math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 8.0s

Alternatives: 16

Speedup: 1.5×

• 49.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \cos re \cdot \cosh im \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(N[Cos[re], $MachinePrecision] * N[Cosh[im], $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. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-exp.f64 N/A

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

   4. lift-exp.f64 N/A

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

   5. lift-neg.f64 N/A

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

   6. cosh-undef N/A

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 \cdot \cosh im\right)} \]

   7. *-commutative N/A

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(\cosh im \cdot 2\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(\cosh im \cdot 2\right)} \]

   9. lower-cosh.f64 100.0

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

4. Applied rewrites 100.0%

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

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\cosh im \cdot 2\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(\cosh im \cdot 2\right) \cdot \left(\frac{1}{2} \cdot \cos re\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\cosh im \cdot 2\right)} \cdot \left(\frac{1}{2} \cdot \cos re\right) \]

   4. *-commutative N/A

      \[\leadsto \color{blue}{\left(2 \cdot \cosh im\right)} \cdot \left(\frac{1}{2} \cdot \cos re\right) \]

   5. lift-cosh.f64 N/A

      \[\leadsto \left(2 \cdot \color{blue}{\cosh im}\right) \cdot \left(\frac{1}{2} \cdot \cos re\right) \]

   6. cosh-undef N/A

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

   7. lift-exp.f64 N/A

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

   8. lift-neg.f64 N/A

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

   9. lift-exp.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-+.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. associate-*r* N/A

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

   14. metadata-eval N/A

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

   15. div-inv N/A

       \[\leadsto \color{blue}{\frac{e^{-im} + e^{im}}{2}} \cdot \cos re \]

   16. lift-+.f64 N/A

       \[\leadsto \frac{\color{blue}{e^{-im} + e^{im}}}{2} \cdot \cos re \]

   17. +-commutative N/A

       \[\leadsto \frac{\color{blue}{e^{im} + e^{-im}}}{2} \cdot \cos re \]

   18. lift-exp.f64 N/A

       \[\leadsto \frac{\color{blue}{e^{im}} + e^{-im}}{2} \cdot \cos re \]

   19. lift-exp.f64 N/A

       \[\leadsto \frac{e^{im} + \color{blue}{e^{-im}}}{2} \cdot \cos re \]

   20. lift-neg.f64 N/A

       \[\leadsto \frac{e^{im} + e^{\color{blue}{\mathsf{neg}\left(im\right)}}}{2} \cdot \cos re \]

   21. cosh-def N/A

       \[\leadsto \color{blue}{\cosh im} \cdot \cos re \]

   22. lift-cosh.f64 N/A

       \[\leadsto \color{blue}{\cosh im} \cdot \cos re \]

   23. lower-*.f64 100.0

       \[\leadsto \color{blue}{\cosh im \cdot \cos re} \]

6. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\cosh im \cdot \cos re} \]

7. Final simplification 100.0%

   \[\leadsto \cos re \cdot \cosh im \]
8. Add Preprocessing

Alternative 2: 81.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ t_1 := \left(e^{im} + e^{-im}\right) \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-0.5, re \cdot re, 1\right) \cdot \cosh im\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 - im\right) + e^{im}\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re))) (t_1 (* (+ (exp im) (exp (- im))) t_0)))
   (if (<= t_1 (- INFINITY))
     (* (fma -0.5 (* re re) 1.0) (cosh im))
     (if (<= t_1 2.0)
       (* (fma im im 2.0) t_0)
       (* (+ (- 1.0 im) (exp im)) 0.5)))))

C

double code(double re, double im) {
	double t_0 = 0.5 * cos(re);
	double t_1 = (exp(im) + exp(-im)) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(-0.5, (re * re), 1.0) * cosh(im);
	} else if (t_1 <= 2.0) {
		tmp = fma(im, im, 2.0) * t_0;
	} else {
		tmp = ((1.0 - im) + exp(im)) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * cos(re))
	t_1 = Float64(Float64(exp(im) + exp(Float64(-im))) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(-0.5, Float64(re * re), 1.0) * cosh(im));
	elseif (t_1 <= 2.0)
		tmp = Float64(fma(im, im, 2.0) * t_0);
	else
		tmp = Float64(Float64(Float64(1.0 - im) + exp(im)) * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(-0.5 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * N[Cosh[im], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(N[(im * im + 2.0), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(N[(1.0 - im), $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -inf.0

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. cosh-undef N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 \cdot \cosh im\right)} \]

      7. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(\cosh im \cdot 2\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(\cosh im \cdot 2\right)} \]

      9. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\cosh im \cdot 2\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\cosh im \cdot 2\right) \cdot \left(\frac{1}{2} \cdot \cos re\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\cosh im \cdot 2\right)} \cdot \left(\frac{1}{2} \cdot \cos re\right) \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(2 \cdot \cosh im\right)} \cdot \left(\frac{1}{2} \cdot \cos re\right) \]

      5. lift-cosh.f64 N/A

         \[\leadsto \left(2 \cdot \color{blue}{\cosh im}\right) \cdot \left(\frac{1}{2} \cdot \cos re\right) \]

      6. cosh-undef N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-exp.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. metadata-eval N/A

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

      15. div-inv N/A

          \[\leadsto \color{blue}{\frac{e^{-im} + e^{im}}{2}} \cdot \cos re \]

      16. lift-+.f64 N/A

          \[\leadsto \frac{\color{blue}{e^{-im} + e^{im}}}{2} \cdot \cos re \]

      17. +-commutative N/A

          \[\leadsto \frac{\color{blue}{e^{im} + e^{-im}}}{2} \cdot \cos re \]

      18. lift-exp.f64 N/A

          \[\leadsto \frac{\color{blue}{e^{im}} + e^{-im}}{2} \cdot \cos re \]

      19. lift-exp.f64 N/A

          \[\leadsto \frac{e^{im} + \color{blue}{e^{-im}}}{2} \cdot \cos re \]

      20. lift-neg.f64 N/A

          \[\leadsto \frac{e^{im} + e^{\color{blue}{\mathsf{neg}\left(im\right)}}}{2} \cdot \cos re \]

      21. cosh-def N/A

          \[\leadsto \color{blue}{\cosh im} \cdot \cos re \]

      22. lift-cosh.f64 N/A

          \[\leadsto \color{blue}{\cosh im} \cdot \cos re \]

      23. lower-*.f64 100.0

          \[\leadsto \color{blue}{\cosh im \cdot \cos re} \]

   6. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\cosh im \cdot \cos re} \]

   7. Taylor expanded in re around 0

      \[\leadsto \cosh im \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \cosh im \cdot \color{blue}{\left(\frac{-1}{2} \cdot {re}^{2} + 1\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \cosh im \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{2}, {re}^{2}, 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \cosh im \cdot \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{re \cdot re}, 1\right) \]

      4. lower-*.f64 100.0

         \[\leadsto \cosh im \cdot \mathsf{fma}\left(-0.5, \color{blue}{re \cdot re}, 1\right) \]

   9. Applied rewrites 100.0%

      \[\leadsto \cosh im \cdot \color{blue}{\mathsf{fma}\left(-0.5, re \cdot re, 1\right)} \]

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 2 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(e^{-im} + e^{im}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in im around 0

      \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left(1 + -1 \cdot im\right)} + e^{im}\right) \]
   7. Step-by-step derivation

      1. neg-mul-1 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(1 + \color{blue}{\left(\mathsf{neg}\left(im\right)\right)}\right) + e^{im}\right) \]

      2. unsub-neg N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left(1 - im\right)} + e^{im}\right) \]

      3. lower--.f64 54.0

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

   8. Applied rewrites 54.0%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(1 - im\right)} + e^{im}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.5%

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

Alternative 3: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ t_1 := \left(e^{im} + e^{-im}\right) \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-0.5, re \cdot re, 1\right) \cdot \cosh im\\ \mathbf{elif}\;t\_1 \leq 0.9908980035424527:\\ \;\;\;\;2 \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \cosh im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re))) (t_1 (* (+ (exp im) (exp (- im))) t_0)))
   (if (<= t_1 (- INFINITY))
     (* (fma -0.5 (* re re) 1.0) (cosh im))
     (if (<= t_1 0.9908980035424527) (* 2.0 t_0) (* 1.0 (cosh im))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * cos(re);
	double t_1 = (exp(im) + exp(-im)) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(-0.5, (re * re), 1.0) * cosh(im);
	} else if (t_1 <= 0.9908980035424527) {
		tmp = 2.0 * t_0;
	} else {
		tmp = 1.0 * cosh(im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * cos(re))
	t_1 = Float64(Float64(exp(im) + exp(Float64(-im))) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(-0.5, Float64(re * re), 1.0) * cosh(im));
	elseif (t_1 <= 0.9908980035424527)
		tmp = Float64(2.0 * t_0);
	else
		tmp = Float64(1.0 * cosh(im));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(-0.5 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * N[Cosh[im], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9908980035424527], N[(2.0 * t$95$0), $MachinePrecision], N[(1.0 * N[Cosh[im], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -inf.0

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. cosh-undef N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 \cdot \cosh im\right)} \]

      7. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(\cosh im \cdot 2\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(\cosh im \cdot 2\right)} \]

      9. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\cosh im \cdot 2\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\cosh im \cdot 2\right) \cdot \left(\frac{1}{2} \cdot \cos re\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\cosh im \cdot 2\right)} \cdot \left(\frac{1}{2} \cdot \cos re\right) \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(2 \cdot \cosh im\right)} \cdot \left(\frac{1}{2} \cdot \cos re\right) \]

      5. lift-cosh.f64 N/A

         \[\leadsto \left(2 \cdot \color{blue}{\cosh im}\right) \cdot \left(\frac{1}{2} \cdot \cos re\right) \]

      6. cosh-undef N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-exp.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. metadata-eval N/A

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

      15. div-inv N/A

          \[\leadsto \color{blue}{\frac{e^{-im} + e^{im}}{2}} \cdot \cos re \]

      16. lift-+.f64 N/A

          \[\leadsto \frac{\color{blue}{e^{-im} + e^{im}}}{2} \cdot \cos re \]

      17. +-commutative N/A

          \[\leadsto \frac{\color{blue}{e^{im} + e^{-im}}}{2} \cdot \cos re \]

      18. lift-exp.f64 N/A

          \[\leadsto \frac{\color{blue}{e^{im}} + e^{-im}}{2} \cdot \cos re \]

      19. lift-exp.f64 N/A

          \[\leadsto \frac{e^{im} + \color{blue}{e^{-im}}}{2} \cdot \cos re \]

      20. lift-neg.f64 N/A

          \[\leadsto \frac{e^{im} + e^{\color{blue}{\mathsf{neg}\left(im\right)}}}{2} \cdot \cos re \]

      21. cosh-def N/A

          \[\leadsto \color{blue}{\cosh im} \cdot \cos re \]

      22. lift-cosh.f64 N/A

          \[\leadsto \color{blue}{\cosh im} \cdot \cos re \]

      23. lower-*.f64 100.0

          \[\leadsto \color{blue}{\cosh im \cdot \cos re} \]

   6. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\cosh im \cdot \cos re} \]

   7. Taylor expanded in re around 0

      \[\leadsto \cosh im \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \cosh im \cdot \color{blue}{\left(\frac{-1}{2} \cdot {re}^{2} + 1\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \cosh im \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{2}, {re}^{2}, 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \cosh im \cdot \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{re \cdot re}, 1\right) \]

      4. lower-*.f64 100.0

         \[\leadsto \cosh im \cdot \mathsf{fma}\left(-0.5, \color{blue}{re \cdot re}, 1\right) \]

   9. Applied rewrites 100.0%

      \[\leadsto \cosh im \cdot \color{blue}{\mathsf{fma}\left(-0.5, re \cdot re, 1\right)} \]

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.990898003542452721

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 100.0%

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

   if 0.990898003542452721 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. cosh-undef N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 \cdot \cosh im\right)} \]

      7. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(\cosh im \cdot 2\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(\cosh im \cdot 2\right)} \]

      9. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\cosh im \cdot 2\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\cosh im \cdot 2\right) \cdot \left(\frac{1}{2} \cdot \cos re\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\cosh im \cdot 2\right)} \cdot \left(\frac{1}{2} \cdot \cos re\right) \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(2 \cdot \cosh im\right)} \cdot \left(\frac{1}{2} \cdot \cos re\right) \]

      5. lift-cosh.f64 N/A

         \[\leadsto \left(2 \cdot \color{blue}{\cosh im}\right) \cdot \left(\frac{1}{2} \cdot \cos re\right) \]

      6. cosh-undef N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-exp.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. metadata-eval N/A

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

      15. div-inv N/A

          \[\leadsto \color{blue}{\frac{e^{-im} + e^{im}}{2}} \cdot \cos re \]

      16. lift-+.f64 N/A

          \[\leadsto \frac{\color{blue}{e^{-im} + e^{im}}}{2} \cdot \cos re \]

      17. +-commutative N/A

          \[\leadsto \frac{\color{blue}{e^{im} + e^{-im}}}{2} \cdot \cos re \]

      18. lift-exp.f64 N/A

          \[\leadsto \frac{\color{blue}{e^{im}} + e^{-im}}{2} \cdot \cos re \]

      19. lift-exp.f64 N/A

          \[\leadsto \frac{e^{im} + \color{blue}{e^{-im}}}{2} \cdot \cos re \]

      20. lift-neg.f64 N/A

          \[\leadsto \frac{e^{im} + e^{\color{blue}{\mathsf{neg}\left(im\right)}}}{2} \cdot \cos re \]

      21. cosh-def N/A

          \[\leadsto \color{blue}{\cosh im} \cdot \cos re \]

      22. lift-cosh.f64 N/A

          \[\leadsto \color{blue}{\cosh im} \cdot \cos re \]

      23. lower-*.f64 100.0

          \[\leadsto \color{blue}{\cosh im \cdot \cos re} \]

   6. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\cosh im \cdot \cos re} \]

   7. Taylor expanded in re around 0

      \[\leadsto \cosh im \cdot \color{blue}{1} \]
   8. Step-by-step derivation

   9. Applied rewrites 99.5%

      \[\leadsto \cosh im \cdot \color{blue}{1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-0.5, re \cdot re, 1\right) \cdot \cosh im\\ \mathbf{elif}\;\left(e^{im} + e^{-im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq 0.9908980035424527:\\ \;\;\;\;2 \cdot \left(0.5 \cdot \cos re\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \cosh im\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ t_1 := \left(e^{im} + e^{-im}\right) \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889 \cdot \left(re \cdot re\right), re \cdot re, -0.5\right), re \cdot re, 1\right) \cdot 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;t\_1 \leq 0.9908980035424527:\\ \;\;\;\;2 \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \cosh im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re))) (t_1 (* (+ (exp im) (exp (- im))) t_0)))
   (if (<= t_1 (- INFINITY))
     (*
      (*
       (fma
        (fma (* -0.001388888888888889 (* re re)) (* re re) -0.5)
        (* re re)
        1.0)
       0.5)
      (fma im im 2.0))
     (if (<= t_1 0.9908980035424527) (* 2.0 t_0) (* 1.0 (cosh im))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * cos(re);
	double t_1 = (exp(im) + exp(-im)) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (fma(fma((-0.001388888888888889 * (re * re)), (re * re), -0.5), (re * re), 1.0) * 0.5) * fma(im, im, 2.0);
	} else if (t_1 <= 0.9908980035424527) {
		tmp = 2.0 * t_0;
	} else {
		tmp = 1.0 * cosh(im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * cos(re))
	t_1 = Float64(Float64(exp(im) + exp(Float64(-im))) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(fma(fma(Float64(-0.001388888888888889 * Float64(re * re)), Float64(re * re), -0.5), Float64(re * re), 1.0) * 0.5) * fma(im, im, 2.0));
	elseif (t_1 <= 0.9908980035424527)
		tmp = Float64(2.0 * t_0);
	else
		tmp = Float64(1.0 * cosh(im));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(N[(-0.001388888888888889 * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(re * re), $MachinePrecision] + -0.5), $MachinePrecision] * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9908980035424527], N[(2.0 * t$95$0), $MachinePrecision], N[(1.0 * N[Cosh[im], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -inf.0

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 61.2

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

   5. Applied rewrites 61.2%

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

   6. Taylor expanded in re around 0

      \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(1 + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right) + 1\right)}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\color{blue}{\left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right) \cdot {re}^{2}} + 1\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\mathsf{fma}\left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}, {re}^{2}, 1\right)}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      4. sub-neg N/A

         \[\leadsto \left(\frac{1}{2} \cdot \mathsf{fma}\left(\color{blue}{{re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, {re}^{2}, 1\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) \cdot {re}^{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), {re}^{2}, 1\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      6. metadata-eval N/A

         \[\leadsto \left(\frac{1}{2} \cdot \mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) \cdot {re}^{2} + \color{blue}{\frac{-1}{2}}, {re}^{2}, 1\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}, {re}^{2}, \frac{-1}{2}\right)}, {re}^{2}, 1\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      8. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{720} \cdot {re}^{2} + \frac{1}{24}}, {re}^{2}, \frac{-1}{2}\right), {re}^{2}, 1\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{720}, {re}^{2}, \frac{1}{24}\right)}, {re}^{2}, \frac{-1}{2}\right), {re}^{2}, 1\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      10. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, \color{blue}{re \cdot re}, \frac{1}{24}\right), {re}^{2}, \frac{-1}{2}\right), {re}^{2}, 1\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, \color{blue}{re \cdot re}, \frac{1}{24}\right), {re}^{2}, \frac{-1}{2}\right), {re}^{2}, 1\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      12. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, re \cdot re, \frac{1}{24}\right), \color{blue}{re \cdot re}, \frac{-1}{2}\right), {re}^{2}, 1\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, re \cdot re, \frac{1}{24}\right), \color{blue}{re \cdot re}, \frac{-1}{2}\right), {re}^{2}, 1\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      14. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{-1}{2}\right), \color{blue}{re \cdot re}, 1\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      15. lower-*.f64 91.7

          \[\leadsto \left(0.5 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, -0.5\right), \color{blue}{re \cdot re}, 1\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   8. Applied rewrites 91.7%

      \[\leadsto \left(0.5 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, -0.5\right), re \cdot re, 1\right)}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   9. Taylor expanded in re around inf

      \[\leadsto \left(\frac{1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720} \cdot {re}^{2}, re \cdot re, \frac{-1}{2}\right), re \cdot re, 1\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 91.7%

       \[\leadsto \left(0.5 \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889 \cdot \left(re \cdot re\right), re \cdot re, -0.5\right), re \cdot re, 1\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.990898003542452721

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 100.0%

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

   if 0.990898003542452721 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. cosh-undef N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 \cdot \cosh im\right)} \]

      7. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(\cosh im \cdot 2\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(\cosh im \cdot 2\right)} \]

      9. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\cosh im \cdot 2\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\cosh im \cdot 2\right) \cdot \left(\frac{1}{2} \cdot \cos re\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\cosh im \cdot 2\right)} \cdot \left(\frac{1}{2} \cdot \cos re\right) \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(2 \cdot \cosh im\right)} \cdot \left(\frac{1}{2} \cdot \cos re\right) \]

      5. lift-cosh.f64 N/A

         \[\leadsto \left(2 \cdot \color{blue}{\cosh im}\right) \cdot \left(\frac{1}{2} \cdot \cos re\right) \]

      6. cosh-undef N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-exp.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. metadata-eval N/A

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

      15. div-inv N/A

          \[\leadsto \color{blue}{\frac{e^{-im} + e^{im}}{2}} \cdot \cos re \]

      16. lift-+.f64 N/A

          \[\leadsto \frac{\color{blue}{e^{-im} + e^{im}}}{2} \cdot \cos re \]

      17. +-commutative N/A

          \[\leadsto \frac{\color{blue}{e^{im} + e^{-im}}}{2} \cdot \cos re \]

      18. lift-exp.f64 N/A

          \[\leadsto \frac{\color{blue}{e^{im}} + e^{-im}}{2} \cdot \cos re \]

      19. lift-exp.f64 N/A

          \[\leadsto \frac{e^{im} + \color{blue}{e^{-im}}}{2} \cdot \cos re \]

      20. lift-neg.f64 N/A

          \[\leadsto \frac{e^{im} + e^{\color{blue}{\mathsf{neg}\left(im\right)}}}{2} \cdot \cos re \]

      21. cosh-def N/A

          \[\leadsto \color{blue}{\cosh im} \cdot \cos re \]

      22. lift-cosh.f64 N/A

          \[\leadsto \color{blue}{\cosh im} \cdot \cos re \]

      23. lower-*.f64 100.0

          \[\leadsto \color{blue}{\cosh im \cdot \cos re} \]

   6. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\cosh im \cdot \cos re} \]

   7. Taylor expanded in re around 0

      \[\leadsto \cosh im \cdot \color{blue}{1} \]
   8. Step-by-step derivation

   9. Applied rewrites 99.5%

      \[\leadsto \cosh im \cdot \color{blue}{1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889 \cdot \left(re \cdot re\right), re \cdot re, -0.5\right), re \cdot re, 1\right) \cdot 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;\left(e^{im} + e^{-im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq 0.9908980035424527:\\ \;\;\;\;2 \cdot \left(0.5 \cdot \cos re\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \cosh im\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq 10:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \cos re\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \cosh im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (+ (exp im) (exp (- im))) (* 0.5 (cos re))) 10.0)
   (*
    (fma
     (fma
      (fma 0.001388888888888889 (* im im) 0.041666666666666664)
      (* im im)
      0.5)
     (* im im)
     1.0)
    (cos re))
   (* 1.0 (cosh im))))

C

double code(double re, double im) {
	double tmp;
	if (((exp(im) + exp(-im)) * (0.5 * cos(re))) <= 10.0) {
		tmp = fma(fma(fma(0.001388888888888889, (im * im), 0.041666666666666664), (im * im), 0.5), (im * im), 1.0) * cos(re);
	} else {
		tmp = 1.0 * cosh(im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(exp(im) + exp(Float64(-im))) * Float64(0.5 * cos(re))) <= 10.0)
		tmp = Float64(fma(fma(fma(0.001388888888888889, Float64(im * im), 0.041666666666666664), Float64(im * im), 0.5), Float64(im * im), 1.0) * cos(re));
	else
		tmp = Float64(1.0 * cosh(im));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 10.0], N[(N[(N[(N[(0.001388888888888889 * N[(im * im), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[Cos[re], $MachinePrecision]), $MachinePrecision], N[(1.0 * N[Cosh[im], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 10

   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

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(e^{-im} + e^{im}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 43.5%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re + {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right)} \]
   7. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \cos re + \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right) + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right)\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\cos re + {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right)\right) + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right)} \]

      3. *-lft-identity N/A

         \[\leadsto \left(\color{blue}{1 \cdot \cos re} + {im}^{2} \cdot \left(\frac{1}{2} \cdot \cos re\right)\right) + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(1 \cdot \cos re + \color{blue}{\left({im}^{2} \cdot \frac{1}{2}\right) \cdot \cos re}\right) + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \left(1 \cdot \cos re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right)} \cdot \cos re\right) + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) \]

      6. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\cos re \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)} + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \cos re \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right) + \color{blue}{\left({im}^{2} \cdot {im}^{2}\right) \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right)} \]

      8. *-commutative N/A

         \[\leadsto \cos re \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right) + \color{blue}{\left(\frac{1}{720} \cdot \left({im}^{2} \cdot \cos re\right) + \frac{1}{24} \cdot \cos re\right) \cdot \left({im}^{2} \cdot {im}^{2}\right)} \]

   8. Applied rewrites 98.9%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right)} \]

   if 10 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. cosh-undef N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 \cdot \cosh im\right)} \]

      7. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(\cosh im \cdot 2\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(\cosh im \cdot 2\right)} \]

      9. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\cosh im \cdot 2\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\cosh im \cdot 2\right) \cdot \left(\frac{1}{2} \cdot \cos re\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\cosh im \cdot 2\right)} \cdot \left(\frac{1}{2} \cdot \cos re\right) \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(2 \cdot \cosh im\right)} \cdot \left(\frac{1}{2} \cdot \cos re\right) \]

      5. lift-cosh.f64 N/A

         \[\leadsto \left(2 \cdot \color{blue}{\cosh im}\right) \cdot \left(\frac{1}{2} \cdot \cos re\right) \]

      6. cosh-undef N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-exp.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. metadata-eval N/A

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

      15. div-inv N/A

          \[\leadsto \color{blue}{\frac{e^{-im} + e^{im}}{2}} \cdot \cos re \]

      16. lift-+.f64 N/A

          \[\leadsto \frac{\color{blue}{e^{-im} + e^{im}}}{2} \cdot \cos re \]

      17. +-commutative N/A

          \[\leadsto \frac{\color{blue}{e^{im} + e^{-im}}}{2} \cdot \cos re \]

      18. lift-exp.f64 N/A

          \[\leadsto \frac{\color{blue}{e^{im}} + e^{-im}}{2} \cdot \cos re \]

      19. lift-exp.f64 N/A

          \[\leadsto \frac{e^{im} + \color{blue}{e^{-im}}}{2} \cdot \cos re \]

      20. lift-neg.f64 N/A

          \[\leadsto \frac{e^{im} + e^{\color{blue}{\mathsf{neg}\left(im\right)}}}{2} \cdot \cos re \]

      21. cosh-def N/A

          \[\leadsto \color{blue}{\cosh im} \cdot \cos re \]

      22. lift-cosh.f64 N/A

          \[\leadsto \color{blue}{\cosh im} \cdot \cos re \]

      23. lower-*.f64 100.0

          \[\leadsto \color{blue}{\cosh im \cdot \cos re} \]

   6. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\cosh im \cdot \cos re} \]

   7. Taylor expanded in re around 0

      \[\leadsto \cosh im \cdot \color{blue}{1} \]
   8. Step-by-step derivation

   9. Applied rewrites 100.0%

      \[\leadsto \cosh im \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq 10:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \cos re\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \cosh im\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq -0.005:\\ \;\;\;\;\left(\left(\left(re \cdot re\right) \cdot -0.5\right) \cdot 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \cosh im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (+ (exp im) (exp (- im))) (* 0.5 (cos re))) -0.005)
   (* (* (* (* re re) -0.5) 0.5) (fma im im 2.0))
   (* 1.0 (cosh im))))

C

double code(double re, double im) {
	double tmp;
	if (((exp(im) + exp(-im)) * (0.5 * cos(re))) <= -0.005) {
		tmp = (((re * re) * -0.5) * 0.5) * fma(im, im, 2.0);
	} else {
		tmp = 1.0 * cosh(im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(exp(im) + exp(Float64(-im))) * Float64(0.5 * cos(re))) <= -0.005)
		tmp = Float64(Float64(Float64(Float64(re * re) * -0.5) * 0.5) * fma(im, im, 2.0));
	else
		tmp = Float64(1.0 * cosh(im));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.005], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.5), $MachinePrecision] * 0.5), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[Cosh[im], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -0.0050000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 81.4

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

   5. Applied rewrites 81.4%

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

   6. Taylor expanded in re around 0

      \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{-1}{2} \cdot {re}^{2} + 1\right)}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{2}, {re}^{2}, 1\right)}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      3. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{re \cdot re}, 1\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      4. lower-*.f64 46.2

         \[\leadsto \left(0.5 \cdot \mathsf{fma}\left(-0.5, \color{blue}{re \cdot re}, 1\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   8. Applied rewrites 46.2%

      \[\leadsto \left(0.5 \cdot \color{blue}{\mathsf{fma}\left(-0.5, re \cdot re, 1\right)}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   9. Taylor expanded in re around inf

      \[\leadsto \left(\frac{1}{2} \cdot \left(\frac{-1}{2} \cdot \color{blue}{{re}^{2}}\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 46.2%

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

   if -0.0050000000000000001 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. cosh-undef N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 \cdot \cosh im\right)} \]

      7. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(\cosh im \cdot 2\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(\cosh im \cdot 2\right)} \]

      9. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\cosh im \cdot 2\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\cosh im \cdot 2\right) \cdot \left(\frac{1}{2} \cdot \cos re\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\cosh im \cdot 2\right)} \cdot \left(\frac{1}{2} \cdot \cos re\right) \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(2 \cdot \cosh im\right)} \cdot \left(\frac{1}{2} \cdot \cos re\right) \]

      5. lift-cosh.f64 N/A

         \[\leadsto \left(2 \cdot \color{blue}{\cosh im}\right) \cdot \left(\frac{1}{2} \cdot \cos re\right) \]

      6. cosh-undef N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-exp.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. metadata-eval N/A

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

      15. div-inv N/A

          \[\leadsto \color{blue}{\frac{e^{-im} + e^{im}}{2}} \cdot \cos re \]

      16. lift-+.f64 N/A

          \[\leadsto \frac{\color{blue}{e^{-im} + e^{im}}}{2} \cdot \cos re \]

      17. +-commutative N/A

          \[\leadsto \frac{\color{blue}{e^{im} + e^{-im}}}{2} \cdot \cos re \]

      18. lift-exp.f64 N/A

          \[\leadsto \frac{\color{blue}{e^{im}} + e^{-im}}{2} \cdot \cos re \]

      19. lift-exp.f64 N/A

          \[\leadsto \frac{e^{im} + \color{blue}{e^{-im}}}{2} \cdot \cos re \]

      20. lift-neg.f64 N/A

          \[\leadsto \frac{e^{im} + e^{\color{blue}{\mathsf{neg}\left(im\right)}}}{2} \cdot \cos re \]

      21. cosh-def N/A

          \[\leadsto \color{blue}{\cosh im} \cdot \cos re \]

      22. lift-cosh.f64 N/A

          \[\leadsto \color{blue}{\cosh im} \cdot \cos re \]

      23. lower-*.f64 100.0

          \[\leadsto \color{blue}{\cosh im \cdot \cos re} \]

   6. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\cosh im \cdot \cos re} \]

   7. Taylor expanded in re around 0

      \[\leadsto \cosh im \cdot \color{blue}{1} \]
   8. Step-by-step derivation

   9. Applied rewrites 91.0%

      \[\leadsto \cosh im \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq -0.005:\\ \;\;\;\;\left(\left(\left(re \cdot re\right) \cdot -0.5\right) \cdot 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \cosh im\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 47.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(re, im):
	tmp = 0
	if ((math.exp(im) + math.exp(-im)) * (0.5 * math.cos(re))) <= 2.0:
		tmp = 2.0 * 0.5
	else:
		tmp = (im * im) * 0.5
	return tmp

Julia

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

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (((exp(im) + exp(-im)) * (0.5 * cos(re))) <= 2.0)
		tmp = 2.0 * 0.5;
	else
		tmp = (im * im) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 2

   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

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(e^{-im} + e^{im}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 43.6%

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

   6. Taylor expanded in im around 0

      \[\leadsto \frac{1}{2} \cdot \color{blue}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 43.5%

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

   if 2 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(e^{-im} + e^{im}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in im around 0

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. unpow2 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{im \cdot im} + 2\right) \]

      3. lower-fma.f64 49.6

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

   8. Applied rewrites 49.6%

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

   9. Taylor expanded in im around inf

      \[\leadsto \frac{1}{2} \cdot {im}^{\color{blue}{2}} \]
   10. Step-by-step derivation

   11. Applied rewrites 49.6%

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

4. Final simplification 45.9%

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

Alternative 8: 53.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.005:\\ \;\;\;\;\left(\left(\left(re \cdot re\right) \cdot -0.5\right) \cdot 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;\cos re \leq 0.66:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, re \cdot re, -0.25\right), re \cdot re, 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, im, 0.5\right), im, 1\right), im, 1\right) + \left(1 - im\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) -0.005)
   (* (* (* (* re re) -0.5) 0.5) (fma im im 2.0))
   (if (<= (cos re) 0.66)
     (*
      (fma (fma 0.020833333333333332 (* re re) -0.25) (* re re) 0.5)
      (fma im im 2.0))
     (*
      (+ (fma (fma (fma 0.16666666666666666 im 0.5) im 1.0) im 1.0) (- 1.0 im))
      0.5))))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= -0.005) {
		tmp = (((re * re) * -0.5) * 0.5) * fma(im, im, 2.0);
	} else if (cos(re) <= 0.66) {
		tmp = fma(fma(0.020833333333333332, (re * re), -0.25), (re * re), 0.5) * fma(im, im, 2.0);
	} else {
		tmp = (fma(fma(fma(0.16666666666666666, im, 0.5), im, 1.0), im, 1.0) + (1.0 - im)) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= -0.005)
		tmp = Float64(Float64(Float64(Float64(re * re) * -0.5) * 0.5) * fma(im, im, 2.0));
	elseif (cos(re) <= 0.66)
		tmp = Float64(fma(fma(0.020833333333333332, Float64(re * re), -0.25), Float64(re * re), 0.5) * fma(im, im, 2.0));
	else
		tmp = Float64(Float64(fma(fma(fma(0.16666666666666666, im, 0.5), im, 1.0), im, 1.0) + Float64(1.0 - im)) * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], -0.005], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.5), $MachinePrecision] * 0.5), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Cos[re], $MachinePrecision], 0.66], N[(N[(N[(0.020833333333333332 * N[(re * re), $MachinePrecision] + -0.25), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.16666666666666666 * im + 0.5), $MachinePrecision] * im + 1.0), $MachinePrecision] * im + 1.0), $MachinePrecision] + N[(1.0 - im), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 re) < -0.0050000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 81.4

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

   5. Applied rewrites 81.4%

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

   6. Taylor expanded in re around 0

      \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{-1}{2} \cdot {re}^{2} + 1\right)}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{2}, {re}^{2}, 1\right)}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      3. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{re \cdot re}, 1\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      4. lower-*.f64 46.2

         \[\leadsto \left(0.5 \cdot \mathsf{fma}\left(-0.5, \color{blue}{re \cdot re}, 1\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   8. Applied rewrites 46.2%

      \[\leadsto \left(0.5 \cdot \color{blue}{\mathsf{fma}\left(-0.5, re \cdot re, 1\right)}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   9. Taylor expanded in re around inf

      \[\leadsto \left(\frac{1}{2} \cdot \left(\frac{-1}{2} \cdot \color{blue}{{re}^{2}}\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 46.2%

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

   if -0.0050000000000000001 < (cos.f64 re) < 0.660000000000000031

   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

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(e^{-im} + e^{im}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 79.3%

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

   6. Taylor expanded in im around 0

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. unpow2 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{im \cdot im} + 2\right) \]

      3. lower-fma.f64 40.5

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

   8. Applied rewrites 40.5%

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

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right)\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right) + \frac{1}{2}\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]

       2. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right) \cdot {re}^{2}} + \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       3. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]

       4. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{48} \cdot {re}^{2} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}, {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{48} \cdot {re}^{2} + \color{blue}{\frac{-1}{4}}, {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       6. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{48}, {re}^{2}, \frac{-1}{4}\right)}, {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, \color{blue}{re \cdot re}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, \color{blue}{re \cdot re}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, re \cdot re, \frac{-1}{4}\right), \color{blue}{re \cdot re}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       10. lower-*.f64 70.2

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, re \cdot re, -0.25\right), \color{blue}{re \cdot re}, 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   11. Applied rewrites 70.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, re \cdot re, -0.25\right), re \cdot re, 0.5\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]

   if 0.660000000000000031 < (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

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(e^{-im} + e^{im}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 93.4%

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

   6. Taylor expanded in im around 0

      \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left(1 + -1 \cdot im\right)} + e^{im}\right) \]
   7. Step-by-step derivation

      1. neg-mul-1 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(1 + \color{blue}{\left(\mathsf{neg}\left(im\right)\right)}\right) + e^{im}\right) \]

      2. unsub-neg N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left(1 - im\right)} + e^{im}\right) \]

      3. lower--.f64 69.0

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

   8. Applied rewrites 69.0%

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

   9. Taylor expanded in im around 0

      \[\leadsto \frac{1}{2} \cdot \left(\left(1 - im\right) + \color{blue}{\left(1 + im \cdot \left(1 + im \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot im\right)\right)\right)}\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{1}{2} \cdot \left(\left(1 - im\right) + \color{blue}{\left(im \cdot \left(1 + im \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot im\right)\right) + 1\right)}\right) \]

       2. *-commutative N/A

          \[\leadsto \frac{1}{2} \cdot \left(\left(1 - im\right) + \left(\color{blue}{\left(1 + im \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot im\right)\right) \cdot im} + 1\right)\right) \]

       3. lower-fma.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\left(1 - im\right) + \color{blue}{\mathsf{fma}\left(1 + im \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot im\right), im, 1\right)}\right) \]

       4. +-commutative N/A

          \[\leadsto \frac{1}{2} \cdot \left(\left(1 - im\right) + \mathsf{fma}\left(\color{blue}{im \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot im\right) + 1}, im, 1\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \frac{1}{2} \cdot \left(\left(1 - im\right) + \mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot im\right) \cdot im} + 1, im, 1\right)\right) \]

       6. lower-fma.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\left(1 - im\right) + \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{6} \cdot im, im, 1\right)}, im, 1\right)\right) \]

       7. +-commutative N/A

          \[\leadsto \frac{1}{2} \cdot \left(\left(1 - im\right) + \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{6} \cdot im + \frac{1}{2}}, im, 1\right), im, 1\right)\right) \]

       8. lower-fma.f64 60.9

          \[\leadsto 0.5 \cdot \left(\left(1 - im\right) + \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.16666666666666666, im, 0.5\right)}, im, 1\right), im, 1\right)\right) \]

   11. Applied rewrites 60.9%

       \[\leadsto 0.5 \cdot \left(\left(1 - im\right) + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, im, 0.5\right), im, 1\right), im, 1\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq -0.005:\\ \;\;\;\;\left(\left(\left(re \cdot re\right) \cdot -0.5\right) \cdot 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;\cos re \leq 0.66:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, re \cdot re, -0.25\right), re \cdot re, 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, im, 0.5\right), im, 1\right), im, 1\right) + \left(1 - im\right)\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 53.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.005:\\ \;\;\;\;\left(\left(\left(re \cdot re\right) \cdot -0.5\right) \cdot 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;\cos re \leq 0.66:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, re \cdot re, -0.25\right), re \cdot re, 0.5\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, im, 0.5\right), im, 1\right), im, 1\right) + \left(1 - im\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) -0.005)
   (* (* (* (* re re) -0.5) 0.5) (fma im im 2.0))
   (if (<= (cos re) 0.66)
     (* (fma (fma 0.020833333333333332 (* re re) -0.25) (* re re) 0.5) 2.0)
     (*
      (+ (fma (fma (fma 0.16666666666666666 im 0.5) im 1.0) im 1.0) (- 1.0 im))
      0.5))))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= -0.005) {
		tmp = (((re * re) * -0.5) * 0.5) * fma(im, im, 2.0);
	} else if (cos(re) <= 0.66) {
		tmp = fma(fma(0.020833333333333332, (re * re), -0.25), (re * re), 0.5) * 2.0;
	} else {
		tmp = (fma(fma(fma(0.16666666666666666, im, 0.5), im, 1.0), im, 1.0) + (1.0 - im)) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= -0.005)
		tmp = Float64(Float64(Float64(Float64(re * re) * -0.5) * 0.5) * fma(im, im, 2.0));
	elseif (cos(re) <= 0.66)
		tmp = Float64(fma(fma(0.020833333333333332, Float64(re * re), -0.25), Float64(re * re), 0.5) * 2.0);
	else
		tmp = Float64(Float64(fma(fma(fma(0.16666666666666666, im, 0.5), im, 1.0), im, 1.0) + Float64(1.0 - im)) * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], -0.005], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.5), $MachinePrecision] * 0.5), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Cos[re], $MachinePrecision], 0.66], N[(N[(N[(0.020833333333333332 * N[(re * re), $MachinePrecision] + -0.25), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(N[(N[(0.16666666666666666 * im + 0.5), $MachinePrecision] * im + 1.0), $MachinePrecision] * im + 1.0), $MachinePrecision] + N[(1.0 - im), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 re) < -0.0050000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 81.4

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

   5. Applied rewrites 81.4%

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

   6. Taylor expanded in re around 0

      \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{-1}{2} \cdot {re}^{2} + 1\right)}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{2}, {re}^{2}, 1\right)}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      3. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{re \cdot re}, 1\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      4. lower-*.f64 46.2

         \[\leadsto \left(0.5 \cdot \mathsf{fma}\left(-0.5, \color{blue}{re \cdot re}, 1\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   8. Applied rewrites 46.2%

      \[\leadsto \left(0.5 \cdot \color{blue}{\mathsf{fma}\left(-0.5, re \cdot re, 1\right)}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   9. Taylor expanded in re around inf

      \[\leadsto \left(\frac{1}{2} \cdot \left(\frac{-1}{2} \cdot \color{blue}{{re}^{2}}\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 46.2%

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

   if -0.0050000000000000001 < (cos.f64 re) < 0.660000000000000031

   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

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(e^{-im} + e^{im}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 79.3%

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

   6. Taylor expanded in im around 0

      \[\leadsto \frac{1}{2} \cdot \color{blue}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 6.7%

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

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right)\right)} \cdot 2 \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right) + \frac{1}{2}\right)} \cdot 2 \]

       2. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right) \cdot {re}^{2}} + \frac{1}{2}\right) \cdot 2 \]

       3. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right)} \cdot 2 \]

       4. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{48} \cdot {re}^{2} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}, {re}^{2}, \frac{1}{2}\right) \cdot 2 \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{48} \cdot {re}^{2} + \color{blue}{\frac{-1}{4}}, {re}^{2}, \frac{1}{2}\right) \cdot 2 \]

       6. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{48}, {re}^{2}, \frac{-1}{4}\right)}, {re}^{2}, \frac{1}{2}\right) \cdot 2 \]

       7. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, \color{blue}{re \cdot re}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot 2 \]

       8. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, \color{blue}{re \cdot re}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot 2 \]

       9. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, re \cdot re, \frac{-1}{4}\right), \color{blue}{re \cdot re}, \frac{1}{2}\right) \cdot 2 \]

       10. lower-*.f64 58.2

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, re \cdot re, -0.25\right), \color{blue}{re \cdot re}, 0.5\right) \cdot 2 \]

   11. Applied rewrites 58.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, re \cdot re, -0.25\right), re \cdot re, 0.5\right)} \cdot 2 \]

   if 0.660000000000000031 < (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

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(e^{-im} + e^{im}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 93.4%

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

   6. Taylor expanded in im around 0

      \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left(1 + -1 \cdot im\right)} + e^{im}\right) \]
   7. Step-by-step derivation

      1. neg-mul-1 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(1 + \color{blue}{\left(\mathsf{neg}\left(im\right)\right)}\right) + e^{im}\right) \]

      2. unsub-neg N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left(1 - im\right)} + e^{im}\right) \]

      3. lower--.f64 69.0

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

   8. Applied rewrites 69.0%

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

   9. Taylor expanded in im around 0

      \[\leadsto \frac{1}{2} \cdot \left(\left(1 - im\right) + \color{blue}{\left(1 + im \cdot \left(1 + im \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot im\right)\right)\right)}\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{1}{2} \cdot \left(\left(1 - im\right) + \color{blue}{\left(im \cdot \left(1 + im \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot im\right)\right) + 1\right)}\right) \]

       2. *-commutative N/A

          \[\leadsto \frac{1}{2} \cdot \left(\left(1 - im\right) + \left(\color{blue}{\left(1 + im \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot im\right)\right) \cdot im} + 1\right)\right) \]

       3. lower-fma.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\left(1 - im\right) + \color{blue}{\mathsf{fma}\left(1 + im \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot im\right), im, 1\right)}\right) \]

       4. +-commutative N/A

          \[\leadsto \frac{1}{2} \cdot \left(\left(1 - im\right) + \mathsf{fma}\left(\color{blue}{im \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot im\right) + 1}, im, 1\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \frac{1}{2} \cdot \left(\left(1 - im\right) + \mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot im\right) \cdot im} + 1, im, 1\right)\right) \]

       6. lower-fma.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\left(1 - im\right) + \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{6} \cdot im, im, 1\right)}, im, 1\right)\right) \]

       7. +-commutative N/A

          \[\leadsto \frac{1}{2} \cdot \left(\left(1 - im\right) + \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{6} \cdot im + \frac{1}{2}}, im, 1\right), im, 1\right)\right) \]

       8. lower-fma.f64 60.9

          \[\leadsto 0.5 \cdot \left(\left(1 - im\right) + \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.16666666666666666, im, 0.5\right)}, im, 1\right), im, 1\right)\right) \]

   11. Applied rewrites 60.9%

       \[\leadsto 0.5 \cdot \left(\left(1 - im\right) + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, im, 0.5\right), im, 1\right), im, 1\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq -0.005:\\ \;\;\;\;\left(\left(\left(re \cdot re\right) \cdot -0.5\right) \cdot 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;\cos re \leq 0.66:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, re \cdot re, -0.25\right), re \cdot re, 0.5\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, im, 0.5\right), im, 1\right), im, 1\right) + \left(1 - im\right)\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.005:\\ \;\;\;\;\left(\left(\left(re \cdot re\right) \cdot -0.5\right) \cdot 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;\cos re \leq 0.93:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, re \cdot re, -0.25\right), re \cdot re, 0.5\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) -0.005)
   (* (* (* (* re re) -0.5) 0.5) (fma im im 2.0))
   (if (<= (cos re) 0.93)
     (* (fma (fma 0.020833333333333332 (* re re) -0.25) (* re re) 0.5) 2.0)
     (* 0.5 (fma im im 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= -0.005) {
		tmp = (((re * re) * -0.5) * 0.5) * fma(im, im, 2.0);
	} else if (cos(re) <= 0.93) {
		tmp = fma(fma(0.020833333333333332, (re * re), -0.25), (re * re), 0.5) * 2.0;
	} else {
		tmp = 0.5 * fma(im, im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= -0.005)
		tmp = Float64(Float64(Float64(Float64(re * re) * -0.5) * 0.5) * fma(im, im, 2.0));
	elseif (cos(re) <= 0.93)
		tmp = Float64(fma(fma(0.020833333333333332, Float64(re * re), -0.25), Float64(re * re), 0.5) * 2.0);
	else
		tmp = Float64(0.5 * fma(im, im, 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], -0.005], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.5), $MachinePrecision] * 0.5), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Cos[re], $MachinePrecision], 0.93], N[(N[(N[(0.020833333333333332 * N[(re * re), $MachinePrecision] + -0.25), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * 2.0), $MachinePrecision], N[(0.5 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 re) < -0.0050000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 81.4

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

   5. Applied rewrites 81.4%

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

   6. Taylor expanded in re around 0

      \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{-1}{2} \cdot {re}^{2} + 1\right)}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{2}, {re}^{2}, 1\right)}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      3. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{re \cdot re}, 1\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      4. lower-*.f64 46.2

         \[\leadsto \left(0.5 \cdot \mathsf{fma}\left(-0.5, \color{blue}{re \cdot re}, 1\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   8. Applied rewrites 46.2%

      \[\leadsto \left(0.5 \cdot \color{blue}{\mathsf{fma}\left(-0.5, re \cdot re, 1\right)}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   9. Taylor expanded in re around inf

      \[\leadsto \left(\frac{1}{2} \cdot \left(\frac{-1}{2} \cdot \color{blue}{{re}^{2}}\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 46.2%

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

   if -0.0050000000000000001 < (cos.f64 re) < 0.930000000000000049

   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

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(e^{-im} + e^{im}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 71.5%

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

   6. Taylor expanded in im around 0

      \[\leadsto \frac{1}{2} \cdot \color{blue}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 8.8%

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

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right)\right)} \cdot 2 \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right) + \frac{1}{2}\right)} \cdot 2 \]

       2. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right) \cdot {re}^{2}} + \frac{1}{2}\right) \cdot 2 \]

       3. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right)} \cdot 2 \]

       4. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{48} \cdot {re}^{2} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}, {re}^{2}, \frac{1}{2}\right) \cdot 2 \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{48} \cdot {re}^{2} + \color{blue}{\frac{-1}{4}}, {re}^{2}, \frac{1}{2}\right) \cdot 2 \]

       6. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{48}, {re}^{2}, \frac{-1}{4}\right)}, {re}^{2}, \frac{1}{2}\right) \cdot 2 \]

       7. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, \color{blue}{re \cdot re}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot 2 \]

       8. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, \color{blue}{re \cdot re}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot 2 \]

       9. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, re \cdot re, \frac{-1}{4}\right), \color{blue}{re \cdot re}, \frac{1}{2}\right) \cdot 2 \]

       10. lower-*.f64 52.8

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, re \cdot re, -0.25\right), \color{blue}{re \cdot re}, 0.5\right) \cdot 2 \]

   11. Applied rewrites 52.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, re \cdot re, -0.25\right), re \cdot re, 0.5\right)} \cdot 2 \]

   if 0.930000000000000049 < (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

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(e^{-im} + e^{im}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 97.8%

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

   6. Taylor expanded in im around 0

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. unpow2 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{im \cdot im} + 2\right) \]

      3. lower-fma.f64 71.6

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

   8. Applied rewrites 71.6%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq -0.005:\\ \;\;\;\;\left(\left(\left(re \cdot re\right) \cdot -0.5\right) \cdot 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;\cos re \leq 0.93:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, re \cdot re, -0.25\right), re \cdot re, 0.5\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 58.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.005:\\ \;\;\;\;\left(\left(\left(re \cdot re\right) \cdot -0.5\right) \cdot 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) -0.005)
   (* (* (* (* re re) -0.5) 0.5) (fma im im 2.0))
   (* 0.5 (fma im im 2.0))))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= -0.005) {
		tmp = (((re * re) * -0.5) * 0.5) * fma(im, im, 2.0);
	} else {
		tmp = 0.5 * fma(im, im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= -0.005)
		tmp = Float64(Float64(Float64(Float64(re * re) * -0.5) * 0.5) * fma(im, im, 2.0));
	else
		tmp = Float64(0.5 * fma(im, im, 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], -0.005], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.5), $MachinePrecision] * 0.5), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -0.0050000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 81.4

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

   5. Applied rewrites 81.4%

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

   6. Taylor expanded in re around 0

      \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{-1}{2} \cdot {re}^{2} + 1\right)}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{2}, {re}^{2}, 1\right)}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      3. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{re \cdot re}, 1\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      4. lower-*.f64 46.2

         \[\leadsto \left(0.5 \cdot \mathsf{fma}\left(-0.5, \color{blue}{re \cdot re}, 1\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   8. Applied rewrites 46.2%

      \[\leadsto \left(0.5 \cdot \color{blue}{\mathsf{fma}\left(-0.5, re \cdot re, 1\right)}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   9. Taylor expanded in re around inf

      \[\leadsto \left(\frac{1}{2} \cdot \left(\frac{-1}{2} \cdot \color{blue}{{re}^{2}}\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 46.2%

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

   if -0.0050000000000000001 < (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

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(e^{-im} + e^{im}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 91.0%

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

   6. Taylor expanded in im around 0

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. unpow2 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{im \cdot im} + 2\right) \]

      3. lower-fma.f64 63.3

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

   8. Applied rewrites 63.3%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq -0.005:\\ \;\;\;\;\left(\left(\left(re \cdot re\right) \cdot -0.5\right) \cdot 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 58.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) -0.005)
   (* (fma (* re re) -0.25 0.5) (fma im im 2.0))
   (* 0.5 (fma im im 2.0))))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= -0.005) {
		tmp = fma((re * re), -0.25, 0.5) * fma(im, im, 2.0);
	} else {
		tmp = 0.5 * fma(im, im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= -0.005)
		tmp = Float64(fma(Float64(re * re), -0.25, 0.5) * fma(im, im, 2.0));
	else
		tmp = Float64(0.5 * fma(im, im, 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], -0.005], N[(N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -0.0050000000000000001

   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

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(e^{-im} + e^{im}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 0.8%

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

   6. Taylor expanded in im around 0

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. unpow2 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{im \cdot im} + 2\right) \]

      3. lower-fma.f64 0.9

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

   8. Applied rewrites 0.9%

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

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{4} \cdot {re}^{2} + \frac{1}{2}\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]

       2. *-commutative N/A

          \[\leadsto \left(\color{blue}{{re}^{2} \cdot \frac{-1}{4}} + \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       3. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{4}, \frac{1}{2}\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{4}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       5. lower-*.f64 46.2

          \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, -0.25, 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   11. Applied rewrites 46.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]

   if -0.0050000000000000001 < (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

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(e^{-im} + e^{im}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 91.0%

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

   6. Taylor expanded in im around 0

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. unpow2 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{im \cdot im} + 2\right) \]

      3. lower-fma.f64 63.3

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

   8. Applied rewrites 63.3%

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

Alternative 13: 54.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.005:\\ \;\;\;\;\left(\left(\left(re \cdot re\right) \cdot -0.5\right) \cdot 0.5\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) -0.005)
   (* (* (* (* re re) -0.5) 0.5) 2.0)
   (* 0.5 (fma im im 2.0))))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= -0.005) {
		tmp = (((re * re) * -0.5) * 0.5) * 2.0;
	} else {
		tmp = 0.5 * fma(im, im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= -0.005)
		tmp = Float64(Float64(Float64(Float64(re * re) * -0.5) * 0.5) * 2.0);
	else
		tmp = Float64(0.5 * fma(im, im, 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], -0.005], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.5), $MachinePrecision] * 0.5), $MachinePrecision] * 2.0), $MachinePrecision], N[(0.5 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -0.0050000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 81.4

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

   5. Applied rewrites 81.4%

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

   6. Taylor expanded in re around 0

      \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{-1}{2} \cdot {re}^{2} + 1\right)}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{2}, {re}^{2}, 1\right)}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      3. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{re \cdot re}, 1\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      4. lower-*.f64 46.2

         \[\leadsto \left(0.5 \cdot \mathsf{fma}\left(-0.5, \color{blue}{re \cdot re}, 1\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   8. Applied rewrites 46.2%

      \[\leadsto \left(0.5 \cdot \color{blue}{\mathsf{fma}\left(-0.5, re \cdot re, 1\right)}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   9. Taylor expanded in re around inf

      \[\leadsto \left(\frac{1}{2} \cdot \left(\frac{-1}{2} \cdot \color{blue}{{re}^{2}}\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 46.2%

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

   12. Taylor expanded in im around 0

       \[\leadsto \left(\frac{1}{2} \cdot \left(\frac{-1}{2} \cdot \left(re \cdot re\right)\right)\right) \cdot \color{blue}{2} \]
   13. Step-by-step derivation

   14. Applied rewrites 28.8%

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

   if -0.0050000000000000001 < (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

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(e^{-im} + e^{im}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 91.0%

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

   6. Taylor expanded in im around 0

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. unpow2 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{im \cdot im} + 2\right) \]

      3. lower-fma.f64 63.3

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

   8. Applied rewrites 63.3%

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

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq -0.005:\\ \;\;\;\;\left(\left(\left(re \cdot re\right) \cdot -0.5\right) \cdot 0.5\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 54.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) -0.005)
   (* (fma (* re re) -0.25 0.5) 2.0)
   (* 0.5 (fma im im 2.0))))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= -0.005) {
		tmp = fma((re * re), -0.25, 0.5) * 2.0;
	} else {
		tmp = 0.5 * fma(im, im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= -0.005)
		tmp = Float64(fma(Float64(re * re), -0.25, 0.5) * 2.0);
	else
		tmp = Float64(0.5 * fma(im, im, 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], -0.005], N[(N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision] * 2.0), $MachinePrecision], N[(0.5 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -0.0050000000000000001

   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

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(e^{-im} + e^{im}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 0.8%

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

   6. Taylor expanded in im around 0

      \[\leadsto \frac{1}{2} \cdot \color{blue}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 1.1%

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

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \cdot 2 \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{4} \cdot {re}^{2} + \frac{1}{2}\right)} \cdot 2 \]

       2. *-commutative N/A

          \[\leadsto \left(\color{blue}{{re}^{2} \cdot \frac{-1}{4}} + \frac{1}{2}\right) \cdot 2 \]

       3. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{4}, \frac{1}{2}\right)} \cdot 2 \]

       4. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{4}, \frac{1}{2}\right) \cdot 2 \]

       5. lower-*.f64 28.8

          \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, -0.25, 0.5\right) \cdot 2 \]

   11. Applied rewrites 28.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)} \cdot 2 \]

   if -0.0050000000000000001 < (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

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(e^{-im} + e^{im}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 91.0%

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

   6. Taylor expanded in im around 0

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. unpow2 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{im \cdot im} + 2\right) \]

      3. lower-fma.f64 63.3

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

   8. Applied rewrites 63.3%

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

Alternative 15: 47.3% accurate, 26.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[re_, im_] := N[(0.5 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in re around 0

   \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(e^{-im} + e^{im}\right) \]
4. Step-by-step derivation

5. Applied rewrites 66.0%

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

6. Taylor expanded in im around 0

   \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

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

   2. unpow2 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{im \cdot im} + 2\right) \]

   3. lower-fma.f64 46.0

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

8. Applied rewrites 46.0%

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

Alternative 16: 28.8% accurate, 52.7× speedup

Math

\[\begin{array}{l} \\ 2 \cdot 0.5 \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 2.0d0 * 0.5d0
end function

Java

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

Python

def code(re, im):
	return 2.0 * 0.5

Julia

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

MATLAB

function tmp = code(re, im)
	tmp = 2.0 * 0.5;
end

Wolfram

code[re_, im_] := N[(2.0 * 0.5), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in re around 0

   \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(e^{-im} + e^{im}\right) \]
4. Step-by-step derivation

5. Applied rewrites 66.0%

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

6. Taylor expanded in im around 0

   \[\leadsto \frac{1}{2} \cdot \color{blue}{2} \]
7. Step-by-step derivation

8. Applied rewrites 27.3%

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

9. Final simplification 27.3%

   \[\leadsto 2 \cdot 0.5 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024295 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))