math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 10.8s

Alternatives: 19

Speedup: 1.5×

• 50.5% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-exp.f64 N/A

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

   10. lift-exp.f64 N/A

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

   11. lift-neg.f64 N/A

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

   12. cosh-undef N/A

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

   13. associate-*r* N/A

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

   14. metadata-eval N/A

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

   15. lower-*.f64 N/A

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

   16. lower-cosh.f64 100.0

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

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \cos re} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-lft-identity 100.0

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

6. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\cosh im} \cdot \cos re \]
7. Add Preprocessing

Alternative 2: 98.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot -0.25, 0.5\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot 0.08333333333333333, im\right), 2\right)\\ \mathbf{elif}\;t\_0 \leq 0.999972512931796:\\ \;\;\;\;\cos re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* (cos re) 0.5) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma re (* re -0.25) 0.5)
      (fma im (fma im (* (* im im) 0.08333333333333333) im) 2.0))
     (if (<= t_0 0.999972512931796)
       (*
        (cos re)
        (fma (* im im) (fma im (* im 0.041666666666666664) 0.5) 1.0))
       (* (cosh im) 1.0)))))

C

double code(double re, double im) {
	double t_0 = (cos(re) * 0.5) * (exp(-im) + exp(im));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(re, (re * -0.25), 0.5) * fma(im, fma(im, ((im * im) * 0.08333333333333333), im), 2.0);
	} else if (t_0 <= 0.999972512931796) {
		tmp = cos(re) * fma((im * im), fma(im, (im * 0.041666666666666664), 0.5), 1.0);
	} else {
		tmp = cosh(im) * 1.0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(cos(re) * 0.5) * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(re, Float64(re * -0.25), 0.5) * fma(im, fma(im, Float64(Float64(im * im) * 0.08333333333333333), im), 2.0));
	elseif (t_0 <= 0.999972512931796)
		tmp = Float64(cos(re) * fma(Float64(im * im), fma(im, Float64(im * 0.041666666666666664), 0.5), 1.0));
	else
		tmp = Float64(cosh(im) * 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(re * N[(re * -0.25), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * 0.08333333333333333), $MachinePrecision] + im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.999972512931796], N[(N[Cos[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Cosh[im], $MachinePrecision] * 1.0), $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

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right) + 2\right)} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right) + 2\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\color{blue}{im \cdot \left(im \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)} + 2\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right), 2\right)} \]

      5. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(im, im \cdot \color{blue}{\left(\frac{1}{12} \cdot {im}^{2} + 1\right)}, 2\right) \]

      6. distribute-lft-in N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(im, \color{blue}{im \cdot \left(\frac{1}{12} \cdot {im}^{2}\right) + im \cdot 1}, 2\right) \]

      7. *-rgt-identity N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(im, im \cdot \left(\frac{1}{12} \cdot {im}^{2}\right) + \color{blue}{im}, 2\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(im, \color{blue}{\mathsf{fma}\left(im, \frac{1}{12} \cdot {im}^{2}, im\right)}, 2\right) \]

      9. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \color{blue}{{im}^{2} \cdot \frac{1}{12}}, im\right), 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \color{blue}{{im}^{2} \cdot \frac{1}{12}}, im\right), 2\right) \]

      11. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \color{blue}{\left(im \cdot im\right)} \cdot \frac{1}{12}, im\right), 2\right) \]

      12. lower-*.f64 77.5

          \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \color{blue}{\left(im \cdot im\right)} \cdot 0.08333333333333333, im\right), 2\right) \]

   5. Applied rewrites 77.5%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot 0.08333333333333333, im\right), 2\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot \frac{1}{12}, im\right), 2\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{4} \cdot {re}^{2} + \frac{1}{2}\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot \frac{1}{12}, im\right), 2\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{{re}^{2} \cdot \frac{-1}{4}} + \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot \frac{1}{12}, im\right), 2\right) \]

      3. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{4} + \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot \frac{1}{12}, im\right), 2\right) \]

      4. associate-*l* N/A

         \[\leadsto \left(\color{blue}{re \cdot \left(re \cdot \frac{-1}{4}\right)} + \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot \frac{1}{12}, im\right), 2\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, re \cdot \frac{-1}{4}, \frac{1}{2}\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot \frac{1}{12}, im\right), 2\right) \]

      6. lower-*.f64 100.0

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{re \cdot -0.25}, 0.5\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot 0.08333333333333333, im\right), 2\right) \]

   8. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, re \cdot -0.25, 0.5\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot 0.08333333333333333, im\right), 2\right) \]

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.99997251293179601

   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

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

   5. Applied rewrites 98.7%

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

   if 0.99997251293179601 < (*.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

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. cosh-undef N/A

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

      13. associate-*r* N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \cos re} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-lft-identity 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 3 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot -0.25, 0.5\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot 0.08333333333333333, im\right), 2\right)\\ \mathbf{elif}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.999972512931796:\\ \;\;\;\;\cos re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos re \cdot 0.5\\ t_1 := t\_0 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot -0.25, 0.5\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot 0.08333333333333333, im\right), 2\right)\\ \mathbf{elif}\;t\_1 \leq 0.999972512931796:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (cos re) 0.5)) (t_1 (* t_0 (+ (exp (- im)) (exp im)))))
   (if (<= t_1 (- INFINITY))
     (*
      (fma re (* re -0.25) 0.5)
      (fma im (fma im (* (* im im) 0.08333333333333333) im) 2.0))
     (if (<= t_1 0.999972512931796)
       (* t_0 (fma im im 2.0))
       (* (cosh im) 1.0)))))

C

double code(double re, double im) {
	double t_0 = cos(re) * 0.5;
	double t_1 = t_0 * (exp(-im) + exp(im));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(re, (re * -0.25), 0.5) * fma(im, fma(im, ((im * im) * 0.08333333333333333), im), 2.0);
	} else if (t_1 <= 0.999972512931796) {
		tmp = t_0 * fma(im, im, 2.0);
	} else {
		tmp = cosh(im) * 1.0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(cos(re) * 0.5)
	t_1 = Float64(t_0 * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(re, Float64(re * -0.25), 0.5) * fma(im, fma(im, Float64(Float64(im * im) * 0.08333333333333333), im), 2.0));
	elseif (t_1 <= 0.999972512931796)
		tmp = Float64(t_0 * fma(im, im, 2.0));
	else
		tmp = Float64(cosh(im) * 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(re * N[(re * -0.25), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * 0.08333333333333333), $MachinePrecision] + im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.999972512931796], N[(t$95$0 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[Cosh[im], $MachinePrecision] * 1.0), $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

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right) + 2\right)} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right) + 2\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\color{blue}{im \cdot \left(im \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)} + 2\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right), 2\right)} \]

      5. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(im, im \cdot \color{blue}{\left(\frac{1}{12} \cdot {im}^{2} + 1\right)}, 2\right) \]

      6. distribute-lft-in N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(im, \color{blue}{im \cdot \left(\frac{1}{12} \cdot {im}^{2}\right) + im \cdot 1}, 2\right) \]

      7. *-rgt-identity N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(im, im \cdot \left(\frac{1}{12} \cdot {im}^{2}\right) + \color{blue}{im}, 2\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(im, \color{blue}{\mathsf{fma}\left(im, \frac{1}{12} \cdot {im}^{2}, im\right)}, 2\right) \]

      9. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \color{blue}{{im}^{2} \cdot \frac{1}{12}}, im\right), 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \color{blue}{{im}^{2} \cdot \frac{1}{12}}, im\right), 2\right) \]

      11. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \color{blue}{\left(im \cdot im\right)} \cdot \frac{1}{12}, im\right), 2\right) \]

      12. lower-*.f64 77.5

          \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \color{blue}{\left(im \cdot im\right)} \cdot 0.08333333333333333, im\right), 2\right) \]

   5. Applied rewrites 77.5%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot 0.08333333333333333, im\right), 2\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot \frac{1}{12}, im\right), 2\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{4} \cdot {re}^{2} + \frac{1}{2}\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot \frac{1}{12}, im\right), 2\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{{re}^{2} \cdot \frac{-1}{4}} + \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot \frac{1}{12}, im\right), 2\right) \]

      3. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{4} + \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot \frac{1}{12}, im\right), 2\right) \]

      4. associate-*l* N/A

         \[\leadsto \left(\color{blue}{re \cdot \left(re \cdot \frac{-1}{4}\right)} + \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot \frac{1}{12}, im\right), 2\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, re \cdot \frac{-1}{4}, \frac{1}{2}\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot \frac{1}{12}, im\right), 2\right) \]

      6. lower-*.f64 100.0

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{re \cdot -0.25}, 0.5\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot 0.08333333333333333, im\right), 2\right) \]

   8. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, re \cdot -0.25, 0.5\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot 0.08333333333333333, im\right), 2\right) \]

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.99997251293179601

   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 98.3

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

   5. Applied rewrites 98.3%

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

   if 0.99997251293179601 < (*.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

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. cosh-undef N/A

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

      13. associate-*r* N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \cos re} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-lft-identity 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 3 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot -0.25, 0.5\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot 0.08333333333333333, im\right), 2\right)\\ \mathbf{elif}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.999972512931796:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot -0.25, 0.5\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot 0.08333333333333333, im\right), 2\right)\\ \mathbf{elif}\;t\_0 \leq 0.999972512931796:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* (cos re) 0.5) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma re (* re -0.25) 0.5)
      (fma im (fma im (* (* im im) 0.08333333333333333) im) 2.0))
     (if (<= t_0 0.999972512931796) (cos re) (* (cosh im) 1.0)))))

C

double code(double re, double im) {
	double t_0 = (cos(re) * 0.5) * (exp(-im) + exp(im));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(re, (re * -0.25), 0.5) * fma(im, fma(im, ((im * im) * 0.08333333333333333), im), 2.0);
	} else if (t_0 <= 0.999972512931796) {
		tmp = cos(re);
	} else {
		tmp = cosh(im) * 1.0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(cos(re) * 0.5) * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(re, Float64(re * -0.25), 0.5) * fma(im, fma(im, Float64(Float64(im * im) * 0.08333333333333333), im), 2.0));
	elseif (t_0 <= 0.999972512931796)
		tmp = cos(re);
	else
		tmp = Float64(cosh(im) * 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(re * N[(re * -0.25), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * 0.08333333333333333), $MachinePrecision] + im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.999972512931796], N[Cos[re], $MachinePrecision], N[(N[Cosh[im], $MachinePrecision] * 1.0), $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

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right) + 2\right)} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right) + 2\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\color{blue}{im \cdot \left(im \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)} + 2\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right), 2\right)} \]

      5. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(im, im \cdot \color{blue}{\left(\frac{1}{12} \cdot {im}^{2} + 1\right)}, 2\right) \]

      6. distribute-lft-in N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(im, \color{blue}{im \cdot \left(\frac{1}{12} \cdot {im}^{2}\right) + im \cdot 1}, 2\right) \]

      7. *-rgt-identity N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(im, im \cdot \left(\frac{1}{12} \cdot {im}^{2}\right) + \color{blue}{im}, 2\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(im, \color{blue}{\mathsf{fma}\left(im, \frac{1}{12} \cdot {im}^{2}, im\right)}, 2\right) \]

      9. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \color{blue}{{im}^{2} \cdot \frac{1}{12}}, im\right), 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \color{blue}{{im}^{2} \cdot \frac{1}{12}}, im\right), 2\right) \]

      11. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \color{blue}{\left(im \cdot im\right)} \cdot \frac{1}{12}, im\right), 2\right) \]

      12. lower-*.f64 77.5

          \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \color{blue}{\left(im \cdot im\right)} \cdot 0.08333333333333333, im\right), 2\right) \]

   5. Applied rewrites 77.5%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot 0.08333333333333333, im\right), 2\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot \frac{1}{12}, im\right), 2\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{4} \cdot {re}^{2} + \frac{1}{2}\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot \frac{1}{12}, im\right), 2\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{{re}^{2} \cdot \frac{-1}{4}} + \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot \frac{1}{12}, im\right), 2\right) \]

      3. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{4} + \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot \frac{1}{12}, im\right), 2\right) \]

      4. associate-*l* N/A

         \[\leadsto \left(\color{blue}{re \cdot \left(re \cdot \frac{-1}{4}\right)} + \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot \frac{1}{12}, im\right), 2\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, re \cdot \frac{-1}{4}, \frac{1}{2}\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot \frac{1}{12}, im\right), 2\right) \]

      6. lower-*.f64 100.0

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{re \cdot -0.25}, 0.5\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot 0.08333333333333333, im\right), 2\right) \]

   8. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, re \cdot -0.25, 0.5\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot 0.08333333333333333, im\right), 2\right) \]

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.99997251293179601

   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

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

      1. lower-cos.f64 97.8

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

   5. Applied rewrites 97.8%

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

   if 0.99997251293179601 < (*.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

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. cosh-undef N/A

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

      13. associate-*r* N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \cos re} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-lft-identity 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 3 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot -0.25, 0.5\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot 0.08333333333333333, im\right), 2\right)\\ \mathbf{elif}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.999972512931796:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right)\\ t_1 := \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot 0.08333333333333333, im\right), 2\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot -0.25, 0.5\right) \cdot t\_1\\ \mathbf{elif}\;t\_0 \leq 0.999972512931796:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, 0.020833333333333332, -0.25\right), 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* (cos re) 0.5) (+ (exp (- im)) (exp im))))
        (t_1 (fma im (fma im (* (* im im) 0.08333333333333333) im) 2.0)))
   (if (<= t_0 (- INFINITY))
     (* (fma re (* re -0.25) 0.5) t_1)
     (if (<= t_0 0.999972512931796)
       (cos re)
       (*
        t_1
        (fma re (* re (fma (* re re) 0.020833333333333332 -0.25)) 0.5))))))

C

double code(double re, double im) {
	double t_0 = (cos(re) * 0.5) * (exp(-im) + exp(im));
	double t_1 = fma(im, fma(im, ((im * im) * 0.08333333333333333), im), 2.0);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(re, (re * -0.25), 0.5) * t_1;
	} else if (t_0 <= 0.999972512931796) {
		tmp = cos(re);
	} else {
		tmp = t_1 * fma(re, (re * fma((re * re), 0.020833333333333332, -0.25)), 0.5);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(cos(re) * 0.5) * Float64(exp(Float64(-im)) + exp(im)))
	t_1 = fma(im, fma(im, Float64(Float64(im * im) * 0.08333333333333333), im), 2.0)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(re, Float64(re * -0.25), 0.5) * t_1);
	elseif (t_0 <= 0.999972512931796)
		tmp = cos(re);
	else
		tmp = Float64(t_1 * fma(re, Float64(re * fma(Float64(re * re), 0.020833333333333332, -0.25)), 0.5));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * 0.08333333333333333), $MachinePrecision] + im), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(re * N[(re * -0.25), $MachinePrecision] + 0.5), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 0.999972512931796], N[Cos[re], $MachinePrecision], N[(t$95$1 * N[(re * N[(re * N[(N[(re * re), $MachinePrecision] * 0.020833333333333332 + -0.25), $MachinePrecision]), $MachinePrecision] + 0.5), $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

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right) + 2\right)} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right) + 2\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\color{blue}{im \cdot \left(im \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)} + 2\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right), 2\right)} \]

      5. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(im, im \cdot \color{blue}{\left(\frac{1}{12} \cdot {im}^{2} + 1\right)}, 2\right) \]

      6. distribute-lft-in N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(im, \color{blue}{im \cdot \left(\frac{1}{12} \cdot {im}^{2}\right) + im \cdot 1}, 2\right) \]

      7. *-rgt-identity N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(im, im \cdot \left(\frac{1}{12} \cdot {im}^{2}\right) + \color{blue}{im}, 2\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(im, \color{blue}{\mathsf{fma}\left(im, \frac{1}{12} \cdot {im}^{2}, im\right)}, 2\right) \]

      9. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \color{blue}{{im}^{2} \cdot \frac{1}{12}}, im\right), 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \color{blue}{{im}^{2} \cdot \frac{1}{12}}, im\right), 2\right) \]

      11. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \color{blue}{\left(im \cdot im\right)} \cdot \frac{1}{12}, im\right), 2\right) \]

      12. lower-*.f64 77.5

          \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \color{blue}{\left(im \cdot im\right)} \cdot 0.08333333333333333, im\right), 2\right) \]

   5. Applied rewrites 77.5%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot 0.08333333333333333, im\right), 2\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot \frac{1}{12}, im\right), 2\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{4} \cdot {re}^{2} + \frac{1}{2}\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot \frac{1}{12}, im\right), 2\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{{re}^{2} \cdot \frac{-1}{4}} + \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot \frac{1}{12}, im\right), 2\right) \]

      3. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{4} + \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot \frac{1}{12}, im\right), 2\right) \]

      4. associate-*l* N/A

         \[\leadsto \left(\color{blue}{re \cdot \left(re \cdot \frac{-1}{4}\right)} + \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot \frac{1}{12}, im\right), 2\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, re \cdot \frac{-1}{4}, \frac{1}{2}\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot \frac{1}{12}, im\right), 2\right) \]

      6. lower-*.f64 100.0

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{re \cdot -0.25}, 0.5\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot 0.08333333333333333, im\right), 2\right) \]

   8. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, re \cdot -0.25, 0.5\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot 0.08333333333333333, im\right), 2\right) \]

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.99997251293179601

   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

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

      1. lower-cos.f64 97.8

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

   5. Applied rewrites 97.8%

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

   if 0.99997251293179601 < (*.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 im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right) + 2\right)} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right) + 2\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\color{blue}{im \cdot \left(im \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)} + 2\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right), 2\right)} \]

      5. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(im, im \cdot \color{blue}{\left(\frac{1}{12} \cdot {im}^{2} + 1\right)}, 2\right) \]

      6. distribute-lft-in N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(im, \color{blue}{im \cdot \left(\frac{1}{12} \cdot {im}^{2}\right) + im \cdot 1}, 2\right) \]

      7. *-rgt-identity N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(im, im \cdot \left(\frac{1}{12} \cdot {im}^{2}\right) + \color{blue}{im}, 2\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(im, \color{blue}{\mathsf{fma}\left(im, \frac{1}{12} \cdot {im}^{2}, im\right)}, 2\right) \]

      9. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \color{blue}{{im}^{2} \cdot \frac{1}{12}}, im\right), 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \color{blue}{{im}^{2} \cdot \frac{1}{12}}, im\right), 2\right) \]

      11. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \color{blue}{\left(im \cdot im\right)} \cdot \frac{1}{12}, im\right), 2\right) \]

      12. lower-*.f64 87.7

          \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \color{blue}{\left(im \cdot im\right)} \cdot 0.08333333333333333, im\right), 2\right) \]

   5. Applied rewrites 87.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot 0.08333333333333333, im\right), 2\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right)\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot \frac{1}{12}, im\right), 2\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right) + \frac{1}{2}\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot \frac{1}{12}, im\right), 2\right) \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(re \cdot re\right)} \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right) + \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot \frac{1}{12}, im\right), 2\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(\color{blue}{re \cdot \left(re \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right)\right)} + \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot \frac{1}{12}, im\right), 2\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, re \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right), \frac{1}{2}\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot \frac{1}{12}, im\right), 2\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{re \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right)}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot \frac{1}{12}, im\right), 2\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(re, re \cdot \color{blue}{\left(\frac{1}{48} \cdot {re}^{2} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot \frac{1}{12}, im\right), 2\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, re \cdot \left(\color{blue}{{re}^{2} \cdot \frac{1}{48}} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right), \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot \frac{1}{12}, im\right), 2\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(re, re \cdot \left({re}^{2} \cdot \frac{1}{48} + \color{blue}{\frac{-1}{4}}\right), \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot \frac{1}{12}, im\right), 2\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, re \cdot \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{48}, \frac{-1}{4}\right)}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot \frac{1}{12}, im\right), 2\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{48}, \frac{-1}{4}\right), \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot \frac{1}{12}, im\right), 2\right) \]

      11. lower-*.f64 91.8

          \[\leadsto \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, 0.020833333333333332, -0.25\right), 0.5\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot 0.08333333333333333, im\right), 2\right) \]

   8. Applied rewrites 91.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, 0.020833333333333332, -0.25\right), 0.5\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot 0.08333333333333333, im\right), 2\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot -0.25, 0.5\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot 0.08333333333333333, im\right), 2\right)\\ \mathbf{elif}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.999972512931796:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot 0.08333333333333333, im\right), 2\right) \cdot \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, 0.020833333333333332, -0.25\right), 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* (cos re) 0.5) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 -0.02)
     (fma re (* re -0.5) 1.0)
     (if (<= t_0 2.0)
       (* (fma im im 2.0) 0.5)
       (* (* im im) (fma (* im im) 0.041666666666666664 0.5))))))

C

double code(double re, double im) {
	double t_0 = (cos(re) * 0.5) * (exp(-im) + exp(im));
	double tmp;
	if (t_0 <= -0.02) {
		tmp = fma(re, (re * -0.5), 1.0);
	} else if (t_0 <= 2.0) {
		tmp = fma(im, im, 2.0) * 0.5;
	} else {
		tmp = (im * im) * fma((im * im), 0.041666666666666664, 0.5);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(cos(re) * 0.5) * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_0 <= -0.02)
		tmp = fma(re, Float64(re * -0.5), 1.0);
	elseif (t_0 <= 2.0)
		tmp = Float64(fma(im, im, 2.0) * 0.5);
	else
		tmp = Float64(Float64(im * im) * fma(Float64(im * im), 0.041666666666666664, 0.5));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.02], N[(re * N[(re * -0.5), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(N[(im * im + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * 0.041666666666666664 + 0.5), $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))) < -0.0200000000000000004

   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

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

      1. lower-cos.f64 53.9

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

   5. Applied rewrites 53.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 25.5%

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

   if -0.0200000000000000004 < (*.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%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 75.0%

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

   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 im around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

   5. Applied rewrites 79.5%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 79.5%

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

   9. Taylor expanded in im around inf

      \[\leadsto {im}^{4} \cdot \left(\frac{1}{24} + \color{blue}{\frac{1}{2} \cdot \frac{1}{{im}^{2}}}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 79.5%

       \[\leadsto \left(im \cdot im\right) \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{0.041666666666666664}, 0.5\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot -0.5, 1\right)\\ \mathbf{elif}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 2:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* (cos re) 0.5) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 -0.02)
     (fma re (* re -0.5) 1.0)
     (if (<= t_0 2.0)
       (* (fma im im 2.0) 0.5)
       (* (* im im) (* (* im im) 0.041666666666666664))))))

C

double code(double re, double im) {
	double t_0 = (cos(re) * 0.5) * (exp(-im) + exp(im));
	double tmp;
	if (t_0 <= -0.02) {
		tmp = fma(re, (re * -0.5), 1.0);
	} else if (t_0 <= 2.0) {
		tmp = fma(im, im, 2.0) * 0.5;
	} else {
		tmp = (im * im) * ((im * im) * 0.041666666666666664);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(cos(re) * 0.5) * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_0 <= -0.02)
		tmp = fma(re, Float64(re * -0.5), 1.0);
	elseif (t_0 <= 2.0)
		tmp = Float64(fma(im, im, 2.0) * 0.5);
	else
		tmp = Float64(Float64(im * im) * Float64(Float64(im * im) * 0.041666666666666664));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.02], N[(re * N[(re * -0.5), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(N[(im * im + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $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))) < -0.0200000000000000004

   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

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

      1. lower-cos.f64 53.9

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

   5. Applied rewrites 53.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 25.5%

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

   if -0.0200000000000000004 < (*.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%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 75.0%

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

   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 im around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

   5. Applied rewrites 79.5%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 79.5%

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

   9. Taylor expanded in im around inf

      \[\leadsto \frac{1}{24} \cdot {im}^{\color{blue}{4}} \]
   10. Step-by-step derivation

   11. Applied rewrites 79.5%

       \[\leadsto \left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \color{blue}{0.041666666666666664}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot -0.5, 1\right)\\ \mathbf{elif}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 2:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.999972512931796:\\ \;\;\;\;\cos re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (cos re) 0.5) (+ (exp (- im)) (exp im))) 0.999972512931796)
   (*
    (cos re)
    (fma
     (* im im)
     (fma
      (* im im)
      (fma (* im im) 0.001388888888888889 0.041666666666666664)
      0.5)
     1.0))
   (* (cosh im) 1.0)))

C

double code(double re, double im) {
	double tmp;
	if (((cos(re) * 0.5) * (exp(-im) + exp(im))) <= 0.999972512931796) {
		tmp = cos(re) * fma((im * im), fma((im * im), fma((im * im), 0.001388888888888889, 0.041666666666666664), 0.5), 1.0);
	} else {
		tmp = cosh(im) * 1.0;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(cos(re) * 0.5) * Float64(exp(Float64(-im)) + exp(im))) <= 0.999972512931796)
		tmp = Float64(cos(re) * fma(Float64(im * im), fma(Float64(im * im), fma(Float64(im * im), 0.001388888888888889, 0.041666666666666664), 0.5), 1.0));
	else
		tmp = Float64(cosh(im) * 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.999972512931796], N[(N[Cos[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Cosh[im], $MachinePrecision] * 1.0), $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))) < 0.99997251293179601

   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

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. cosh-undef N/A

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

      13. associate-*r* N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 96.1

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

   7. Applied rewrites 96.1%

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

   if 0.99997251293179601 < (*.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

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. cosh-undef N/A

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

      13. associate-*r* N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \cos re} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-lft-identity 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 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.999972512931796:\\ \;\;\;\;\cos re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.999972512931796:\\ \;\;\;\;\cos re \cdot \mathsf{fma}\left(im \cdot im, im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (cos re) 0.5) (+ (exp (- im)) (exp im))) 0.999972512931796)
   (*
    (cos re)
    (fma
     (* im im)
     (* im (* im (fma (* im im) 0.001388888888888889 0.041666666666666664)))
     1.0))
   (* (cosh im) 1.0)))

C

double code(double re, double im) {
	double tmp;
	if (((cos(re) * 0.5) * (exp(-im) + exp(im))) <= 0.999972512931796) {
		tmp = cos(re) * fma((im * im), (im * (im * fma((im * im), 0.001388888888888889, 0.041666666666666664))), 1.0);
	} else {
		tmp = cosh(im) * 1.0;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(cos(re) * 0.5) * Float64(exp(Float64(-im)) + exp(im))) <= 0.999972512931796)
		tmp = Float64(cos(re) * fma(Float64(im * im), Float64(im * Float64(im * fma(Float64(im * im), 0.001388888888888889, 0.041666666666666664))), 1.0));
	else
		tmp = Float64(cosh(im) * 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.999972512931796], N[(N[Cos[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Cosh[im], $MachinePrecision] * 1.0), $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))) < 0.99997251293179601

   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

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. cosh-undef N/A

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

      13. associate-*r* N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 96.1

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

   7. Applied rewrites 96.1%

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

   8. Taylor expanded in im around inf

      \[\leadsto \mathsf{fma}\left(im \cdot im, {im}^{4} \cdot \color{blue}{\left(\frac{1}{720} + \frac{1}{24} \cdot \frac{1}{{im}^{2}}\right)}, 1\right) \cdot \cos re \]
   9. Step-by-step derivation

   10. Applied rewrites 95.5%

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

   if 0.99997251293179601 < (*.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

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. cosh-undef N/A

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

      13. associate-*r* N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \cos re} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-lft-identity 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 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.999972512931796:\\ \;\;\;\;\cos re \cdot \mathsf{fma}\left(im \cdot im, im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.999972512931796:\\ \;\;\;\;\cos re \cdot \mathsf{fma}\left(im \cdot im, im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.001388888888888889\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (cos re) 0.5) (+ (exp (- im)) (exp im))) 0.999972512931796)
   (*
    (cos re)
    (fma (* im im) (* im (* im (* (* im im) 0.001388888888888889))) 1.0))
   (* (cosh im) 1.0)))

C

double code(double re, double im) {
	double tmp;
	if (((cos(re) * 0.5) * (exp(-im) + exp(im))) <= 0.999972512931796) {
		tmp = cos(re) * fma((im * im), (im * (im * ((im * im) * 0.001388888888888889))), 1.0);
	} else {
		tmp = cosh(im) * 1.0;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(cos(re) * 0.5) * Float64(exp(Float64(-im)) + exp(im))) <= 0.999972512931796)
		tmp = Float64(cos(re) * fma(Float64(im * im), Float64(im * Float64(im * Float64(Float64(im * im) * 0.001388888888888889))), 1.0));
	else
		tmp = Float64(cosh(im) * 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.999972512931796], N[(N[Cos[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Cosh[im], $MachinePrecision] * 1.0), $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))) < 0.99997251293179601

   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

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. cosh-undef N/A

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

      13. associate-*r* N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 96.1

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

   7. Applied rewrites 96.1%

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

   8. Taylor expanded in im around inf

      \[\leadsto \mathsf{fma}\left(im \cdot im, \frac{1}{720} \cdot \color{blue}{{im}^{4}}, 1\right) \cdot \cos re \]
   9. Step-by-step derivation

   10. Applied rewrites 95.5%

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

   if 0.99997251293179601 < (*.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

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. cosh-undef N/A

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

      13. associate-*r* N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \cos re} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-lft-identity 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 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.999972512931796:\\ \;\;\;\;\cos re \cdot \mathsf{fma}\left(im \cdot im, im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.001388888888888889\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 71.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot -0.25, 0.5\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot 0.08333333333333333, im\right), 2\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (cos re) 0.5) (+ (exp (- im)) (exp im))) -0.02)
   (*
    (fma re (* re -0.25) 0.5)
    (fma im (fma im (* (* im im) 0.08333333333333333) im) 2.0))
   (*
    1.0
    (fma
     im
     (*
      im
      (fma
       (* im im)
       (fma (* im im) 0.001388888888888889 0.041666666666666664)
       0.5))
     1.0))))

C

double code(double re, double im) {
	double tmp;
	if (((cos(re) * 0.5) * (exp(-im) + exp(im))) <= -0.02) {
		tmp = fma(re, (re * -0.25), 0.5) * fma(im, fma(im, ((im * im) * 0.08333333333333333), im), 2.0);
	} else {
		tmp = 1.0 * fma(im, (im * fma((im * im), fma((im * im), 0.001388888888888889, 0.041666666666666664), 0.5)), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(cos(re) * 0.5) * Float64(exp(Float64(-im)) + exp(im))) <= -0.02)
		tmp = Float64(fma(re, Float64(re * -0.25), 0.5) * fma(im, fma(im, Float64(Float64(im * im) * 0.08333333333333333), im), 2.0));
	else
		tmp = Float64(1.0 * fma(im, Float64(im * fma(Float64(im * im), fma(Float64(im * im), 0.001388888888888889, 0.041666666666666664), 0.5)), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.02], N[(N[(re * N[(re * -0.25), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * 0.08333333333333333), $MachinePrecision] + im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $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))) < -0.0200000000000000004

   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

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right) + 2\right)} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right) + 2\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\color{blue}{im \cdot \left(im \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)} + 2\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right), 2\right)} \]

      5. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(im, im \cdot \color{blue}{\left(\frac{1}{12} \cdot {im}^{2} + 1\right)}, 2\right) \]

      6. distribute-lft-in N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(im, \color{blue}{im \cdot \left(\frac{1}{12} \cdot {im}^{2}\right) + im \cdot 1}, 2\right) \]

      7. *-rgt-identity N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(im, im \cdot \left(\frac{1}{12} \cdot {im}^{2}\right) + \color{blue}{im}, 2\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(im, \color{blue}{\mathsf{fma}\left(im, \frac{1}{12} \cdot {im}^{2}, im\right)}, 2\right) \]

      9. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \color{blue}{{im}^{2} \cdot \frac{1}{12}}, im\right), 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \color{blue}{{im}^{2} \cdot \frac{1}{12}}, im\right), 2\right) \]

      11. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \color{blue}{\left(im \cdot im\right)} \cdot \frac{1}{12}, im\right), 2\right) \]

      12. lower-*.f64 88.5

          \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \color{blue}{\left(im \cdot im\right)} \cdot 0.08333333333333333, im\right), 2\right) \]

   5. Applied rewrites 88.5%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot 0.08333333333333333, im\right), 2\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot \frac{1}{12}, im\right), 2\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{4} \cdot {re}^{2} + \frac{1}{2}\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot \frac{1}{12}, im\right), 2\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{{re}^{2} \cdot \frac{-1}{4}} + \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot \frac{1}{12}, im\right), 2\right) \]

      3. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{4} + \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot \frac{1}{12}, im\right), 2\right) \]

      4. associate-*l* N/A

         \[\leadsto \left(\color{blue}{re \cdot \left(re \cdot \frac{-1}{4}\right)} + \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot \frac{1}{12}, im\right), 2\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, re \cdot \frac{-1}{4}, \frac{1}{2}\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot \frac{1}{12}, im\right), 2\right) \]

      6. lower-*.f64 47.7

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{re \cdot -0.25}, 0.5\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot 0.08333333333333333, im\right), 2\right) \]

   8. Applied rewrites 47.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, re \cdot -0.25, 0.5\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot 0.08333333333333333, im\right), 2\right) \]

   if -0.0200000000000000004 < (*.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

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. cosh-undef N/A

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

      13. associate-*r* N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 87.7%

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

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 80.3

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

   10. Applied rewrites 80.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)} \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot -0.25, 0.5\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot 0.08333333333333333, im\right), 2\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 34.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (cos re) 0.5) (+ (exp (- im)) (exp im))) -0.02)
   (fma re (* re -0.5) 1.0)
   1.0))

C

double code(double re, double im) {
	double tmp;
	if (((cos(re) * 0.5) * (exp(-im) + exp(im))) <= -0.02) {
		tmp = fma(re, (re * -0.5), 1.0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(cos(re) * 0.5) * Float64(exp(Float64(-im)) + exp(im))) <= -0.02)
		tmp = fma(re, Float64(re * -0.5), 1.0);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.02], N[(re * N[(re * -0.5), $MachinePrecision] + 1.0), $MachinePrecision], 1.0]

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))) < -0.0200000000000000004

   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

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

      1. lower-cos.f64 53.9

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

   5. Applied rewrites 53.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 25.5%

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

   if -0.0200000000000000004 < (*.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 im around 0

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

      1. lower-cos.f64 50.8

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

   5. Applied rewrites 50.8%

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

   6. Taylor expanded in re around 0

      \[\leadsto 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 38.4%

      \[\leadsto 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 70.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot -0.25, 0.5\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot 0.08333333333333333, im\right), 2\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(im, im \cdot \left(im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right)\right)\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) -0.02)
   (*
    (fma re (* re -0.25) 0.5)
    (fma im (fma im (* (* im im) 0.08333333333333333) im) 2.0))
   (*
    1.0
    (fma
     im
     (*
      im
      (* im (* im (fma (* im im) 0.001388888888888889 0.041666666666666664))))
     1.0))))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= -0.02) {
		tmp = fma(re, (re * -0.25), 0.5) * fma(im, fma(im, ((im * im) * 0.08333333333333333), im), 2.0);
	} else {
		tmp = 1.0 * fma(im, (im * (im * (im * fma((im * im), 0.001388888888888889, 0.041666666666666664)))), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= -0.02)
		tmp = Float64(fma(re, Float64(re * -0.25), 0.5) * fma(im, fma(im, Float64(Float64(im * im) * 0.08333333333333333), im), 2.0));
	else
		tmp = Float64(1.0 * fma(im, Float64(im * Float64(im * Float64(im * fma(Float64(im * im), 0.001388888888888889, 0.041666666666666664)))), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], -0.02], N[(N[(re * N[(re * -0.25), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * 0.08333333333333333), $MachinePrecision] + im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(im * N[(im * N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -0.0200000000000000004

   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

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right) + 2\right)} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right) + 2\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\color{blue}{im \cdot \left(im \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)} + 2\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right), 2\right)} \]

      5. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(im, im \cdot \color{blue}{\left(\frac{1}{12} \cdot {im}^{2} + 1\right)}, 2\right) \]

      6. distribute-lft-in N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(im, \color{blue}{im \cdot \left(\frac{1}{12} \cdot {im}^{2}\right) + im \cdot 1}, 2\right) \]

      7. *-rgt-identity N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(im, im \cdot \left(\frac{1}{12} \cdot {im}^{2}\right) + \color{blue}{im}, 2\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(im, \color{blue}{\mathsf{fma}\left(im, \frac{1}{12} \cdot {im}^{2}, im\right)}, 2\right) \]

      9. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \color{blue}{{im}^{2} \cdot \frac{1}{12}}, im\right), 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \color{blue}{{im}^{2} \cdot \frac{1}{12}}, im\right), 2\right) \]

      11. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \color{blue}{\left(im \cdot im\right)} \cdot \frac{1}{12}, im\right), 2\right) \]

      12. lower-*.f64 88.5

          \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \color{blue}{\left(im \cdot im\right)} \cdot 0.08333333333333333, im\right), 2\right) \]

   5. Applied rewrites 88.5%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot 0.08333333333333333, im\right), 2\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot \frac{1}{12}, im\right), 2\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{4} \cdot {re}^{2} + \frac{1}{2}\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot \frac{1}{12}, im\right), 2\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{{re}^{2} \cdot \frac{-1}{4}} + \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot \frac{1}{12}, im\right), 2\right) \]

      3. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{4} + \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot \frac{1}{12}, im\right), 2\right) \]

      4. associate-*l* N/A

         \[\leadsto \left(\color{blue}{re \cdot \left(re \cdot \frac{-1}{4}\right)} + \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot \frac{1}{12}, im\right), 2\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, re \cdot \frac{-1}{4}, \frac{1}{2}\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot \frac{1}{12}, im\right), 2\right) \]

      6. lower-*.f64 47.7

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{re \cdot -0.25}, 0.5\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot 0.08333333333333333, im\right), 2\right) \]

   8. Applied rewrites 47.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, re \cdot -0.25, 0.5\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot 0.08333333333333333, im\right), 2\right) \]

   if -0.0200000000000000004 < (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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. cosh-undef N/A

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

      13. associate-*r* N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 87.7%

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

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 80.3

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

   10. Applied rewrites 80.3%

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

   11. Taylor expanded in im around inf

       \[\leadsto \mathsf{fma}\left(im, im \cdot \left({im}^{4} \cdot \color{blue}{\left(\frac{1}{720} + \frac{1}{24} \cdot \frac{1}{{im}^{2}}\right)}\right), 1\right) \cdot 1 \]
   12. Step-by-step derivation

   13. Applied rewrites 80.0%

       \[\leadsto \mathsf{fma}\left(im, im \cdot \left(im \cdot \color{blue}{\left(im \cdot \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right)\right)}\right), 1\right) \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot -0.25, 0.5\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot 0.08333333333333333, im\right), 2\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(im, im \cdot \left(im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right)\right)\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 69.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot -0.25, 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(im, im \cdot \left(im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right)\right)\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) -0.02)
   (* (fma re (* re -0.25) 0.5) (fma im im 2.0))
   (*
    1.0
    (fma
     im
     (*
      im
      (* im (* im (fma (* im im) 0.001388888888888889 0.041666666666666664))))
     1.0))))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= -0.02) {
		tmp = fma(re, (re * -0.25), 0.5) * fma(im, im, 2.0);
	} else {
		tmp = 1.0 * fma(im, (im * (im * (im * fma((im * im), 0.001388888888888889, 0.041666666666666664)))), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= -0.02)
		tmp = Float64(fma(re, Float64(re * -0.25), 0.5) * fma(im, im, 2.0));
	else
		tmp = Float64(1.0 * fma(im, Float64(im * Float64(im * Float64(im * fma(Float64(im * im), 0.001388888888888889, 0.041666666666666664)))), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], -0.02], N[(N[(re * N[(re * -0.25), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(im * N[(im * N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -0.0200000000000000004

   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 76.0

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

   5. Applied rewrites 76.0%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]
   7. 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. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 44.8

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

   8. Applied rewrites 44.8%

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

   if -0.0200000000000000004 < (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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. cosh-undef N/A

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

      13. associate-*r* N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 87.7%

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

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 80.3

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

   10. Applied rewrites 80.3%

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

   11. Taylor expanded in im around inf

       \[\leadsto \mathsf{fma}\left(im, im \cdot \left({im}^{4} \cdot \color{blue}{\left(\frac{1}{720} + \frac{1}{24} \cdot \frac{1}{{im}^{2}}\right)}\right), 1\right) \cdot 1 \]
   12. Step-by-step derivation

   13. Applied rewrites 80.0%

       \[\leadsto \mathsf{fma}\left(im, im \cdot \left(im \cdot \color{blue}{\left(im \cdot \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right)\right)}\right), 1\right) \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot -0.25, 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(im, im \cdot \left(im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right)\right)\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 69.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot -0.25, 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(im, im \cdot \left(im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) -0.02)
   (* (fma re (* re -0.25) 0.5) (fma im im 2.0))
   (*
    1.0
    (fma im (* im (* im (* im (* (* im im) 0.001388888888888889)))) 1.0))))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= -0.02) {
		tmp = fma(re, (re * -0.25), 0.5) * fma(im, im, 2.0);
	} else {
		tmp = 1.0 * fma(im, (im * (im * (im * ((im * im) * 0.001388888888888889)))), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= -0.02)
		tmp = Float64(fma(re, Float64(re * -0.25), 0.5) * fma(im, im, 2.0));
	else
		tmp = Float64(1.0 * fma(im, Float64(im * Float64(im * Float64(im * Float64(Float64(im * im) * 0.001388888888888889)))), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], -0.02], N[(N[(re * N[(re * -0.25), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(im * N[(im * N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -0.0200000000000000004

   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 76.0

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

   5. Applied rewrites 76.0%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]
   7. 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. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 44.8

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

   8. Applied rewrites 44.8%

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

   if -0.0200000000000000004 < (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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. cosh-undef N/A

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

      13. associate-*r* N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 87.7%

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

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 80.3

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

   10. Applied rewrites 80.3%

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

   11. Taylor expanded in im around inf

       \[\leadsto \mathsf{fma}\left(im, im \cdot \left(\frac{1}{720} \cdot \color{blue}{{im}^{4}}\right), 1\right) \cdot 1 \]
   12. Step-by-step derivation

   13. Applied rewrites 80.0%

       \[\leadsto \mathsf{fma}\left(im, im \cdot \left(im \cdot \color{blue}{\left(im \cdot \left(\left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)}\right), 1\right) \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot -0.25, 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(im, im \cdot \left(im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 67.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot -0.25, 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) -0.02)
   (* (fma re (* re -0.25) 0.5) (fma im im 2.0))
   (fma (* im im) (fma (* im im) 0.041666666666666664 0.5) 1.0)))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= -0.02) {
		tmp = fma(re, (re * -0.25), 0.5) * fma(im, im, 2.0);
	} else {
		tmp = fma((im * im), fma((im * im), 0.041666666666666664, 0.5), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= -0.02)
		tmp = Float64(fma(re, Float64(re * -0.25), 0.5) * fma(im, im, 2.0));
	else
		tmp = fma(Float64(im * im), fma(Float64(im * im), 0.041666666666666664, 0.5), 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], -0.02], N[(N[(re * N[(re * -0.25), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -0.0200000000000000004

   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 76.0

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

   5. Applied rewrites 76.0%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]
   7. 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. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 44.8

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

   8. Applied rewrites 44.8%

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

   if -0.0200000000000000004 < (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 im around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

   5. Applied rewrites 89.6%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 77.3%

      \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right)}, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 62.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) -0.02)
   (fma re (* re -0.5) 1.0)
   (fma (* im im) (fma (* im im) 0.041666666666666664 0.5) 1.0)))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= -0.02) {
		tmp = fma(re, (re * -0.5), 1.0);
	} else {
		tmp = fma((im * im), fma((im * im), 0.041666666666666664, 0.5), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= -0.02)
		tmp = fma(re, Float64(re * -0.5), 1.0);
	else
		tmp = fma(Float64(im * im), fma(Float64(im * im), 0.041666666666666664, 0.5), 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], -0.02], N[(re * N[(re * -0.5), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -0.0200000000000000004

   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

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

      1. lower-cos.f64 53.9

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

   5. Applied rewrites 53.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 25.5%

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

   if -0.0200000000000000004 < (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 im around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

   5. Applied rewrites 89.6%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 77.3%

      \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right)}, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 53.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) -0.02) (fma re (* re -0.5) 1.0) (* (fma im im 2.0) 0.5)))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= -0.02) {
		tmp = fma(re, (re * -0.5), 1.0);
	} else {
		tmp = fma(im, im, 2.0) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= -0.02)
		tmp = fma(re, Float64(re * -0.5), 1.0);
	else
		tmp = Float64(fma(im, im, 2.0) * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], -0.02], N[(re * N[(re * -0.5), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(im * im + 2.0), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -0.0200000000000000004

   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

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

      1. lower-cos.f64 53.9

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

   5. Applied rewrites 53.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 25.5%

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

   if -0.0200000000000000004 < (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 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 81.2

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

   5. Applied rewrites 81.2%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 68.9%

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

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 27.9% accurate, 316.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return 1.0

Julia

function code(re, im)
	return 1.0
end

MATLAB

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

Wolfram

code[re_, im_] := 1.0

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0

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

   1. lower-cos.f64 51.6

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

5. Applied rewrites 51.6%

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

6. Taylor expanded in re around 0

   \[\leadsto 1 \]
7. Step-by-step derivation

8. Applied rewrites 29.1%

   \[\leadsto 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024221 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))