math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 6.3s

Alternatives: 9

Speedup: 1.0×

• 49.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 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: 75.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(\cos re \cdot \mathsf{fma}\left(im, im, 2\right)\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* 0.5 (* (cos re) (fma im im 2.0))))

C

double code(double re, double im) {
	return 0.5 * (cos(re) * fma(im, im, 2.0));
}

Julia

function code(re, im)
	return Float64(0.5 * Float64(cos(re) * fma(im, im, 2.0)))
end

Wolfram

code[re_, im_] := N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im * im + 2.0), $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. Taylor expanded in im around 0 73.8%

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

4. Taylor expanded in re around inf 73.8%

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

   1. *-commutative 73.8%

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

   2. +-commutative 73.8%

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

   3. unpow2 73.8%

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

   4. fma-udef 73.8%

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

6. Simplified 73.8%

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

7. Final simplification 73.8%

   \[\leadsto 0.5 \cdot \left(\cos re \cdot \mathsf{fma}\left(im, im, 2\right)\right) \]
8. Add Preprocessing

Alternative 3: 65.8% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 (* 0.5 (+ (exp (- im)) (exp im))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(0.5 * 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. Taylor expanded in re around 0 63.8%

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

4. Final simplification 63.8%

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

Alternative 4: 69.0% 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 73.8%

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

4. Taylor expanded in re around 0 64.8%

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

   1. +-commutative 64.8%

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

   2. unpow2 64.8%

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

   3. associate-*r* 64.8%

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

   4. fma-def 64.8%

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

6. Applied egg-rr 64.8%

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

7. Final simplification 64.8%

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

Alternative 5: 47.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \mathsf{fma}\left(im, im, 2\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* 0.5 (fma im im 2.0)))

C

double code(double re, double im) {
	return 0.5 * fma(im, im, 2.0);
}

Julia

function code(re, im)
	return Float64(0.5 * fma(im, im, 2.0))
end

Wolfram

code[re_, im_] := N[(0.5 * N[(im * im + 2.0), $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 re around 0 63.8%

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

4. Taylor expanded in im around 0 45.0%

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

   1. +-commutative 45.0%

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

   2. unpow2 45.0%

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

   3. fma-def 45.0%

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

6. Simplified 45.0%

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

7. Final simplification 45.0%

   \[\leadsto 0.5 \cdot \mathsf{fma}\left(im, im, 2\right) \]
8. Add Preprocessing

Alternative 6: 50.7% 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 47.6%

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

4. Final simplification 47.6%

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

Alternative 7: 3.7% 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 73.8%

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

5. Applied egg-rr 2.9%

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

6. Taylor expanded in re around 0 3.2%

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

7. Final simplification 3.2%

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

Alternative 8: 8.1% 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 8.0%

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

5. Taylor expanded in re around 0 8.0%

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

6. Final simplification 8.0%

   \[\leadsto 0.25 \]
7. Add Preprocessing

Alternative 9: 28.5% 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 47.6%

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

4. Taylor expanded in re around 0 27.4%

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

5. Final simplification 27.4%

   \[\leadsto 1 \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024033 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))