math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 6.4s

Alternatives: 6

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: 87.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.3:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.3)
   (* (* 0.5 (cos re)) (fma im im 2.0))
   (* 0.5 (* (cos re) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.3) {
		tmp = (0.5 * cos(re)) * fma(im, im, 2.0);
	} else {
		tmp = 0.5 * (cos(re) * exp(im));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.3)
		tmp = Float64(Float64(0.5 * cos(re)) * fma(im, im, 2.0));
	else
		tmp = Float64(0.5 * Float64(cos(re) * exp(im)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.3], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.30000000000000004

   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 84.1%

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

      1. +-commutative 84.1%

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

      2. unpow2 84.1%

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

      3. fma-def 84.1%

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

   5. Simplified 84.1%

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

   if 1.30000000000000004 < im

   1. Initial program 100.0%

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

      1. cos-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-rgt-in 100.0%

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

      6. cos-neg 100.0%

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

      7. distribute-rgt-in 100.0%

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

      8. distribute-lft-in 100.0%

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

      9. *-commutative 100.0%

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

      10. fma-def 100.0%

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

      11. exp-neg 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

      12. associate-*l/ 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{0}\right) \]

   7. Taylor expanded in re around inf 100.0%

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

      1. *-commutative 100.0%

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

   9. Simplified 100.0%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.3:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot e^{im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.7:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.7) (cos re) (* 0.5 (* (cos re) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.7) {
		tmp = cos(re);
	} else {
		tmp = 0.5 * (cos(re) * 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 <= 0.7d0) then
        tmp = cos(re)
    else
        tmp = 0.5d0 * (cos(re) * exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.7) {
		tmp = Math.cos(re);
	} else {
		tmp = 0.5 * (Math.cos(re) * Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.7:
		tmp = math.cos(re)
	else:
		tmp = 0.5 * (math.cos(re) * math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.7)
		tmp = cos(re);
	else
		tmp = Float64(0.5 * Float64(cos(re) * exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.7)
		tmp = cos(re);
	else
		tmp = 0.5 * (cos(re) * exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.7], N[Cos[re], $MachinePrecision], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 0.69999999999999996

   1. Initial program 100.0%

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

      1. cos-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-rgt-in 100.0%

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

      6. cos-neg 100.0%

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

      7. distribute-rgt-in 100.0%

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

      8. distribute-lft-in 100.0%

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

      9. *-commutative 100.0%

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

      10. fma-def 100.0%

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

      11. exp-neg 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

      12. associate-*l/ 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 67.6%

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

   if 0.69999999999999996 < im

   1. Initial program 100.0%

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

      1. cos-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-rgt-in 100.0%

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

      6. cos-neg 100.0%

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

      7. distribute-rgt-in 100.0%

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

      8. distribute-lft-in 100.0%

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

      9. *-commutative 100.0%

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

      10. fma-def 100.0%

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

      11. exp-neg 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

      12. associate-*l/ 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{0}\right) \]

   7. Taylor expanded in re around inf 100.0%

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

      1. *-commutative 100.0%

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

   9. Simplified 100.0%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.7:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot e^{im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.3:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot e^{im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 1.3) (cos re) (* 0.5 (exp im))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.3) {
		tmp = cos(re);
	} else {
		tmp = 0.5 * exp(im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.3d0) then
        tmp = cos(re)
    else
        tmp = 0.5d0 * exp(im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.3) {
		tmp = Math.cos(re);
	} else {
		tmp = 0.5 * Math.exp(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.3:
		tmp = math.cos(re)
	else:
		tmp = 0.5 * math.exp(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.3)
		tmp = cos(re);
	else
		tmp = Float64(0.5 * exp(im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.3)
		tmp = cos(re);
	else
		tmp = 0.5 * exp(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.3], N[Cos[re], $MachinePrecision], N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.30000000000000004

   1. Initial program 100.0%

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

      1. cos-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-rgt-in 100.0%

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

      6. cos-neg 100.0%

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

      7. distribute-rgt-in 100.0%

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

      8. distribute-lft-in 100.0%

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

      9. *-commutative 100.0%

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

      10. fma-def 100.0%

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

      11. exp-neg 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

      12. associate-*l/ 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 67.6%

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

   if 1.30000000000000004 < im

   1. Initial program 100.0%

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

      1. cos-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-rgt-in 100.0%

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

      6. cos-neg 100.0%

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

      7. distribute-rgt-in 100.0%

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

      8. distribute-lft-in 100.0%

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

      9. *-commutative 100.0%

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

      10. fma-def 100.0%

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

      11. exp-neg 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

      12. associate-*l/ 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{0}\right) \]

   7. Taylor expanded in re around 0 76.2%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.3:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot e^{im}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 50.8% 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. Step-by-step derivation

   1. cos-neg 100.0%

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

   2. *-commutative 100.0%

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

   3. associate-*l* 100.0%

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

   4. +-commutative 100.0%

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

   5. distribute-rgt-in 100.0%

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

   6. cos-neg 100.0%

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

   7. distribute-rgt-in 100.0%

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

   8. distribute-lft-in 100.0%

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

   9. *-commutative 100.0%

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

   10. fma-def 100.0%

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

   11. exp-neg 100.0%

       \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

   12. associate-*l/ 100.0%

       \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

   13. metadata-eval 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in im around 0 51.8%

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

6. Final simplification 51.8%

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

Alternative 6: 28.8% 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. Step-by-step derivation

   1. cos-neg 100.0%

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

   2. *-commutative 100.0%

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

   3. associate-*l* 100.0%

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

   4. +-commutative 100.0%

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

   5. distribute-rgt-in 100.0%

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

   6. cos-neg 100.0%

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

   7. distribute-rgt-in 100.0%

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

   8. distribute-lft-in 100.0%

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

   9. *-commutative 100.0%

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

   10. fma-def 100.0%

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

   11. exp-neg 100.0%

       \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1}{e^{im}}} \cdot 0.5\right) \]

   12. associate-*l/ 100.0%

       \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{1 \cdot 0.5}{e^{im}}}\right) \]

   13. metadata-eval 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in im around 0 51.8%

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

7. Applied egg-rr 33.5%

   \[\leadsto \color{blue}{{\cos re}^{-1}} \]
8. Step-by-step derivation

   1. unpow-1 33.5%

      \[\leadsto \color{blue}{\frac{1}{\cos re}} \]

9. Simplified 33.5%

   \[\leadsto \color{blue}{\frac{1}{\cos re}} \]

10. Taylor expanded in re around 0 31.9%

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

11. Final simplification 31.9%

    \[\leadsto 1 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024019 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))