math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 6.5s

Alternatives: 13

Speedup: 1.0×

• 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, 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: 83.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.0134 \lor \neg \left(im \leq 1.5 \cdot 10^{+154}\right):\\ \;\;\;\;\mathsf{fma}\left(0.5, \cos re \cdot \left(im \cdot im\right), \cos re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 0.0134) (not (<= im 1.5e+154)))
   (fma 0.5 (* (cos re) (* im im)) (cos re))
   (* (+ (exp (- im)) (exp im)) (+ 0.5 (* -0.25 (* re re))))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 0.0134) || !(im <= 1.5e+154)) {
		tmp = fma(0.5, (cos(re) * (im * im)), cos(re));
	} else {
		tmp = (exp(-im) + exp(im)) * (0.5 + (-0.25 * (re * re)));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 0.0134) || !(im <= 1.5e+154))
		tmp = fma(0.5, Float64(cos(re) * Float64(im * im)), cos(re));
	else
		tmp = Float64(Float64(exp(Float64(-im)) + exp(im)) * Float64(0.5 + Float64(-0.25 * Float64(re * re))));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 0.0134], N[Not[LessEqual[im, 1.5e+154]], $MachinePrecision]], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] + N[Cos[re], $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 0.0134000000000000005 or 1.50000000000000013e154 < 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 83.5%

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

   4. Simplified 83.5%

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

   if 0.0134000000000000005 < im < 1.50000000000000013e154

   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 3.2%

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

   4. Simplified 72.0%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.0134 \lor \neg \left(im \leq 1.5 \cdot 10^{+154}\right):\\ \;\;\;\;\mathsf{fma}\left(0.5, \cos re \cdot \left(im \cdot im\right), \cos re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]

Alternative 3: 77.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} + e^{im}\\ \mathbf{if}\;im \leq 0.0065:\\ \;\;\;\;\cos re + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;im \leq 6.6 \cdot 10^{+122} \lor \neg \left(im \leq 2 \cdot 10^{+198}\right):\\ \;\;\;\;t_0 \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ (exp (- im)) (exp im))))
   (if (<= im 0.0065)
     (+ (cos re) (* 0.5 (* im im)))
     (if (or (<= im 6.6e+122) (not (<= im 2e+198)))
       (* t_0 (+ 0.5 (* -0.25 (* re re))))
       (* 0.5 t_0)))))

C

double code(double re, double im) {
	double t_0 = exp(-im) + exp(im);
	double tmp;
	if (im <= 0.0065) {
		tmp = cos(re) + (0.5 * (im * im));
	} else if ((im <= 6.6e+122) || !(im <= 2e+198)) {
		tmp = t_0 * (0.5 + (-0.25 * (re * re)));
	} else {
		tmp = 0.5 * 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 = exp(-im) + exp(im)
    if (im <= 0.0065d0) then
        tmp = cos(re) + (0.5d0 * (im * im))
    else if ((im <= 6.6d+122) .or. (.not. (im <= 2d+198))) then
        tmp = t_0 * (0.5d0 + ((-0.25d0) * (re * re)))
    else
        tmp = 0.5d0 * t_0
    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 <= 0.0065) {
		tmp = Math.cos(re) + (0.5 * (im * im));
	} else if ((im <= 6.6e+122) || !(im <= 2e+198)) {
		tmp = t_0 * (0.5 + (-0.25 * (re * re)));
	} else {
		tmp = 0.5 * t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) + math.exp(im)
	tmp = 0
	if im <= 0.0065:
		tmp = math.cos(re) + (0.5 * (im * im))
	elif (im <= 6.6e+122) or not (im <= 2e+198):
		tmp = t_0 * (0.5 + (-0.25 * (re * re)))
	else:
		tmp = 0.5 * t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) + exp(im))
	tmp = 0.0
	if (im <= 0.0065)
		tmp = Float64(cos(re) + Float64(0.5 * Float64(im * im)));
	elseif ((im <= 6.6e+122) || !(im <= 2e+198))
		tmp = Float64(t_0 * Float64(0.5 + Float64(-0.25 * Float64(re * re))));
	else
		tmp = Float64(0.5 * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) + exp(im);
	tmp = 0.0;
	if (im <= 0.0065)
		tmp = cos(re) + (0.5 * (im * im));
	elseif ((im <= 6.6e+122) || ~((im <= 2e+198)))
		tmp = t_0 * (0.5 + (-0.25 * (re * re)));
	else
		tmp = 0.5 * t_0;
	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, 0.0065], N[(N[Cos[re], $MachinePrecision] + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[im, 6.6e+122], N[Not[LessEqual[im, 2e+198]], $MachinePrecision]], N[(t$95$0 * N[(0.5 + N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.0064999999999999997

   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 80.8%

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

   4. Simplified 80.8%

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

   5. Taylor expanded in re around 0 78.3%

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

      1. unpow2 78.3%

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

   7. Simplified 78.3%

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

      1. fma-udef 78.3%

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

   9. Applied egg-rr 78.3%

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

   if 0.0064999999999999997 < im < 6.5999999999999998e122 or 2.00000000000000004e198 < 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 2.2%

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

   4. Simplified 85.5%

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

   if 6.5999999999999998e122 < im < 2.00000000000000004e198

   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 75.0%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.0065:\\ \;\;\;\;\cos re + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;im \leq 6.6 \cdot 10^{+122} \lor \neg \left(im \leq 2 \cdot 10^{+198}\right):\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \end{array} \]

Alternative 4: 77.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.00185:\\ \;\;\;\;\cos re + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.00185)
   (+ (cos re) (* 0.5 (* im im)))
   (* 0.5 (+ (exp (- im)) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.00185) {
		tmp = cos(re) + (0.5 * (im * im));
	} 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 <= 0.00185d0) then
        tmp = cos(re) + (0.5d0 * (im * im))
    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 <= 0.00185) {
		tmp = Math.cos(re) + (0.5 * (im * im));
	} else {
		tmp = 0.5 * (Math.exp(-im) + Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.00185:
		tmp = math.cos(re) + (0.5 * (im * im))
	else:
		tmp = 0.5 * (math.exp(-im) + math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.00185)
		tmp = Float64(cos(re) + Float64(0.5 * Float64(im * im)));
	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 <= 0.00185)
		tmp = cos(re) + (0.5 * (im * im));
	else
		tmp = 0.5 * (exp(-im) + exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.00185], N[(N[Cos[re], $MachinePrecision] + N[(0.5 * N[(im * im), $MachinePrecision]), $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 < 0.0018500000000000001

   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 80.8%

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

   4. Simplified 80.8%

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

   5. Taylor expanded in re around 0 78.3%

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

      1. unpow2 78.3%

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

   7. Simplified 78.3%

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

      1. fma-udef 78.3%

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

   9. Applied egg-rr 78.3%

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

   if 0.0018500000000000001 < 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 68.8%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.00185:\\ \;\;\;\;\cos re + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \end{array} \]

Alternative 5: 77.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{if}\;im \leq 380:\\ \;\;\;\;\cos re + t_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(t_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (* im im))))
   (if (<= im 380.0) (+ (cos re) t_0) (log1p (expm1 t_0)))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (im * im);
	double tmp;
	if (im <= 380.0) {
		tmp = cos(re) + t_0;
	} else {
		tmp = log1p(expm1(t_0));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * (im * im);
	double tmp;
	if (im <= 380.0) {
		tmp = Math.cos(re) + t_0;
	} else {
		tmp = Math.log1p(Math.expm1(t_0));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * (im * im)
	tmp = 0
	if im <= 380.0:
		tmp = math.cos(re) + t_0
	else:
		tmp = math.log1p(math.expm1(t_0))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(im * im))
	tmp = 0.0
	if (im <= 380.0)
		tmp = Float64(cos(re) + t_0);
	else
		tmp = log1p(expm1(t_0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 380.0], N[(N[Cos[re], $MachinePrecision] + t$95$0), $MachinePrecision], N[Log[1 + N[(Exp[t$95$0] - 1), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if im < 380

   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 80.3%

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

   4. Simplified 80.3%

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

   5. Taylor expanded in re around 0 77.7%

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

      1. unpow2 77.7%

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

   7. Simplified 77.7%

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

      1. fma-udef 77.7%

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

   9. Applied egg-rr 77.7%

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

   if 380 < 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 54.3%

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

   4. Simplified 54.3%

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

   5. Taylor expanded in re around 0 37.3%

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

      1. unpow2 37.3%

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

   7. Simplified 37.3%

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

   8. Taylor expanded in im around inf 37.3%

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

      1. unpow2 37.3%

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

   10. Simplified 37.3%

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

       1. log1p-expm1-u 69.4%

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

   12. Applied egg-rr 69.4%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(0.5 \cdot \left(im \cdot im\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 380:\\ \;\;\;\;\cos re + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(0.5 \cdot \left(im \cdot im\right)\right)\right)\\ \end{array} \]

Alternative 6: 74.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 10^{+51}:\\ \;\;\;\;\cos re + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{0.125 \cdot {im}^{6}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1e+51)
   (+ (cos re) (* 0.5 (* im im)))
   (cbrt (* 0.125 (pow im 6.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1e+51) {
		tmp = cos(re) + (0.5 * (im * im));
	} else {
		tmp = cbrt((0.125 * pow(im, 6.0)));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1e+51) {
		tmp = Math.cos(re) + (0.5 * (im * im));
	} else {
		tmp = Math.cbrt((0.125 * Math.pow(im, 6.0)));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1e+51)
		tmp = Float64(cos(re) + Float64(0.5 * Float64(im * im)));
	else
		tmp = cbrt(Float64(0.125 * (im ^ 6.0)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1e+51], N[(N[Cos[re], $MachinePrecision] + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(0.125 * N[Power[im, 6.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1e51

   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 75.4%

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

   4. Simplified 75.4%

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

   5. Taylor expanded in re around 0 73.1%

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

      1. unpow2 73.1%

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

   7. Simplified 73.1%

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

      1. fma-udef 73.1%

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

   9. Applied egg-rr 73.1%

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

   if 1e51 < 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 67.8%

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

   4. Simplified 67.8%

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

   5. Taylor expanded in re around 0 46.4%

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

      1. unpow2 46.4%

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

   7. Simplified 46.4%

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

   8. Taylor expanded in im around inf 46.4%

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

      1. unpow2 46.4%

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

   10. Simplified 46.4%

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

       1. add-cbrt-cube 65.3%

          \[\leadsto \color{blue}{\sqrt[3]{\left(\left(0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\right) \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)}} \]

       2. pow1/3 65.3%

          \[\leadsto \color{blue}{{\left(\left(\left(0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\right) \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\right)}^{0.3333333333333333}} \]

       3. pow3 65.3%

          \[\leadsto {\color{blue}{\left({\left(0.5 \cdot \left(im \cdot im\right)\right)}^{3}\right)}}^{0.3333333333333333} \]

       4. unpow-prod-down 65.3%

          \[\leadsto {\color{blue}{\left({0.5}^{3} \cdot {\left(im \cdot im\right)}^{3}\right)}}^{0.3333333333333333} \]

       5. metadata-eval 65.3%

          \[\leadsto {\left(\color{blue}{0.125} \cdot {\left(im \cdot im\right)}^{3}\right)}^{0.3333333333333333} \]

       6. pow-prod-down 65.3%

          \[\leadsto {\left(0.125 \cdot \color{blue}{\left({im}^{3} \cdot {im}^{3}\right)}\right)}^{0.3333333333333333} \]

       7. pow-prod-up 65.3%

          \[\leadsto {\left(0.125 \cdot \color{blue}{{im}^{\left(3 + 3\right)}}\right)}^{0.3333333333333333} \]

       8. metadata-eval 65.3%

          \[\leadsto {\left(0.125 \cdot {im}^{\color{blue}{6}}\right)}^{0.3333333333333333} \]

   12. Applied egg-rr 65.3%

       \[\leadsto \color{blue}{{\left(0.125 \cdot {im}^{6}\right)}^{0.3333333333333333}} \]
   13. Step-by-step derivation

       1. unpow1/3 65.3%

          \[\leadsto \color{blue}{\sqrt[3]{0.125 \cdot {im}^{6}}} \]

   14. Simplified 65.3%

       \[\leadsto \color{blue}{\sqrt[3]{0.125 \cdot {im}^{6}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 10^{+51}:\\ \;\;\;\;\cos re + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{0.125 \cdot {im}^{6}}\\ \end{array} \]

Alternative 7: 69.1% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in im around 0 74.0%

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

4. Simplified 74.0%

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

5. Taylor expanded in re around 0 68.0%

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

   1. unpow2 68.0%

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

7. Simplified 68.0%

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

   1. fma-udef 68.0%

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

9. Applied egg-rr 68.0%

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

10. Final simplification 68.0%

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

Alternative 8: 60.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 680:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{+131}:\\ \;\;\;\;1 + re \cdot \left(re \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot im\right) + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 680.0)
   (cos re)
   (if (<= im 2.6e+131) (+ 1.0 (* re (* re -0.5))) (+ (* 0.5 (* im im)) 1.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 680.0) {
		tmp = cos(re);
	} else if (im <= 2.6e+131) {
		tmp = 1.0 + (re * (re * -0.5));
	} else {
		tmp = (0.5 * (im * im)) + 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 <= 680.0d0) then
        tmp = cos(re)
    else if (im <= 2.6d+131) then
        tmp = 1.0d0 + (re * (re * (-0.5d0)))
    else
        tmp = (0.5d0 * (im * im)) + 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 680.0) {
		tmp = Math.cos(re);
	} else if (im <= 2.6e+131) {
		tmp = 1.0 + (re * (re * -0.5));
	} else {
		tmp = (0.5 * (im * im)) + 1.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 680.0:
		tmp = math.cos(re)
	elif im <= 2.6e+131:
		tmp = 1.0 + (re * (re * -0.5))
	else:
		tmp = (0.5 * (im * im)) + 1.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 680.0)
		tmp = cos(re);
	elseif (im <= 2.6e+131)
		tmp = Float64(1.0 + Float64(re * Float64(re * -0.5)));
	else
		tmp = Float64(Float64(0.5 * Float64(im * im)) + 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 680.0)
		tmp = cos(re);
	elseif (im <= 2.6e+131)
		tmp = 1.0 + (re * (re * -0.5));
	else
		tmp = (0.5 * (im * im)) + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 680.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 2.6e+131], N[(1.0 + N[(re * N[(re * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 680

   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 63.4%

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

   if 680 < im < 2.6e131

   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 4.6%

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

   4. Simplified 4.6%

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

   5. Taylor expanded in re around 0 3.0%

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

      1. unpow2 3.0%

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

   7. Simplified 3.0%

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

   8. Taylor expanded in re around 0 18.8%

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

      1. fma-def 18.8%

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

      2. unpow2 18.8%

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

      3. unpow2 18.8%

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

   10. Simplified 18.8%

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

   11. Taylor expanded in im around 0 18.3%

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

       1. unpow2 18.3%

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

       2. associate-*r* 18.3%

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

       3. *-commutative 18.3%

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

   13. Simplified 18.3%

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

   if 2.6e131 < 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 87.9%

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

   4. Simplified 87.9%

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

   5. Taylor expanded in re around 0 60.6%

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

      1. unpow2 60.6%

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

   7. Simplified 60.6%

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

   8. Taylor expanded in re around 0 60.6%

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

      1. unpow2 60.6%

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

   10. Simplified 60.6%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 680:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{+131}:\\ \;\;\;\;1 + re \cdot \left(re \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot im\right) + 1\\ \end{array} \]

Alternative 9: 48.0% accurate, 27.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.22 \cdot 10^{+14} \lor \neg \left(im \leq 1.95 \cdot 10^{+152}\right):\\ \;\;\;\;0.5 \cdot \left(im \cdot im\right) + 1\\ \mathbf{else}:\\ \;\;\;\;0.25 + \left(re \cdot re\right) \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 1.22e+14) (not (<= im 1.95e+152)))
   (+ (* 0.5 (* im im)) 1.0)
   (+ 0.25 (* (* re re) 0.25))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 1.22e+14) || !(im <= 1.95e+152)) {
		tmp = (0.5 * (im * im)) + 1.0;
	} 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 ((im <= 1.22d+14) .or. (.not. (im <= 1.95d+152))) then
        tmp = (0.5d0 * (im * im)) + 1.0d0
    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 ((im <= 1.22e+14) || !(im <= 1.95e+152)) {
		tmp = (0.5 * (im * im)) + 1.0;
	} else {
		tmp = 0.25 + ((re * re) * 0.25);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= 1.22e+14) or not (im <= 1.95e+152):
		tmp = (0.5 * (im * im)) + 1.0
	else:
		tmp = 0.25 + ((re * re) * 0.25)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 1.22e+14) || !(im <= 1.95e+152))
		tmp = Float64(Float64(0.5 * Float64(im * im)) + 1.0);
	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 ((im <= 1.22e+14) || ~((im <= 1.95e+152)))
		tmp = (0.5 * (im * im)) + 1.0;
	else
		tmp = 0.25 + ((re * re) * 0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 1.22e+14], N[Not[LessEqual[im, 1.95e+152]], $MachinePrecision]], N[(N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(0.25 + N[(N[(re * re), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.22e14 or 1.95000000000000006e152 < 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 82.4%

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

   4. Simplified 82.4%

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

   5. Taylor expanded in re around 0 75.9%

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

      1. unpow2 75.9%

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

   7. Simplified 75.9%

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

   8. Taylor expanded in re around 0 51.3%

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

      1. unpow2 51.3%

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

   10. Simplified 51.3%

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

   if 1.22e14 < im < 1.95000000000000006e152

   1. Initial program 100.0%

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

   3. Applied egg-rr 2.3%

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

   4. Taylor expanded in re around 0 16.3%

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

      1. *-commutative 16.3%

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

      2. unpow2 16.3%

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

   6. Simplified 16.3%

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

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.22 \cdot 10^{+14} \lor \neg \left(im \leq 1.95 \cdot 10^{+152}\right):\\ \;\;\;\;0.5 \cdot \left(im \cdot im\right) + 1\\ \mathbf{else}:\\ \;\;\;\;0.25 + \left(re \cdot re\right) \cdot 0.25\\ \end{array} \]

Alternative 10: 47.7% accurate, 27.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 86 \lor \neg \left(im \leq 4.2 \cdot 10^{+130}\right):\\ \;\;\;\;0.5 \cdot \left(im \cdot im\right) + 1\\ \mathbf{else}:\\ \;\;\;\;1 + re \cdot \left(re \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 86.0) (not (<= im 4.2e+130)))
   (+ (* 0.5 (* im im)) 1.0)
   (+ 1.0 (* re (* re -0.5)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 86.0) || !(im <= 4.2e+130)) {
		tmp = (0.5 * (im * im)) + 1.0;
	} else {
		tmp = 1.0 + (re * (re * -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= 86.0d0) .or. (.not. (im <= 4.2d+130))) then
        tmp = (0.5d0 * (im * im)) + 1.0d0
    else
        tmp = 1.0d0 + (re * (re * (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= 86.0) || !(im <= 4.2e+130)) {
		tmp = (0.5 * (im * im)) + 1.0;
	} else {
		tmp = 1.0 + (re * (re * -0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= 86.0) or not (im <= 4.2e+130):
		tmp = (0.5 * (im * im)) + 1.0
	else:
		tmp = 1.0 + (re * (re * -0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 86.0) || !(im <= 4.2e+130))
		tmp = Float64(Float64(0.5 * Float64(im * im)) + 1.0);
	else
		tmp = Float64(1.0 + Float64(re * Float64(re * -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= 86.0) || ~((im <= 4.2e+130)))
		tmp = (0.5 * (im * im)) + 1.0;
	else
		tmp = 1.0 + (re * (re * -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 86.0], N[Not[LessEqual[im, 4.2e+130]], $MachinePrecision]], N[(N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(1.0 + N[(re * N[(re * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 86 or 4.19999999999999981e130 < 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 81.8%

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

   4. Simplified 81.8%

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

   5. Taylor expanded in re around 0 75.3%

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

      1. unpow2 75.3%

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

   7. Simplified 75.3%

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

   8. Taylor expanded in re around 0 50.9%

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

      1. unpow2 50.9%

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

   10. Simplified 50.9%

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

   if 86 < im < 4.19999999999999981e130

   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 4.6%

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

   4. Simplified 4.6%

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

   5. Taylor expanded in re around 0 2.9%

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

      1. unpow2 2.9%

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

   7. Simplified 2.9%

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

   8. Taylor expanded in re around 0 18.2%

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

      1. fma-def 18.2%

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

      2. unpow2 18.2%

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

      3. unpow2 18.2%

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

   10. Simplified 18.2%

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

   11. Taylor expanded in im around 0 17.7%

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

       1. unpow2 17.7%

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

       2. associate-*r* 17.7%

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

       3. *-commutative 17.7%

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

   13. Simplified 17.7%

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

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 86 \lor \neg \left(im \leq 4.2 \cdot 10^{+130}\right):\\ \;\;\;\;0.5 \cdot \left(im \cdot im\right) + 1\\ \mathbf{else}:\\ \;\;\;\;1 + re \cdot \left(re \cdot -0.5\right)\\ \end{array} \]

Alternative 11: 18.2% accurate, 27.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.22 \cdot 10^{+14}:\\ \;\;\;\;0.25\\ \mathbf{elif}\;im \leq 1.5 \cdot 10^{+152}:\\ \;\;\;\;0.25 + \left(re \cdot re\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.22e+14)
   0.25
   (if (<= im 1.5e+152) (+ 0.25 (* (* re re) 0.25)) (* 0.5 (* im im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.22e+14) {
		tmp = 0.25;
	} else if (im <= 1.5e+152) {
		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 <= 1.22d+14) then
        tmp = 0.25d0
    else if (im <= 1.5d+152) 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 <= 1.22e+14) {
		tmp = 0.25;
	} else if (im <= 1.5e+152) {
		tmp = 0.25 + ((re * re) * 0.25);
	} else {
		tmp = 0.5 * (im * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.22e+14:
		tmp = 0.25
	elif im <= 1.5e+152:
		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 <= 1.22e+14)
		tmp = 0.25;
	elseif (im <= 1.5e+152)
		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 <= 1.22e+14)
		tmp = 0.25;
	elseif (im <= 1.5e+152)
		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, 1.22e+14], 0.25, If[LessEqual[im, 1.5e+152], 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 < 1.22e14

   1. Initial program 100.0%

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

   3. Applied egg-rr 9.5%

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

   4. Taylor expanded in re around 0 9.7%

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

   if 1.22e14 < im < 1.49999999999999995e152

   1. Initial program 100.0%

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

   3. Applied egg-rr 2.3%

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

   4. Taylor expanded in re around 0 16.3%

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

      1. *-commutative 16.3%

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

      2. unpow2 16.3%

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

   6. Simplified 16.3%

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

   if 1.49999999999999995e152 < 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 97.4%

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

   4. Simplified 97.4%

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

   5. Taylor expanded in re around 0 67.1%

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

      1. unpow2 67.1%

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

   7. Simplified 67.1%

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

   8. Taylor expanded in im around inf 67.1%

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

      1. unpow2 67.1%

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

   10. Simplified 67.1%

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

4. Final simplification 17.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.22 \cdot 10^{+14}:\\ \;\;\;\;0.25\\ \mathbf{elif}\;im \leq 1.5 \cdot 10^{+152}:\\ \;\;\;\;0.25 + \left(re \cdot re\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot im\right)\\ \end{array} \]

Alternative 12: 17.0% accurate, 43.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 155:\\ \;\;\;\;0.25\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 155.0) 0.25 (* 0.5 (* im im))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 155.0) {
		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 <= 155.0d0) 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 <= 155.0) {
		tmp = 0.25;
	} else {
		tmp = 0.5 * (im * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 155.0:
		tmp = 0.25
	else:
		tmp = 0.5 * (im * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 155.0)
		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 <= 155.0)
		tmp = 0.25;
	else
		tmp = 0.5 * (im * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 155.0], 0.25, N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 155

   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 9.7%

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

   if 155 < 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 54.3%

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

   4. Simplified 54.3%

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

   5. Taylor expanded in re around 0 37.3%

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

      1. unpow2 37.3%

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

   7. Simplified 37.3%

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

   8. Taylor expanded in im around inf 37.3%

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

      1. unpow2 37.3%

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

   10. Simplified 37.3%

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

4. Final simplification 16.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 155:\\ \;\;\;\;0.25\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot im\right)\\ \end{array} \]

Alternative 13: 8.0% 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 7.8%

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

4. Taylor expanded in re around 0 7.9%

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

5. Final simplification 7.9%

   \[\leadsto 0.25 \]

Reproduce

herbie shell --seed 2023240 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))