math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 5.3s

Alternatives: 13

Speedup: 1.0×

• 49.7% 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. Add Preprocessing

3. Final simplification 100.0%

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

Alternative 2: 81.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.0102:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot im, im, \cos re\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot {im}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.0102)
   (fma (* 0.5 im) im (cos re))
   (if (<= im 1.35e+154)
     (* 0.5 (+ (exp (- im)) (exp im)))
     (* (* 0.5 (cos re)) (pow im 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.0102) {
		tmp = fma((0.5 * im), im, cos(re));
	} else if (im <= 1.35e+154) {
		tmp = 0.5 * (exp(-im) + exp(im));
	} else {
		tmp = (0.5 * cos(re)) * pow(im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.0102)
		tmp = fma(Float64(0.5 * im), im, cos(re));
	elseif (im <= 1.35e+154)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64(Float64(0.5 * cos(re)) * (im ^ 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.0102], N[(N[(0.5 * im), $MachinePrecision] * im + N[Cos[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.010200000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 85.0%

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

   4. Taylor expanded in re around 0 79.0%

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

      1. *-commutative 79.0%

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

   6. Simplified 79.0%

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

      1. +-commutative 79.0%

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

      2. *-commutative 79.0%

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

      3. unpow2 79.0%

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

      4. associate-*r* 79.0%

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

      5. fma-def 79.0%

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

   8. Applied egg-rr 79.0%

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

   if 0.010200000000000001 < im < 1.35000000000000003e154

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 78.1%

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

   if 1.35000000000000003e154 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

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

   4. Taylor expanded in im around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.0102:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot im, im, \cos re\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot {im}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ \mathbf{if}\;im \leq 0.024:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot {im}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re))))
   (if (<= im 0.024)
     (* t_0 (fma im im 2.0))
     (if (<= im 1.35e+154)
       (* 0.5 (+ (exp (- im)) (exp im)))
       (* t_0 (pow im 2.0))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * cos(re);
	double tmp;
	if (im <= 0.024) {
		tmp = t_0 * fma(im, im, 2.0);
	} else if (im <= 1.35e+154) {
		tmp = 0.5 * (exp(-im) + exp(im));
	} else {
		tmp = t_0 * pow(im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * cos(re))
	tmp = 0.0
	if (im <= 0.024)
		tmp = Float64(t_0 * fma(im, im, 2.0));
	elseif (im <= 1.35e+154)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64(t_0 * (im ^ 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 0.024], N[(t$95$0 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.024

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 85.0%

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

   5. Simplified 85.0%

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

   if 0.024 < im < 1.35000000000000003e154

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 78.1%

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

   if 1.35000000000000003e154 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

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

   4. Taylor expanded in im around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.024:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot {im}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.0072:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot im, im, \cos re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.0072)
   (fma (* 0.5 im) im (cos re))
   (* 0.5 (+ (exp (- im)) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.0072) {
		tmp = fma((0.5 * im), im, cos(re));
	} else {
		tmp = 0.5 * (exp(-im) + exp(im));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.0072)
		tmp = fma(Float64(0.5 * im), im, cos(re));
	else
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.0072], N[(N[(0.5 * im), $MachinePrecision] * im + 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 < 0.0071999999999999998

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 85.0%

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

   4. Taylor expanded in re around 0 79.0%

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

      1. *-commutative 79.0%

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

   6. Simplified 79.0%

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

      1. +-commutative 79.0%

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

      2. *-commutative 79.0%

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

      3. unpow2 79.0%

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

      4. associate-*r* 79.0%

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

      5. fma-def 79.0%

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

   8. Applied egg-rr 79.0%

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

   if 0.0071999999999999998 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 80.3%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.0072:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot im, im, \cos re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0 77.2%

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

4. Taylor expanded in re around 0 70.2%

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

   1. *-commutative 70.2%

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

6. Simplified 70.2%

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

   1. +-commutative 70.2%

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

   2. *-commutative 70.2%

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

   3. unpow2 70.2%

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

   4. associate-*r* 70.2%

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

   5. fma-def 70.2%

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

8. Applied egg-rr 70.2%

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

9. Final simplification 70.2%

   \[\leadsto \mathsf{fma}\left(0.5 \cdot im, im, \cos re\right) \]
10. Add Preprocessing

Alternative 6: 61.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 820000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 4 \cdot 10^{+153}:\\ \;\;\;\;0.25 + {re}^{2} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 820000.0)
   (cos re)
   (if (<= im 4e+153) (+ 0.25 (* (pow re 2.0) 0.25)) (* 0.5 (fma im im 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 820000.0) {
		tmp = cos(re);
	} else if (im <= 4e+153) {
		tmp = 0.25 + (pow(re, 2.0) * 0.25);
	} else {
		tmp = 0.5 * fma(im, im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 820000.0)
		tmp = cos(re);
	elseif (im <= 4e+153)
		tmp = Float64(0.25 + Float64((re ^ 2.0) * 0.25));
	else
		tmp = Float64(0.5 * fma(im, im, 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 820000.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 4e+153], N[(0.25 + N[(N[Power[re, 2.0], $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 8.2e5

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 66.7%

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

   if 8.2e5 < im < 4e153

   1. Initial program 100.0%

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

   4. Applied egg-rr 2.6%

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

   5. Taylor expanded in re around 0 18.1%

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

      1. *-commutative 18.1%

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

   7. Simplified 18.1%

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

   if 4e153 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 82.9%

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

   4. Taylor expanded in im around 0 80.4%

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

      1. +-commutative 80.4%

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

      2. unpow2 80.4%

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

      3. fma-def 80.4%

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

   6. Simplified 80.4%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 820000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 4 \cdot 10^{+153}:\\ \;\;\;\;0.25 + {re}^{2} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 6.5 \cdot 10^{+42}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{+152}:\\ \;\;\;\;-0.5 \cdot {re}^{2} + -1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 6.5e+42)
   (cos re)
   (if (<= im 4.2e+152)
     (+ (* -0.5 (pow re 2.0)) -1.0)
     (* 0.5 (fma im im 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 6.5e+42) {
		tmp = cos(re);
	} else if (im <= 4.2e+152) {
		tmp = (-0.5 * pow(re, 2.0)) + -1.0;
	} else {
		tmp = 0.5 * fma(im, im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 6.5e+42)
		tmp = cos(re);
	elseif (im <= 4.2e+152)
		tmp = Float64(Float64(-0.5 * (re ^ 2.0)) + -1.0);
	else
		tmp = Float64(0.5 * fma(im, im, 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 6.5e+42], N[Cos[re], $MachinePrecision], If[LessEqual[im, 4.2e+152], N[(N[(-0.5 * N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(0.5 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 6.50000000000000052e42

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 65.7%

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

   if 6.50000000000000052e42 < im < 4.2000000000000003e152

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 6.4%

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

   5. Applied egg-rr 1.3%

      \[\leadsto \cos re + 0.5 \cdot \color{blue}{-4} \]

   6. Taylor expanded in re around 0 22.9%

      \[\leadsto \color{blue}{-0.5 \cdot {re}^{2} - 1} \]

   if 4.2000000000000003e152 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 83.3%

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

   4. Taylor expanded in im around 0 78.6%

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

      1. +-commutative 78.6%

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

      2. unpow2 78.6%

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

      3. fma-def 78.6%

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

   6. Simplified 78.6%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 6.5 \cdot 10^{+42}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{+152}:\\ \;\;\;\;-0.5 \cdot {re}^{2} + -1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 9.2 \cdot 10^{+51}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 9.2e+51) (cos re) (* 0.5 (fma im im 2.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 9.2e+51) {
		tmp = cos(re);
	} else {
		tmp = 0.5 * fma(im, im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 9.2e+51)
		tmp = cos(re);
	else
		tmp = Float64(0.5 * fma(im, im, 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 9.2e+51], N[Cos[re], $MachinePrecision], N[(0.5 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 9.2000000000000002e51

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 64.8%

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

   if 9.2000000000000002e51 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 80.0%

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

   4. Taylor expanded in im around 0 49.1%

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

      1. +-commutative 49.1%

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

      2. unpow2 49.1%

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

      3. fma-def 49.1%

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

   6. Simplified 49.1%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 9.2 \cdot 10^{+51}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 60.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.8 \cdot 10^{+51}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {im}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.8e+51) (cos re) (* 0.5 (pow im 2.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.8e+51) {
		tmp = cos(re);
	} else {
		tmp = 0.5 * pow(im, 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 3.8d+51) then
        tmp = cos(re)
    else
        tmp = 0.5d0 * (im ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3.8e+51) {
		tmp = Math.cos(re);
	} else {
		tmp = 0.5 * Math.pow(im, 2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3.8e+51:
		tmp = math.cos(re)
	else:
		tmp = 0.5 * math.pow(im, 2.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.8e+51)
		tmp = cos(re);
	else
		tmp = Float64(0.5 * (im ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3.8e+51)
		tmp = cos(re);
	else
		tmp = 0.5 * (im ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.8e+51], N[Cos[re], $MachinePrecision], N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 3.7999999999999997e51

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 64.8%

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

   if 3.7999999999999997e51 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 59.8%

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

   4. Taylor expanded in re around 0 49.1%

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

      1. *-commutative 49.1%

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

   6. Simplified 49.1%

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

      1. +-commutative 49.1%

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

      2. *-commutative 49.1%

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

      3. unpow2 49.1%

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

      4. associate-*r* 49.1%

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

      5. fma-def 49.1%

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

   8. Applied egg-rr 49.1%

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

   9. Taylor expanded in im around inf 49.1%

      \[\leadsto \color{blue}{0.5 \cdot {im}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.8 \cdot 10^{+51}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {im}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 51.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \cos re \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (cos re))

C

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

Fortran

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

Java

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

Python

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

Julia

function code(re, im)
	return cos(re)
end

MATLAB

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

Wolfram

code[re_, im_] := N[Cos[re], $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0 50.3%

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

4. Final simplification 50.3%

   \[\leadsto \cos re \]
5. Add Preprocessing

Alternative 11: 4.0% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 -1.0)

C

double code(double re, double im) {
	return -1.0;
}

Fortran

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

Java

public static double code(double re, double im) {
	return -1.0;
}

Python

def code(re, im):
	return -1.0

Julia

function code(re, im)
	return -1.0
end

MATLAB

function tmp = code(re, im)
	tmp = -1.0;
end

Wolfram

code[re_, im_] := -1.0

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0 77.2%

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

5. Applied egg-rr 3.2%

   \[\leadsto \cos re + 0.5 \cdot \color{blue}{-4} \]

6. Taylor expanded in re around 0 3.6%

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

7. Final simplification 3.6%

   \[\leadsto -1 \]
8. Add Preprocessing

Alternative 12: 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%

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

4. Applied egg-rr 7.9%

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

5. Taylor expanded in re around 0 7.9%

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

6. Final simplification 7.9%

   \[\leadsto 0.25 \]
7. Add Preprocessing

Alternative 13: 28.3% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return 1.0

Julia

function code(re, im)
	return 1.0
end

MATLAB

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

Wolfram

code[re_, im_] := 1.0

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in re around 0 70.4%

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

4. Taylor expanded in im around 0 32.2%

   \[\leadsto 0.5 \cdot \color{blue}{2} \]

5. Final simplification 32.2%

   \[\leadsto 1 \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024027 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))