math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 5.7s

Alternatives: 13

Speedup: 1.0×

• 48.7% 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, 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. Final simplification 100.0%

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

Alternative 2: 95.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq 0.9999:\\ \;\;\;\;\cos re \cdot \left(1 + 0.001388888888888889 \cdot {im}^{6}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot e^{im} + 0.5 \cdot \frac{1}{e^{im}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) 0.9999)
   (* (cos re) (+ 1.0 (* 0.001388888888888889 (pow im 6.0))))
   (+ (* 0.5 (exp im)) (* 0.5 (/ 1.0 (exp im))))))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= 0.9999) {
		tmp = cos(re) * (1.0 + (0.001388888888888889 * pow(im, 6.0)));
	} else {
		tmp = (0.5 * exp(im)) + (0.5 * (1.0 / 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 (cos(re) <= 0.9999d0) then
        tmp = cos(re) * (1.0d0 + (0.001388888888888889d0 * (im ** 6.0d0)))
    else
        tmp = (0.5d0 * exp(im)) + (0.5d0 * (1.0d0 / exp(im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.cos(re) <= 0.9999) {
		tmp = Math.cos(re) * (1.0 + (0.001388888888888889 * Math.pow(im, 6.0)));
	} else {
		tmp = (0.5 * Math.exp(im)) + (0.5 * (1.0 / Math.exp(im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.cos(re) <= 0.9999:
		tmp = math.cos(re) * (1.0 + (0.001388888888888889 * math.pow(im, 6.0)))
	else:
		tmp = (0.5 * math.exp(im)) + (0.5 * (1.0 / math.exp(im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= 0.9999)
		tmp = Float64(cos(re) * Float64(1.0 + Float64(0.001388888888888889 * (im ^ 6.0))));
	else
		tmp = Float64(Float64(0.5 * exp(im)) + Float64(0.5 * Float64(1.0 / exp(im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (cos(re) <= 0.9999)
		tmp = cos(re) * (1.0 + (0.001388888888888889 * (im ^ 6.0)));
	else
		tmp = (0.5 * exp(im)) + (0.5 * (1.0 / exp(im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], 0.9999], N[(N[Cos[re], $MachinePrecision] * N[(1.0 + N[(0.001388888888888889 * N[Power[im, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < 0.99990000000000001

   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 91.7%

      \[\leadsto \cos re \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(0.5 + {im}^{2} \cdot \left(0.041666666666666664 + 0.001388888888888889 \cdot {im}^{2}\right)\right)\right)} \]

   6. Taylor expanded in im around inf 91.6%

      \[\leadsto \cos re \cdot \left(1 + \color{blue}{0.001388888888888889 \cdot {im}^{6}}\right) \]

   if 0.99990000000000001 < (cos.f64 re)

   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 re around 0 100.0%

      \[\leadsto \color{blue}{0.5 \cdot e^{im} + 0.5 \cdot \frac{1}{e^{im}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq 0.9999:\\ \;\;\;\;\cos re \cdot \left(1 + 0.001388888888888889 \cdot {im}^{6}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot e^{im} + 0.5 \cdot \frac{1}{e^{im}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq 0.9999:\\ \;\;\;\;\cos re \cdot \left(1 + 0.001388888888888889 \cdot {im}^{6}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{im} + e^{-im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) 0.9999)
   (* (cos re) (+ 1.0 (* 0.001388888888888889 (pow im 6.0))))
   (* 0.5 (+ (exp im) (exp (- im))))))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= 0.9999) {
		tmp = cos(re) * (1.0 + (0.001388888888888889 * pow(im, 6.0)));
	} else {
		tmp = 0.5 * (exp(im) + exp(-im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (cos(re) <= 0.9999d0) then
        tmp = cos(re) * (1.0d0 + (0.001388888888888889d0 * (im ** 6.0d0)))
    else
        tmp = 0.5d0 * (exp(im) + exp(-im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.cos(re) <= 0.9999) {
		tmp = Math.cos(re) * (1.0 + (0.001388888888888889 * Math.pow(im, 6.0)));
	} else {
		tmp = 0.5 * (Math.exp(im) + Math.exp(-im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.cos(re) <= 0.9999:
		tmp = math.cos(re) * (1.0 + (0.001388888888888889 * math.pow(im, 6.0)))
	else:
		tmp = 0.5 * (math.exp(im) + math.exp(-im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= 0.9999)
		tmp = Float64(cos(re) * Float64(1.0 + Float64(0.001388888888888889 * (im ^ 6.0))));
	else
		tmp = Float64(0.5 * Float64(exp(im) + exp(Float64(-im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (cos(re) <= 0.9999)
		tmp = cos(re) * (1.0 + (0.001388888888888889 * (im ^ 6.0)));
	else
		tmp = 0.5 * (exp(im) + exp(-im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], 0.9999], N[(N[Cos[re], $MachinePrecision] * N[(1.0 + N[(0.001388888888888889 * N[Power[im, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < 0.99990000000000001

   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 91.7%

      \[\leadsto \cos re \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(0.5 + {im}^{2} \cdot \left(0.041666666666666664 + 0.001388888888888889 \cdot {im}^{2}\right)\right)\right)} \]

   6. Taylor expanded in im around inf 91.6%

      \[\leadsto \cos re \cdot \left(1 + \color{blue}{0.001388888888888889 \cdot {im}^{6}}\right) \]

   if 0.99990000000000001 < (cos.f64 re)

   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 0 100.0%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq 0.9999:\\ \;\;\;\;\cos re \cdot \left(1 + 0.001388888888888889 \cdot {im}^{6}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{im} + e^{-im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.35:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, 0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.35)
   (* (cos re) (* 0.5 (fma im im 2.0)))
   (* (cos re) (fma 0.5 (exp im) 0.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.35) {
		tmp = cos(re) * (0.5 * fma(im, im, 2.0));
	} else {
		tmp = cos(re) * fma(0.5, exp(im), 0.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.35)
		tmp = Float64(cos(re) * Float64(0.5 * fma(im, im, 2.0)));
	else
		tmp = Float64(cos(re) * fma(0.5, exp(im), 0.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.35], N[(N[Cos[re], $MachinePrecision] * N[(0.5 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(0.5 * N[Exp[im], $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.3500000000000001

   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 92.7%

      \[\leadsto \cos re \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(0.5 + {im}^{2} \cdot \left(0.041666666666666664 + 0.001388888888888889 \cdot {im}^{2}\right)\right)\right)} \]

   6. Taylor expanded in im around 0 81.3%

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

      1. associate-*r* 81.3%

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

      2. distribute-rgt1-in 81.3%

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

      3. metadata-eval 81.3%

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

      4. distribute-lft-in 81.3%

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

      5. +-commutative 81.3%

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

      6. *-commutative 81.3%

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

      7. +-commutative 81.3%

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

      8. unpow2 81.3%

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

      9. fma-define 81.3%

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

   8. Simplified 81.3%

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

   if 1.3500000000000001 < im

   1. Initial program 99.9%

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

      1. cos-neg 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-*l* 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-rgt-in 99.9%

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

      6. cancel-sign-sub 99.9%

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

      7. distribute-lft-neg-in 99.9%

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

      8. cos-neg 99.9%

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

      9. *-commutative 99.9%

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

      10. fma-neg 99.9%

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

      11. remove-double-neg 99.9%

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

      12. exp-neg 99.9%

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

      13. associate-*l/ 99.9%

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

      14. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   6. Applied egg-rr 98.8%

      \[\leadsto \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{0}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.35:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, 0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 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 6: 72.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.3 \cdot 10^{+19}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 10^{+48}:\\ \;\;\;\;\sqrt{{im}^{12} \cdot 1.9290123456790124 \cdot 10^{-6}}\\ \mathbf{else}:\\ \;\;\;\;0.001388888888888889 \cdot \left(\cos re \cdot {im}^{6}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.3e+19)
   (cos re)
   (if (<= im 1e+48)
     (sqrt (* (pow im 12.0) 1.9290123456790124e-6))
     (* 0.001388888888888889 (* (cos re) (pow im 6.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.3e+19) {
		tmp = cos(re);
	} else if (im <= 1e+48) {
		tmp = sqrt((pow(im, 12.0) * 1.9290123456790124e-6));
	} else {
		tmp = 0.001388888888888889 * (cos(re) * pow(im, 6.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.3d+19) then
        tmp = cos(re)
    else if (im <= 1d+48) then
        tmp = sqrt(((im ** 12.0d0) * 1.9290123456790124d-6))
    else
        tmp = 0.001388888888888889d0 * (cos(re) * (im ** 6.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.3e+19) {
		tmp = Math.cos(re);
	} else if (im <= 1e+48) {
		tmp = Math.sqrt((Math.pow(im, 12.0) * 1.9290123456790124e-6));
	} else {
		tmp = 0.001388888888888889 * (Math.cos(re) * Math.pow(im, 6.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.3e+19:
		tmp = math.cos(re)
	elif im <= 1e+48:
		tmp = math.sqrt((math.pow(im, 12.0) * 1.9290123456790124e-6))
	else:
		tmp = 0.001388888888888889 * (math.cos(re) * math.pow(im, 6.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.3e+19)
		tmp = cos(re);
	elseif (im <= 1e+48)
		tmp = sqrt(Float64((im ^ 12.0) * 1.9290123456790124e-6));
	else
		tmp = Float64(0.001388888888888889 * Float64(cos(re) * (im ^ 6.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.3e+19)
		tmp = cos(re);
	elseif (im <= 1e+48)
		tmp = sqrt(((im ^ 12.0) * 1.9290123456790124e-6));
	else
		tmp = 0.001388888888888889 * (cos(re) * (im ^ 6.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.3e+19], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1e+48], N[Sqrt[N[(N[Power[im, 12.0], $MachinePrecision] * 1.9290123456790124e-6), $MachinePrecision]], $MachinePrecision], N[(0.001388888888888889 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.3e19

   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 63.6%

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

   if 1.3e19 < im < 1.00000000000000004e48

   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 5.1%

      \[\leadsto \cos re \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(0.5 + {im}^{2} \cdot \left(0.041666666666666664 + 0.001388888888888889 \cdot {im}^{2}\right)\right)\right)} \]

   6. Taylor expanded in im around inf 5.1%

      \[\leadsto \color{blue}{0.001388888888888889 \cdot \left({im}^{6} \cdot \cos re\right)} \]

   7. Taylor expanded in re around 0 5.1%

      \[\leadsto \color{blue}{0.001388888888888889 \cdot {im}^{6}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 5.1%

         \[\leadsto \color{blue}{\sqrt{0.001388888888888889 \cdot {im}^{6}} \cdot \sqrt{0.001388888888888889 \cdot {im}^{6}}} \]

      2. sqrt-unprod 42.6%

         \[\leadsto \color{blue}{\sqrt{\left(0.001388888888888889 \cdot {im}^{6}\right) \cdot \left(0.001388888888888889 \cdot {im}^{6}\right)}} \]

      3. *-commutative 42.6%

         \[\leadsto \sqrt{\color{blue}{\left({im}^{6} \cdot 0.001388888888888889\right)} \cdot \left(0.001388888888888889 \cdot {im}^{6}\right)} \]

      4. *-commutative 42.6%

         \[\leadsto \sqrt{\left({im}^{6} \cdot 0.001388888888888889\right) \cdot \color{blue}{\left({im}^{6} \cdot 0.001388888888888889\right)}} \]

      5. swap-sqr 42.6%

         \[\leadsto \sqrt{\color{blue}{\left({im}^{6} \cdot {im}^{6}\right) \cdot \left(0.001388888888888889 \cdot 0.001388888888888889\right)}} \]

      6. pow-prod-up 42.6%

         \[\leadsto \sqrt{\color{blue}{{im}^{\left(6 + 6\right)}} \cdot \left(0.001388888888888889 \cdot 0.001388888888888889\right)} \]

      7. metadata-eval 42.6%

         \[\leadsto \sqrt{{im}^{\color{blue}{12}} \cdot \left(0.001388888888888889 \cdot 0.001388888888888889\right)} \]

      8. metadata-eval 42.6%

         \[\leadsto \sqrt{{im}^{12} \cdot \color{blue}{1.9290123456790124 \cdot 10^{-6}}} \]

   9. Applied egg-rr 42.6%

      \[\leadsto \color{blue}{\sqrt{{im}^{12} \cdot 1.9290123456790124 \cdot 10^{-6}}} \]

   if 1.00000000000000004e48 < 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 98.2%

      \[\leadsto \cos re \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(0.5 + {im}^{2} \cdot \left(0.041666666666666664 + 0.001388888888888889 \cdot {im}^{2}\right)\right)\right)} \]

   6. Taylor expanded in im around inf 98.2%

      \[\leadsto \color{blue}{0.001388888888888889 \cdot \left({im}^{6} \cdot \cos re\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.3 \cdot 10^{+19}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 10^{+48}:\\ \;\;\;\;\sqrt{{im}^{12} \cdot 1.9290123456790124 \cdot 10^{-6}}\\ \mathbf{else}:\\ \;\;\;\;0.001388888888888889 \cdot \left(\cos re \cdot {im}^{6}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 6.8 \cdot 10^{-7}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 10^{+48}:\\ \;\;\;\;0.5 \cdot \left(e^{im} + e^{-im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.001388888888888889 \cdot \left(\cos re \cdot {im}^{6}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 6.8e-7)
   (cos re)
   (if (<= im 1e+48)
     (* 0.5 (+ (exp im) (exp (- im))))
     (* 0.001388888888888889 (* (cos re) (pow im 6.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 6.8e-7) {
		tmp = cos(re);
	} else if (im <= 1e+48) {
		tmp = 0.5 * (exp(im) + exp(-im));
	} else {
		tmp = 0.001388888888888889 * (cos(re) * pow(im, 6.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 6.8d-7) then
        tmp = cos(re)
    else if (im <= 1d+48) then
        tmp = 0.5d0 * (exp(im) + exp(-im))
    else
        tmp = 0.001388888888888889d0 * (cos(re) * (im ** 6.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 6.8e-7) {
		tmp = Math.cos(re);
	} else if (im <= 1e+48) {
		tmp = 0.5 * (Math.exp(im) + Math.exp(-im));
	} else {
		tmp = 0.001388888888888889 * (Math.cos(re) * Math.pow(im, 6.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 6.8e-7:
		tmp = math.cos(re)
	elif im <= 1e+48:
		tmp = 0.5 * (math.exp(im) + math.exp(-im))
	else:
		tmp = 0.001388888888888889 * (math.cos(re) * math.pow(im, 6.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 6.8e-7)
		tmp = cos(re);
	elseif (im <= 1e+48)
		tmp = Float64(0.5 * Float64(exp(im) + exp(Float64(-im))));
	else
		tmp = Float64(0.001388888888888889 * Float64(cos(re) * (im ^ 6.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 6.8e-7)
		tmp = cos(re);
	elseif (im <= 1e+48)
		tmp = 0.5 * (exp(im) + exp(-im));
	else
		tmp = 0.001388888888888889 * (cos(re) * (im ^ 6.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 6.8e-7], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1e+48], N[(0.5 * N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.001388888888888889 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 6.79999999999999948e-7

   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 64.6%

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

   if 6.79999999999999948e-7 < im < 1.00000000000000004e48

   1. Initial program 99.5%

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

   3. Taylor expanded in re around 0 79.7%

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

   if 1.00000000000000004e48 < 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 98.2%

      \[\leadsto \cos re \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(0.5 + {im}^{2} \cdot \left(0.041666666666666664 + 0.001388888888888889 \cdot {im}^{2}\right)\right)\right)} \]

   6. Taylor expanded in im around inf 98.2%

      \[\leadsto \color{blue}{0.001388888888888889 \cdot \left({im}^{6} \cdot \cos re\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 6.8 \cdot 10^{-7}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 10^{+48}:\\ \;\;\;\;0.5 \cdot \left(e^{im} + e^{-im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.001388888888888889 \cdot \left(\cos re \cdot {im}^{6}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 86.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 6.8 \cdot 10^{-7}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\right)\\ \mathbf{elif}\;im \leq 10^{+48}:\\ \;\;\;\;0.5 \cdot \left(e^{im} + e^{-im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.001388888888888889 \cdot \left(\cos re \cdot {im}^{6}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 6.8e-7)
   (* (cos re) (* 0.5 (fma im im 2.0)))
   (if (<= im 1e+48)
     (* 0.5 (+ (exp im) (exp (- im))))
     (* 0.001388888888888889 (* (cos re) (pow im 6.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 6.8e-7) {
		tmp = cos(re) * (0.5 * fma(im, im, 2.0));
	} else if (im <= 1e+48) {
		tmp = 0.5 * (exp(im) + exp(-im));
	} else {
		tmp = 0.001388888888888889 * (cos(re) * pow(im, 6.0));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 6.8e-7)
		tmp = Float64(cos(re) * Float64(0.5 * fma(im, im, 2.0)));
	elseif (im <= 1e+48)
		tmp = Float64(0.5 * Float64(exp(im) + exp(Float64(-im))));
	else
		tmp = Float64(0.001388888888888889 * Float64(cos(re) * (im ^ 6.0)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 6.8e-7], N[(N[Cos[re], $MachinePrecision] * N[(0.5 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1e+48], N[(0.5 * N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.001388888888888889 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 6.79999999999999948e-7

   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 92.8%

      \[\leadsto \cos re \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(0.5 + {im}^{2} \cdot \left(0.041666666666666664 + 0.001388888888888889 \cdot {im}^{2}\right)\right)\right)} \]

   6. Taylor expanded in im around 0 81.5%

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

      1. associate-*r* 81.5%

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

      2. distribute-rgt1-in 81.5%

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

      3. metadata-eval 81.5%

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

      4. distribute-lft-in 81.5%

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

      5. +-commutative 81.5%

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

      6. *-commutative 81.5%

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

      7. +-commutative 81.5%

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

      8. unpow2 81.5%

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

      9. fma-define 81.5%

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

   8. Simplified 81.5%

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

   if 6.79999999999999948e-7 < im < 1.00000000000000004e48

   1. Initial program 99.5%

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

   3. Taylor expanded in re around 0 79.7%

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

   if 1.00000000000000004e48 < 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 98.2%

      \[\leadsto \cos re \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(0.5 + {im}^{2} \cdot \left(0.041666666666666664 + 0.001388888888888889 \cdot {im}^{2}\right)\right)\right)} \]

   6. Taylor expanded in im around inf 98.2%

      \[\leadsto \color{blue}{0.001388888888888889 \cdot \left({im}^{6} \cdot \cos re\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 6.8 \cdot 10^{-7}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\right)\\ \mathbf{elif}\;im \leq 10^{+48}:\\ \;\;\;\;0.5 \cdot \left(e^{im} + e^{-im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.001388888888888889 \cdot \left(\cos re \cdot {im}^{6}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.5 \cdot 10^{+19}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{im}^{12} \cdot 1.9290123456790124 \cdot 10^{-6}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.5e+19) (cos re) (sqrt (* (pow im 12.0) 1.9290123456790124e-6))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.5e+19) {
		tmp = cos(re);
	} else {
		tmp = sqrt((pow(im, 12.0) * 1.9290123456790124e-6));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.5d+19) then
        tmp = cos(re)
    else
        tmp = sqrt(((im ** 12.0d0) * 1.9290123456790124d-6))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.5e+19) {
		tmp = Math.cos(re);
	} else {
		tmp = Math.sqrt((Math.pow(im, 12.0) * 1.9290123456790124e-6));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.5e+19:
		tmp = math.cos(re)
	else:
		tmp = math.sqrt((math.pow(im, 12.0) * 1.9290123456790124e-6))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.5e+19)
		tmp = cos(re);
	else
		tmp = sqrt(Float64((im ^ 12.0) * 1.9290123456790124e-6));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.5e+19)
		tmp = cos(re);
	else
		tmp = sqrt(((im ^ 12.0) * 1.9290123456790124e-6));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.5e+19], N[Cos[re], $MachinePrecision], N[Sqrt[N[(N[Power[im, 12.0], $MachinePrecision] * 1.9290123456790124e-6), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.5e19

   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 63.6%

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

   if 1.5e19 < 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 89.7%

      \[\leadsto \cos re \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(0.5 + {im}^{2} \cdot \left(0.041666666666666664 + 0.001388888888888889 \cdot {im}^{2}\right)\right)\right)} \]

   6. Taylor expanded in im around inf 89.7%

      \[\leadsto \color{blue}{0.001388888888888889 \cdot \left({im}^{6} \cdot \cos re\right)} \]

   7. Taylor expanded in re around 0 71.4%

      \[\leadsto \color{blue}{0.001388888888888889 \cdot {im}^{6}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 71.4%

         \[\leadsto \color{blue}{\sqrt{0.001388888888888889 \cdot {im}^{6}} \cdot \sqrt{0.001388888888888889 \cdot {im}^{6}}} \]

      2. sqrt-unprod 74.8%

         \[\leadsto \color{blue}{\sqrt{\left(0.001388888888888889 \cdot {im}^{6}\right) \cdot \left(0.001388888888888889 \cdot {im}^{6}\right)}} \]

      3. *-commutative 74.8%

         \[\leadsto \sqrt{\color{blue}{\left({im}^{6} \cdot 0.001388888888888889\right)} \cdot \left(0.001388888888888889 \cdot {im}^{6}\right)} \]

      4. *-commutative 74.8%

         \[\leadsto \sqrt{\left({im}^{6} \cdot 0.001388888888888889\right) \cdot \color{blue}{\left({im}^{6} \cdot 0.001388888888888889\right)}} \]

      5. swap-sqr 74.8%

         \[\leadsto \sqrt{\color{blue}{\left({im}^{6} \cdot {im}^{6}\right) \cdot \left(0.001388888888888889 \cdot 0.001388888888888889\right)}} \]

      6. pow-prod-up 74.8%

         \[\leadsto \sqrt{\color{blue}{{im}^{\left(6 + 6\right)}} \cdot \left(0.001388888888888889 \cdot 0.001388888888888889\right)} \]

      7. metadata-eval 74.8%

         \[\leadsto \sqrt{{im}^{\color{blue}{12}} \cdot \left(0.001388888888888889 \cdot 0.001388888888888889\right)} \]

      8. metadata-eval 74.8%

         \[\leadsto \sqrt{{im}^{12} \cdot \color{blue}{1.9290123456790124 \cdot 10^{-6}}} \]

   9. Applied egg-rr 74.8%

      \[\leadsto \color{blue}{\sqrt{{im}^{12} \cdot 1.9290123456790124 \cdot 10^{-6}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.5 \cdot 10^{+19}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{im}^{12} \cdot 1.9290123456790124 \cdot 10^{-6}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 66.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.1 \cdot 10^{+20}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;1 + 0.001388888888888889 \cdot {im}^{6}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.1e+20) (cos re) (+ 1.0 (* 0.001388888888888889 (pow im 6.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.1e+20) {
		tmp = cos(re);
	} else {
		tmp = 1.0 + (0.001388888888888889 * pow(im, 6.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 2.1d+20) then
        tmp = cos(re)
    else
        tmp = 1.0d0 + (0.001388888888888889d0 * (im ** 6.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.1e+20) {
		tmp = Math.cos(re);
	} else {
		tmp = 1.0 + (0.001388888888888889 * Math.pow(im, 6.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.1e+20:
		tmp = math.cos(re)
	else:
		tmp = 1.0 + (0.001388888888888889 * math.pow(im, 6.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.1e+20)
		tmp = cos(re);
	else
		tmp = Float64(1.0 + Float64(0.001388888888888889 * (im ^ 6.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.1e+20)
		tmp = cos(re);
	else
		tmp = 1.0 + (0.001388888888888889 * (im ^ 6.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.1e+20], N[Cos[re], $MachinePrecision], N[(1.0 + N[(0.001388888888888889 * N[Power[im, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2.1e20

   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 63.6%

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

   if 2.1e20 < 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 89.7%

      \[\leadsto \cos re \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(0.5 + {im}^{2} \cdot \left(0.041666666666666664 + 0.001388888888888889 \cdot {im}^{2}\right)\right)\right)} \]

   6. Taylor expanded in im around inf 89.7%

      \[\leadsto \cos re \cdot \left(1 + \color{blue}{0.001388888888888889 \cdot {im}^{6}}\right) \]

   7. Taylor expanded in re around 0 71.4%

      \[\leadsto \color{blue}{1 + 0.001388888888888889 \cdot {im}^{6}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.1 \cdot 10^{+20}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;1 + 0.001388888888888889 \cdot {im}^{6}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 66.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.3 \cdot 10^{+19}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.001388888888888889 \cdot {im}^{6}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.3e+19) (cos re) (* 0.001388888888888889 (pow im 6.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.3e+19) {
		tmp = cos(re);
	} else {
		tmp = 0.001388888888888889 * pow(im, 6.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.3d+19) then
        tmp = cos(re)
    else
        tmp = 0.001388888888888889d0 * (im ** 6.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.3e+19) {
		tmp = Math.cos(re);
	} else {
		tmp = 0.001388888888888889 * Math.pow(im, 6.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.3e+19:
		tmp = math.cos(re)
	else:
		tmp = 0.001388888888888889 * math.pow(im, 6.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.3e+19)
		tmp = cos(re);
	else
		tmp = Float64(0.001388888888888889 * (im ^ 6.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.3e+19)
		tmp = cos(re);
	else
		tmp = 0.001388888888888889 * (im ^ 6.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.3e+19], N[Cos[re], $MachinePrecision], N[(0.001388888888888889 * N[Power[im, 6.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.3e19

   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 63.6%

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

   if 1.3e19 < 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 89.7%

      \[\leadsto \cos re \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(0.5 + {im}^{2} \cdot \left(0.041666666666666664 + 0.001388888888888889 \cdot {im}^{2}\right)\right)\right)} \]

   6. Taylor expanded in im around inf 89.7%

      \[\leadsto \color{blue}{0.001388888888888889 \cdot \left({im}^{6} \cdot \cos re\right)} \]

   7. Taylor expanded in re around 0 71.4%

      \[\leadsto \color{blue}{0.001388888888888889 \cdot {im}^{6}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.3 \cdot 10^{+19}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.001388888888888889 \cdot {im}^{6}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 52.0% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
2. 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 50.6%

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

6. Final simplification 50.6%

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

Alternative 13: 29.0% accurate, 308.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return 1.0

Julia

function code(re, im)
	return 1.0
end

MATLAB

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

Wolfram

code[re_, im_] := 1.0

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
2. 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 91.1%

   \[\leadsto \cos re \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(0.5 + {im}^{2} \cdot \left(0.041666666666666664 + 0.001388888888888889 \cdot {im}^{2}\right)\right)\right)} \]

7. Applied egg-rr 26.4%

   \[\leadsto \color{blue}{\frac{\cos re - 0.043055555555555555 \cdot \cos re}{\cos re - 0.043055555555555555 \cdot \cos re}} \]
8. Step-by-step derivation

   1. *-inverses 26.4%

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

9. Simplified 26.4%

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

10. Final simplification 26.4%

    \[\leadsto 1 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024077 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))