math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 7.8s

Alternatives: 11

Speedup: 1.0×

• 49.6% 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, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[re_, im_] := N[(N[Cos[re], $MachinePrecision] * N[(0.5 * N[Exp[im], $MachinePrecision] + N[(0.5 / N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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. cancel-sign-sub 100.0%

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

   7. distribute-lft-neg-in 100.0%

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

   8. cos-neg 100.0%

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

   9. *-commutative 100.0%

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

   10. fma-neg 100.0%

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

   11. remove-double-neg 100.0%

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

   12. exp-neg 100.0%

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

   13. associate-*l/ 100.0%

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

   14. 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. Add Preprocessing

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 3: 74.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(N[Cos[re], $MachinePrecision] * N[(0.5 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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. cancel-sign-sub 100.0%

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

   7. distribute-lft-neg-in 100.0%

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

   8. cos-neg 100.0%

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

   9. *-commutative 100.0%

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

   10. fma-neg 100.0%

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

   11. remove-double-neg 100.0%

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

   12. exp-neg 100.0%

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

   13. associate-*l/ 100.0%

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

   14. 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 75.1%

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

   1. fma-undefine 75.1%

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

7. Applied egg-rr 75.1%

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

8. Final simplification 75.1%

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

Alternative 4: 73.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.15:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 3.1 \cdot 10^{+102}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{im}\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(1 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.15)
   (cos re)
   (if (<= im 3.1e+102)
     (+ 0.5 (* 0.5 (exp im)))
     (*
      (cos re)
      (+ 1.0 (* im (+ 0.5 (* im (+ 0.25 (* im 0.08333333333333333))))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.15) {
		tmp = cos(re);
	} else if (im <= 3.1e+102) {
		tmp = 0.5 + (0.5 * exp(im));
	} else {
		tmp = cos(re) * (1.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333))))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 2.15d0) then
        tmp = cos(re)
    else if (im <= 3.1d+102) then
        tmp = 0.5d0 + (0.5d0 * exp(im))
    else
        tmp = cos(re) * (1.0d0 + (im * (0.5d0 + (im * (0.25d0 + (im * 0.08333333333333333d0))))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.15) {
		tmp = Math.cos(re);
	} else if (im <= 3.1e+102) {
		tmp = 0.5 + (0.5 * Math.exp(im));
	} else {
		tmp = Math.cos(re) * (1.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333))))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.15:
		tmp = math.cos(re)
	elif im <= 3.1e+102:
		tmp = 0.5 + (0.5 * math.exp(im))
	else:
		tmp = math.cos(re) * (1.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333))))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.15)
		tmp = cos(re);
	elseif (im <= 3.1e+102)
		tmp = Float64(0.5 + Float64(0.5 * exp(im)));
	else
		tmp = Float64(cos(re) * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * Float64(0.25 + Float64(im * 0.08333333333333333)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.15)
		tmp = cos(re);
	elseif (im <= 3.1e+102)
		tmp = 0.5 + (0.5 * exp(im));
	else
		tmp = cos(re) * (1.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.15], N[Cos[re], $MachinePrecision], If[LessEqual[im, 3.1e+102], N[(0.5 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(1.0 + N[(im * N[(0.5 + N[(im * N[(0.25 + N[(im * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 2.14999999999999991

   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. cancel-sign-sub 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. cos-neg 100.0%

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

      9. *-commutative 100.0%

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

      10. fma-neg 100.0%

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

      11. remove-double-neg 100.0%

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

      12. exp-neg 100.0%

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

      13. associate-*l/ 100.0%

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

      14. 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 66.3%

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

   if 2.14999999999999991 < im < 3.09999999999999987e102

   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. cancel-sign-sub 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. cos-neg 100.0%

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

      9. *-commutative 100.0%

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

      10. fma-neg 100.0%

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

      11. remove-double-neg 100.0%

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

      12. exp-neg 100.0%

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

      13. associate-*l/ 100.0%

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

      14. 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 0 100.0%

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

   6. Taylor expanded in re around 0 88.2%

      \[\leadsto \color{blue}{0.5 + 0.5 \cdot e^{im}} \]

   if 3.09999999999999987e102 < 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. cancel-sign-sub 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. cos-neg 100.0%

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

      9. *-commutative 100.0%

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

      10. fma-neg 100.0%

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

      11. remove-double-neg 100.0%

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

      12. exp-neg 100.0%

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

      13. associate-*l/ 100.0%

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

      14. 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 0 100.0%

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

   6. Taylor expanded in im around 0 98.4%

      \[\leadsto \cos re \cdot \color{blue}{\left(1 + im \cdot \left(0.5 + im \cdot \left(0.25 + 0.08333333333333333 \cdot im\right)\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative 98.4%

         \[\leadsto \cos re \cdot \left(1 + im \cdot \left(0.5 + im \cdot \left(0.25 + \color{blue}{im \cdot 0.08333333333333333}\right)\right)\right) \]

   8. Simplified 98.4%

      \[\leadsto \cos re \cdot \color{blue}{\left(1 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 72.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.92:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2.65 \cdot 10^{+154}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{im}\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.92)
   (cos re)
   (if (<= im 2.65e+154)
     (+ 0.5 (* 0.5 (exp im)))
     (* (cos re) (+ 1.0 (* im (+ 0.5 (* im 0.25))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.92) {
		tmp = cos(re);
	} else if (im <= 2.65e+154) {
		tmp = 0.5 + (0.5 * exp(im));
	} else {
		tmp = cos(re) * (1.0 + (im * (0.5 + (im * 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 <= 1.92d0) then
        tmp = cos(re)
    else if (im <= 2.65d+154) then
        tmp = 0.5d0 + (0.5d0 * exp(im))
    else
        tmp = cos(re) * (1.0d0 + (im * (0.5d0 + (im * 0.25d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.92) {
		tmp = Math.cos(re);
	} else if (im <= 2.65e+154) {
		tmp = 0.5 + (0.5 * Math.exp(im));
	} else {
		tmp = Math.cos(re) * (1.0 + (im * (0.5 + (im * 0.25))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.92:
		tmp = math.cos(re)
	elif im <= 2.65e+154:
		tmp = 0.5 + (0.5 * math.exp(im))
	else:
		tmp = math.cos(re) * (1.0 + (im * (0.5 + (im * 0.25))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.92)
		tmp = cos(re);
	elseif (im <= 2.65e+154)
		tmp = Float64(0.5 + Float64(0.5 * exp(im)));
	else
		tmp = Float64(cos(re) * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.25)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.92)
		tmp = cos(re);
	elseif (im <= 2.65e+154)
		tmp = 0.5 + (0.5 * exp(im));
	else
		tmp = cos(re) * (1.0 + (im * (0.5 + (im * 0.25))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.92], N[Cos[re], $MachinePrecision], If[LessEqual[im, 2.65e+154], N[(0.5 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.9199999999999999

   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. cancel-sign-sub 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. cos-neg 100.0%

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

      9. *-commutative 100.0%

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

      10. fma-neg 100.0%

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

      11. remove-double-neg 100.0%

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

      12. exp-neg 100.0%

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

      13. associate-*l/ 100.0%

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

      14. 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 66.3%

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

   if 1.9199999999999999 < im < 2.65000000000000012e154

   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. cancel-sign-sub 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. cos-neg 100.0%

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

      9. *-commutative 100.0%

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

      10. fma-neg 100.0%

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

      11. remove-double-neg 100.0%

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

      12. exp-neg 100.0%

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

      13. associate-*l/ 100.0%

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

      14. 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 0 100.0%

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

   6. Taylor expanded in re around 0 79.3%

      \[\leadsto \color{blue}{0.5 + 0.5 \cdot e^{im}} \]

   if 2.65000000000000012e154 < 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. cancel-sign-sub 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. cos-neg 100.0%

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

      9. *-commutative 100.0%

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

      10. fma-neg 100.0%

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

      11. remove-double-neg 100.0%

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

      12. exp-neg 100.0%

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

      13. associate-*l/ 100.0%

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

      14. 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 0 100.0%

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

   6. Taylor expanded in im around 0 100.0%

      \[\leadsto \cos re \cdot \color{blue}{\left(1 + im \cdot \left(0.5 + 0.25 \cdot im\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \cos re \cdot \left(1 + im \cdot \left(0.5 + \color{blue}{im \cdot 0.25}\right)\right) \]

   8. Simplified 100.0%

      \[\leadsto \cos re \cdot \color{blue}{\left(1 + im \cdot \left(0.5 + im \cdot 0.25\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 69.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 5.1:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 5.1) (cos re) (+ 0.5 (* 0.5 (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 5.1) {
		tmp = cos(re);
	} else {
		tmp = 0.5 + (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 <= 5.1d0) then
        tmp = cos(re)
    else
        tmp = 0.5d0 + (0.5d0 * exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 5.1) {
		tmp = Math.cos(re);
	} else {
		tmp = 0.5 + (0.5 * Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 5.1:
		tmp = math.cos(re)
	else:
		tmp = 0.5 + (0.5 * math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 5.1)
		tmp = cos(re);
	else
		tmp = Float64(0.5 + Float64(0.5 * exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 5.1)
		tmp = cos(re);
	else
		tmp = 0.5 + (0.5 * exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 5.1], N[Cos[re], $MachinePrecision], N[(0.5 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 5.0999999999999996

   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. cancel-sign-sub 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. cos-neg 100.0%

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

      9. *-commutative 100.0%

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

      10. fma-neg 100.0%

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

      11. remove-double-neg 100.0%

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

      12. exp-neg 100.0%

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

      13. associate-*l/ 100.0%

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

      14. 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 66.3%

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

   if 5.0999999999999996 < 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. cancel-sign-sub 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. cos-neg 100.0%

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

      9. *-commutative 100.0%

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

      10. fma-neg 100.0%

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

      11. remove-double-neg 100.0%

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

      12. exp-neg 100.0%

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

      13. associate-*l/ 100.0%

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

      14. 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 0 100.0%

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

   6. Taylor expanded in re around 0 76.8%

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

Alternative 7: 63.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1450000000000:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(0.5 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1450000000000.0)
   (cos re)
   (+ 0.5 (+ 0.5 (* im (+ 0.5 (* im (+ 0.25 (* im 0.08333333333333333)))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1450000000000.0) {
		tmp = cos(re);
	} else {
		tmp = 0.5 + (0.5 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333))))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1450000000000.0d0) then
        tmp = cos(re)
    else
        tmp = 0.5d0 + (0.5d0 + (im * (0.5d0 + (im * (0.25d0 + (im * 0.08333333333333333d0))))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1450000000000.0) {
		tmp = Math.cos(re);
	} else {
		tmp = 0.5 + (0.5 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333))))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1450000000000.0:
		tmp = math.cos(re)
	else:
		tmp = 0.5 + (0.5 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333))))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1450000000000.0)
		tmp = cos(re);
	else
		tmp = Float64(0.5 + Float64(0.5 + Float64(im * Float64(0.5 + Float64(im * Float64(0.25 + Float64(im * 0.08333333333333333)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1450000000000.0)
		tmp = cos(re);
	else
		tmp = 0.5 + (0.5 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1450000000000.0], N[Cos[re], $MachinePrecision], N[(0.5 + N[(0.5 + N[(im * N[(0.5 + N[(im * N[(0.25 + N[(im * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.45e12

   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. cancel-sign-sub 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. cos-neg 100.0%

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

      9. *-commutative 100.0%

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

      10. fma-neg 100.0%

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

      11. remove-double-neg 100.0%

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

      12. exp-neg 100.0%

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

      13. associate-*l/ 100.0%

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

      14. 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 65.7%

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

   if 1.45e12 < 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. cancel-sign-sub 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. cos-neg 100.0%

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

      9. *-commutative 100.0%

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

      10. fma-neg 100.0%

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

      11. remove-double-neg 100.0%

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

      12. exp-neg 100.0%

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

      13. associate-*l/ 100.0%

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

      14. 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 0 100.0%

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

   6. Taylor expanded in re around 0 77.6%

      \[\leadsto \color{blue}{0.5 + 0.5 \cdot e^{im}} \]

   7. Taylor expanded in im around 0 57.9%

      \[\leadsto 0.5 + \color{blue}{\left(0.5 + im \cdot \left(0.5 + im \cdot \left(0.25 + 0.08333333333333333 \cdot im\right)\right)\right)} \]
   8. Step-by-step derivation

      1. *-commutative 57.9%

         \[\leadsto 0.5 + \left(0.5 + im \cdot \left(0.5 + im \cdot \left(0.25 + \color{blue}{im \cdot 0.08333333333333333}\right)\right)\right) \]

   9. Simplified 57.9%

      \[\leadsto 0.5 + \color{blue}{\left(0.5 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 44.3% accurate, 20.5× speedup

Math

\[\begin{array}{l} \\ 0.5 + \left(0.5 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (+ 0.5 (+ 0.5 (* im (+ 0.5 (* im (+ 0.25 (* im 0.08333333333333333))))))))

C

double code(double re, double im) {
	return 0.5 + (0.5 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333))))));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 + (0.5d0 + (im * (0.5d0 + (im * (0.25d0 + (im * 0.08333333333333333d0))))))
end function

Java

public static double code(double re, double im) {
	return 0.5 + (0.5 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333))))));
}

Python

def code(re, im):
	return 0.5 + (0.5 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333))))))

Julia

function code(re, im)
	return Float64(0.5 + Float64(0.5 + Float64(im * Float64(0.5 + Float64(im * Float64(0.25 + Float64(im * 0.08333333333333333)))))))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 + (0.5 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333))))));
end

Wolfram

code[re_, im_] := N[(0.5 + N[(0.5 + N[(im * N[(0.5 + N[(im * N[(0.25 + N[(im * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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. cancel-sign-sub 100.0%

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

   7. distribute-lft-neg-in 100.0%

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

   8. cos-neg 100.0%

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

   9. *-commutative 100.0%

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

   10. fma-neg 100.0%

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

   11. remove-double-neg 100.0%

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

   12. exp-neg 100.0%

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

   13. associate-*l/ 100.0%

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

   14. 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 75.1%

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

6. Taylor expanded in re around 0 45.6%

   \[\leadsto \color{blue}{0.5 + 0.5 \cdot e^{im}} \]

7. Taylor expanded in im around 0 43.0%

   \[\leadsto 0.5 + \color{blue}{\left(0.5 + im \cdot \left(0.5 + im \cdot \left(0.25 + 0.08333333333333333 \cdot im\right)\right)\right)} \]
8. Step-by-step derivation

   1. *-commutative 43.0%

      \[\leadsto 0.5 + \left(0.5 + im \cdot \left(0.5 + im \cdot \left(0.25 + \color{blue}{im \cdot 0.08333333333333333}\right)\right)\right) \]

9. Simplified 43.0%

   \[\leadsto 0.5 + \color{blue}{\left(0.5 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)\right)} \]
10. Add Preprocessing

Alternative 9: 46.5% accurate, 34.2× speedup

Math

\[\begin{array}{l} \\ 1 + im \cdot \left(0.5 + im \cdot 0.25\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (+ 1.0 (* im (+ 0.5 (* im 0.25)))))

C

double code(double re, double im) {
	return 1.0 + (im * (0.5 + (im * 0.25)));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 1.0d0 + (im * (0.5d0 + (im * 0.25d0)))
end function

Java

public static double code(double re, double im) {
	return 1.0 + (im * (0.5 + (im * 0.25)));
}

Python

def code(re, im):
	return 1.0 + (im * (0.5 + (im * 0.25)))

Julia

function code(re, im)
	return Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.25))))
end

MATLAB

function tmp = code(re, im)
	tmp = 1.0 + (im * (0.5 + (im * 0.25)));
end

Wolfram

code[re_, im_] := N[(1.0 + N[(im * N[(0.5 + N[(im * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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. cancel-sign-sub 100.0%

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

   7. distribute-lft-neg-in 100.0%

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

   8. cos-neg 100.0%

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

   9. *-commutative 100.0%

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

   10. fma-neg 100.0%

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

   11. remove-double-neg 100.0%

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

   12. exp-neg 100.0%

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

   13. associate-*l/ 100.0%

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

   14. 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 75.1%

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

6. Taylor expanded in im around 0 74.2%

   \[\leadsto \cos re \cdot \color{blue}{\left(1 + im \cdot \left(0.5 + 0.25 \cdot im\right)\right)} \]
7. Step-by-step derivation

   1. *-commutative 74.2%

      \[\leadsto \cos re \cdot \left(1 + im \cdot \left(0.5 + \color{blue}{im \cdot 0.25}\right)\right) \]

8. Simplified 74.2%

   \[\leadsto \cos re \cdot \color{blue}{\left(1 + im \cdot \left(0.5 + im \cdot 0.25\right)\right)} \]

9. Taylor expanded in re around 0 44.9%

   \[\leadsto \color{blue}{1 + im \cdot \left(0.5 + 0.25 \cdot im\right)} \]

10. Final simplification 44.9%

    \[\leadsto 1 + im \cdot \left(0.5 + im \cdot 0.25\right) \]
11. Add Preprocessing

Alternative 10: 28.3% accurate, 61.6× speedup

Math

\[\begin{array}{l} \\ 1 + 0.5 \cdot im \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (+ 1.0 (* 0.5 im)))

C

double code(double re, double im) {
	return 1.0 + (0.5 * im);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 1.0d0 + (0.5d0 * im)
end function

Java

public static double code(double re, double im) {
	return 1.0 + (0.5 * im);
}

Python

def code(re, im):
	return 1.0 + (0.5 * im)

Julia

function code(re, im)
	return Float64(1.0 + Float64(0.5 * im))
end

MATLAB

function tmp = code(re, im)
	tmp = 1.0 + (0.5 * im);
end

Wolfram

code[re_, im_] := N[(1.0 + N[(0.5 * im), $MachinePrecision]), $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. cancel-sign-sub 100.0%

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

   7. distribute-lft-neg-in 100.0%

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

   8. cos-neg 100.0%

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

   9. *-commutative 100.0%

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

   10. fma-neg 100.0%

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

   11. remove-double-neg 100.0%

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

   12. exp-neg 100.0%

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

   13. associate-*l/ 100.0%

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

   14. 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 75.1%

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

6. Taylor expanded in im around 0 74.2%

   \[\leadsto \cos re \cdot \color{blue}{\left(1 + im \cdot \left(0.5 + 0.25 \cdot im\right)\right)} \]
7. Step-by-step derivation

   1. *-commutative 74.2%

      \[\leadsto \cos re \cdot \left(1 + im \cdot \left(0.5 + \color{blue}{im \cdot 0.25}\right)\right) \]

8. Simplified 74.2%

   \[\leadsto \cos re \cdot \color{blue}{\left(1 + im \cdot \left(0.5 + im \cdot 0.25\right)\right)} \]

9. Taylor expanded in im around 0 49.0%

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

    1. associate-*r* 49.0%

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

    2. *-commutative 49.0%

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

    3. distribute-rgt1-in 49.0%

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

    4. *-commutative 49.0%

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

11. Simplified 49.0%

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

12. Taylor expanded in re around 0 25.8%

    \[\leadsto \color{blue}{1 + 0.5 \cdot im} \]
13. Add Preprocessing

Alternative 11: 28.4% 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. cancel-sign-sub 100.0%

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

   7. distribute-lft-neg-in 100.0%

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

   8. cos-neg 100.0%

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

   9. *-commutative 100.0%

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

   10. fma-neg 100.0%

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

   11. remove-double-neg 100.0%

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

   12. exp-neg 100.0%

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

   13. associate-*l/ 100.0%

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

   14. 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 49.3%

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

7. Applied egg-rr 27.9%

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

   1. unpow-1 27.9%

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

9. Simplified 27.9%

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

10. Taylor expanded in re around 0 25.8%

    \[\leadsto \color{blue}{1} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024111 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))