[Power[im, 4.0], $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if im < 2.5999999999999998e25

   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 61.5%

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

   if 2.5999999999999998e25 < im < 1.35000000000000003e154

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 5.4%

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

   4. Simplified 5.4%

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

   5. Taylor expanded in re around 0 4.8%

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

      1. unpow2 4.8%

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

   7. Simplified 4.8%

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

   8. Taylor expanded in re around 0 10.8%

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

      1. +-commutative 10.8%

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

      2. *-commutative 10.8%

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

      3. unpow2 10.8%

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

      4. fma-def 10.8%

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

      5. unpow2 10.8%

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

   10. Simplified 10.8%

       \[\leadsto \color{blue}{1 + \mathsf{fma}\left(re \cdot re, -0.5, 0.5 \cdot \left(im \cdot im\right)\right)} \]
   11. Step-by-step derivation

       1. fma-udef 10.8%

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

       2. flip-+ 24.3%

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

       3. associate-*l* 24.3%

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

       4. associate-*l* 24.3%

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

       5. *-commutative 24.3%

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

       6. *-commutative 24.3%

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

       7. swap-sqr 24.3%

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

       8. pow2 24.3%

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

       9. pow2 24.3%

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

       10. pow-prod-up 24.3%

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

       11. metadata-eval 24.3%

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

       12. metadata-eval 24.3%

           \[\leadsto 1 + \frac{\left(re \cdot \left(re \cdot -0.5\right)\right) \cdot \left(re \cdot \left(re \cdot -0.5\right)\right) - {im}^{4} \cdot \color{blue}{0.25}}{\left(re \cdot re\right) \cdot -0.5 - 0.5 \cdot \left(im \cdot im\right)} \]

       13. associate-*l* 24.3%

           \[\leadsto 1 + \frac{\left(re \cdot \left(re \cdot -0.5\right)\right) \cdot \left(re \cdot \left(re \cdot -0.5\right)\right) - {im}^{4} \cdot 0.25}{\color{blue}{re \cdot \left(re \cdot -0.5\right)} - 0.5 \cdot \left(im \cdot im\right)} \]

   12. Applied egg-rr 24.3%

       \[\leadsto 1 + \color{blue}{\frac{\left(re \cdot \left(re \cdot -0.5\right)\right) \cdot \left(re \cdot \left(re \cdot -0.5\right)\right) - {im}^{4} \cdot 0.25}{re \cdot \left(re \cdot -0.5\right) - 0.5 \cdot \left(im \cdot im\right)}} \]

   if 1.35000000000000003e154 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in re around 0 55.6%

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

      1. unpow2 55.6%

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

   7. Simplified 55.6%

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

   8. Taylor expanded in re around 0 47.2%

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

      1. +-commutative 47.2%

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

      2. *-commutative 47.2%

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

      3. unpow2 47.2%

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

      4. fma-def 47.2%

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

      5. unpow2 47.2%

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

   10. Simplified 47.2%

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

   11. Taylor expanded in re around 0 55.6%

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

       1. unpow2 55.6%

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

   13. Simplified 55.6%

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

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.6 \cdot 10^{+25}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;1 + \frac{\left(re \cdot \left(re \cdot -0.5\right)\right) \cdot \left(re \cdot \left(re \cdot -0.5\right)\right) - {im}^{4} \cdot 0.25}{re \cdot \left(re \cdot -0.5\right) - 0.5 \cdot \left(im \cdot im\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \end{array} \]

Alternative 7: 62.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 720:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 3.5 \cdot 10^{+151}:\\ \;\;\;\;0.25 + \left(re \cdot re\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 720.0)
   (cos re)
   (if (<= im 3.5e+151)
     (+ 0.25 (* (* re re) 0.25))
     (+ 1.0 (* 0.5 (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 720.0) {
		tmp = cos(re);
	} else if (im <= 3.5e+151) {
		tmp = 0.25 + ((re * re) * 0.25);
	} else {
		tmp = 1.0 + (0.5 * (im * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 720.0d0) then
        tmp = cos(re)
    else if (im <= 3.5d+151) then
        tmp = 0.25d0 + ((re * re) * 0.25d0)
    else
        tmp = 1.0d0 + (0.5d0 * (im * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 720.0) {
		tmp = Math.cos(re);
	} else if (im <= 3.5e+151) {
		tmp = 0.25 + ((re * re) * 0.25);
	} else {
		tmp = 1.0 + (0.5 * (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 720.0:
		tmp = math.cos(re)
	elif im <= 3.5e+151:
		tmp = 0.25 + ((re * re) * 0.25)
	else:
		tmp = 1.0 + (0.5 * (im * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 720.0)
		tmp = cos(re);
	elseif (im <= 3.5e+151)
		tmp = Float64(0.25 + Float64(Float64(re * re) * 0.25));
	else
		tmp = Float64(1.0 + Float64(0.5 * Float64(im * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 720.0)
		tmp = cos(re);
	elseif (im <= 3.5e+151)
		tmp = 0.25 + ((re * re) * 0.25);
	else
		tmp = 1.0 + (0.5 * (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 720.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 3.5e+151], N[(0.25 + N[(N[(re * re), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 720

   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 63.0%

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

   if 720 < im < 3.5000000000000003e151

   1. Initial program 100.0%

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

   3. Applied egg-rr 2.9%

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

   4. Taylor expanded in re around 0 22.4%

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

      1. *-commutative 22.4%

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

      2. unpow2 22.4%

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

   6. Simplified 22.4%

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

   if 3.5000000000000003e151 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in re around 0 55.6%

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

      1. unpow2 55.6%

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

   7. Simplified 55.6%

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

   8. Taylor expanded in re around 0 47.2%

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

      1. +-commutative 47.2%

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

      2. *-commutative 47.2%

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

      3. unpow2 47.2%

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

      4. fma-def 47.2%

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

      5. unpow2 47.2%

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

   10. Simplified 47.2%

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

   11. Taylor expanded in re around 0 55.6%

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

       1. unpow2 55.6%

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

   13. Simplified 55.6%

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

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 720:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 3.5 \cdot 10^{+151}:\\ \;\;\;\;0.25 + \left(re \cdot re\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \end{array} \]

Alternative 8: 47.7% accurate, 23.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 5.8 \cdot 10^{+165}:\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 6.5 \cdot 10^{+238} \lor \neg \left(re \leq 1.8 \cdot 10^{+290}\right):\\ \;\;\;\;1 + \left(re \cdot re\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;0.25 + \left(re \cdot re\right) \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 5.8e+165)
   (+ 1.0 (* 0.5 (* im im)))
   (if (or (<= re 6.5e+238) (not (<= re 1.8e+290)))
     (+ 1.0 (* (* re re) -0.5))
     (+ 0.25 (* (* re re) 0.25)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 5.8e+165) {
		tmp = 1.0 + (0.5 * (im * im));
	} else if ((re <= 6.5e+238) || !(re <= 1.8e+290)) {
		tmp = 1.0 + ((re * re) * -0.5);
	} else {
		tmp = 0.25 + ((re * re) * 0.25);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 5.8d+165) then
        tmp = 1.0d0 + (0.5d0 * (im * im))
    else if ((re <= 6.5d+238) .or. (.not. (re <= 1.8d+290))) then
        tmp = 1.0d0 + ((re * re) * (-0.5d0))
    else
        tmp = 0.25d0 + ((re * re) * 0.25d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 5.8e+165) {
		tmp = 1.0 + (0.5 * (im * im));
	} else if ((re <= 6.5e+238) || !(re <= 1.8e+290)) {
		tmp = 1.0 + ((re * re) * -0.5);
	} else {
		tmp = 0.25 + ((re * re) * 0.25);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 5.8e+165:
		tmp = 1.0 + (0.5 * (im * im))
	elif (re <= 6.5e+238) or not (re <= 1.8e+290):
		tmp = 1.0 + ((re * re) * -0.5)
	else:
		tmp = 0.25 + ((re * re) * 0.25)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 5.8e+165)
		tmp = Float64(1.0 + Float64(0.5 * Float64(im * im)));
	elseif ((re <= 6.5e+238) || !(re <= 1.8e+290))
		tmp = Float64(1.0 + Float64(Float64(re * re) * -0.5));
	else
		tmp = Float64(0.25 + Float64(Float64(re * re) * 0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 5.8e+165)
		tmp = 1.0 + (0.5 * (im * im));
	elseif ((re <= 6.5e+238) || ~((re <= 1.8e+290)))
		tmp = 1.0 + ((re * re) * -0.5);
	else
		tmp = 0.25 + ((re * re) * 0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 5.8e+165], N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, 6.5e+238], N[Not[LessEqual[re, 1.8e+290]], $MachinePrecision]], N[(1.0 + N[(N[(re * re), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], N[(0.25 + N[(N[(re * re), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < 5.80000000000000011e165

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 71.2%

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

   4. Simplified 71.2%

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

   5. Taylor expanded in re around 0 65.5%

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

      1. unpow2 65.5%

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

   7. Simplified 65.5%

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

   8. Taylor expanded in re around 0 49.0%

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

      1. +-commutative 49.0%

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

      2. *-commutative 49.0%

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

      3. unpow2 49.0%

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

      4. fma-def 49.0%

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

      5. unpow2 49.0%

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

   10. Simplified 49.0%

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

   11. Taylor expanded in re around 0 49.1%

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

       1. unpow2 49.1%

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

   13. Simplified 49.1%

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

   if 5.80000000000000011e165 < re < 6.5000000000000005e238 or 1.79999999999999994e290 < re

   1. Initial program 99.9%

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

   2. Taylor expanded in im around 0 92.9%

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

   4. Simplified 92.9%

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

   5. Taylor expanded in re around 0 61.0%

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

      1. unpow2 61.0%

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

   7. Simplified 61.0%

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

   8. Taylor expanded in re around 0 9.2%

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

      1. +-commutative 9.2%

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

      2. *-commutative 9.2%

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

      3. unpow2 9.2%

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

      4. fma-def 9.2%

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

      5. unpow2 9.2%

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

   10. Simplified 9.2%

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

   11. Taylor expanded in re around inf 40.0%

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

       1. unpow2 40.0%

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

   13. Simplified 40.0%

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

   if 6.5000000000000005e238 < re < 1.79999999999999994e290

   1. Initial program 100.0%

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

   3. Applied egg-rr 4.4%

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

   4. Taylor expanded in re around 0 56.0%

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

      1. *-commutative 56.0%

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

      2. unpow2 56.0%

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

   6. Simplified 56.0%

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

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 5.8 \cdot 10^{+165}:\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;re \leq 6.5 \cdot 10^{+238} \lor \neg \left(re \leq 1.8 \cdot 10^{+290}\right):\\ \;\;\;\;1 + \left(re \cdot re\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;0.25 + \left(re \cdot re\right) \cdot 0.25\\ \end{array} \]

Alternative 9: 40.2% accurate, 27.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 24.5:\\ \;\;\;\;1 + \left(re \cdot re\right) \cdot -0.5\\ \mathbf{elif}\;im \leq 2.8 \cdot 10^{+151}:\\ \;\;\;\;0.25 + \left(re \cdot re\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 24.5)
   (+ 1.0 (* (* re re) -0.5))
   (if (<= im 2.8e+151) (+ 0.25 (* (* re re) 0.25)) (* 0.5 (* im im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 24.5) {
		tmp = 1.0 + ((re * re) * -0.5);
	} else if (im <= 2.8e+151) {
		tmp = 0.25 + ((re * re) * 0.25);
	} else {
		tmp = 0.5 * (im * im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 24.5d0) then
        tmp = 1.0d0 + ((re * re) * (-0.5d0))
    else if (im <= 2.8d+151) then
        tmp = 0.25d0 + ((re * re) * 0.25d0)
    else
        tmp = 0.5d0 * (im * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 24.5) {
		tmp = 1.0 + ((re * re) * -0.5);
	} else if (im <= 2.8e+151) {
		tmp = 0.25 + ((re * re) * 0.25);
	} else {
		tmp = 0.5 * (im * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 24.5:
		tmp = 1.0 + ((re * re) * -0.5)
	elif im <= 2.8e+151:
		tmp = 0.25 + ((re * re) * 0.25)
	else:
		tmp = 0.5 * (im * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 24.5)
		tmp = Float64(1.0 + Float64(Float64(re * re) * -0.5));
	elseif (im <= 2.8e+151)
		tmp = Float64(0.25 + Float64(Float64(re * re) * 0.25));
	else
		tmp = Float64(0.5 * Float64(im * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 24.5)
		tmp = 1.0 + ((re * re) * -0.5);
	elseif (im <= 2.8e+151)
		tmp = 0.25 + ((re * re) * 0.25);
	else
		tmp = 0.5 * (im * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 24.5], N[(1.0 + N[(N[(re * re), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.8e+151], N[(0.25 + N[(N[(re * re), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 24.5

   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 77.5%

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

   4. Simplified 77.5%

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

   5. Taylor expanded in re around 0 75.6%

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

      1. unpow2 75.6%

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

   7. Simplified 75.6%

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

   8. Taylor expanded in re around 0 50.8%

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

      1. +-commutative 50.8%

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

      2. *-commutative 50.8%

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

      3. unpow2 50.8%

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

      4. fma-def 50.8%

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

      5. unpow2 50.8%

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

   10. Simplified 50.8%

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

   11. Taylor expanded in re around inf 39.2%

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

       1. unpow2 39.2%

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

   13. Simplified 39.2%

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

   if 24.5 < im < 2.79999999999999987e151

   1. Initial program 100.0%

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

   3. Applied egg-rr 2.9%

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

   4. Taylor expanded in re around 0 22.4%

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

      1. *-commutative 22.4%

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

      2. unpow2 22.4%

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

   6. Simplified 22.4%

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

   if 2.79999999999999987e151 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in re around 0 55.6%

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

      1. unpow2 55.6%

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

   7. Simplified 55.6%

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

   8. Taylor expanded in im around inf 55.6%

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

      1. unpow2 55.6%

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

   10. Simplified 55.6%

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

4. Final simplification 39.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 24.5:\\ \;\;\;\;1 + \left(re \cdot re\right) \cdot -0.5\\ \mathbf{elif}\;im \leq 2.8 \cdot 10^{+151}:\\ \;\;\;\;0.25 + \left(re \cdot re\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot im\right)\\ \end{array} \]

Alternative 10: 20.0% accurate, 34.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 7.5 \cdot 10^{+150}:\\ \;\;\;\;0.25 + \left(re \cdot re\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 7.5e+150) (+ 0.25 (* (* re re) 0.25)) (* 0.5 (* im im))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 7.5e+150) {
		tmp = 0.25 + ((re * re) * 0.25);
	} else {
		tmp = 0.5 * (im * im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 7.5d+150) then
        tmp = 0.25d0 + ((re * re) * 0.25d0)
    else
        tmp = 0.5d0 * (im * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 7.5e+150) {
		tmp = 0.25 + ((re * re) * 0.25);
	} else {
		tmp = 0.5 * (im * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 7.5e+150:
		tmp = 0.25 + ((re * re) * 0.25)
	else:
		tmp = 0.5 * (im * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 7.5e+150)
		tmp = Float64(0.25 + Float64(Float64(re * re) * 0.25));
	else
		tmp = Float64(0.5 * Float64(im * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 7.5e+150)
		tmp = 0.25 + ((re * re) * 0.25);
	else
		tmp = 0.5 * (im * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 7.5e+150], N[(0.25 + N[(N[(re * re), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 7.4999999999999998e150

   1. Initial program 100.0%

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

   3. Applied egg-rr 8.6%

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

   4. Taylor expanded in re around 0 12.5%

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

      1. *-commutative 12.5%

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

      2. unpow2 12.5%

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

   6. Simplified 12.5%

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

   if 7.4999999999999998e150 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in re around 0 55.6%

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

      1. unpow2 55.6%

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

   7. Simplified 55.6%

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

   8. Taylor expanded in im around inf 55.6%

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

      1. unpow2 55.6%

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

   10. Simplified 55.6%

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

4. Final simplification 18.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 7.5 \cdot 10^{+150}:\\ \;\;\;\;0.25 + \left(re \cdot re\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot im\right)\\ \end{array} \]

Alternative 11: 17.4% accurate, 43.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.71:\\ \;\;\;\;0.25\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 0.71) 0.25 (* 0.5 (* im im))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.71) {
		tmp = 0.25;
	} else {
		tmp = 0.5 * (im * im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.71d0) then
        tmp = 0.25d0
    else
        tmp = 0.5d0 * (im * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.71) {
		tmp = 0.25;
	} else {
		tmp = 0.5 * (im * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.71:
		tmp = 0.25
	else:
		tmp = 0.5 * (im * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.71)
		tmp = 0.25;
	else
		tmp = Float64(0.5 * Float64(im * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.71)
		tmp = 0.25;
	else
		tmp = 0.5 * (im * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.71], 0.25, N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 0.70999999999999996

   1. Initial program 100.0%

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

   3. Applied egg-rr 9.6%

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

   4. Taylor expanded in re around 0 9.6%

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

   if 0.70999999999999996 < 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 56.1%

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

   4. Simplified 56.1%

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

   5. Taylor expanded in re around 0 32.0%

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

      1. unpow2 32.0%

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

   7. Simplified 32.0%

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

   8. Taylor expanded in im around inf 32.0%

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

      1. unpow2 32.0%

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

   10. Simplified 32.0%

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

4. Final simplification 15.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.71:\\ \;\;\;\;0.25\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot im\right)\\ \end{array} \]

Alternative 12: 8.2% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ 0.25 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.25)

C

double code(double re, double im) {
	return 0.25;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.25d0
end function

Java

public static double code(double re, double im) {
	return 0.25;
}

Python

def code(re, im):
	return 0.25

Julia

function code(re, im)
	return 0.25
end

MATLAB

function tmp = code(re, im)
	tmp = 0.25;
end

Wolfram

code[re_, im_] := 0.25

Derivation

1. Initial program 100.0%

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

3. Applied egg-rr 7.7%

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

4. Taylor expanded in re around 0 7.7%

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

5. Final simplification 7.7%

   \[\leadsto 0.25 \]

Reproduce

herbie shell --seed 2023257 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))