math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 10.1s

Alternatives: 18

Speedup: 1.5×

• 49.3% 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: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right)\\ t_1 := \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot -0.5, 1\right) \cdot t\_1\\ \mathbf{elif}\;t\_0 \leq 0.9999999928091802:\\ \;\;\;\;\cos re \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* (cos re) 0.5) (+ (exp (- im)) (exp im))))
        (t_1 (fma (* im im) (fma (* im im) 0.041666666666666664 0.5) 1.0)))
   (if (<= t_0 (- INFINITY))
     (* (fma re (* re -0.5) 1.0) t_1)
     (if (<= t_0 0.9999999928091802) (* (cos re) t_1) (* (cosh im) 1.0)))))

C

double code(double re, double im) {
	double t_0 = (cos(re) * 0.5) * (exp(-im) + exp(im));
	double t_1 = fma((im * im), fma((im * im), 0.041666666666666664, 0.5), 1.0);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(re, (re * -0.5), 1.0) * t_1;
	} else if (t_0 <= 0.9999999928091802) {
		tmp = cos(re) * t_1;
	} 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)))
	t_1 = fma(Float64(im * im), fma(Float64(im * im), 0.041666666666666664, 0.5), 1.0)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(re, Float64(re * -0.5), 1.0) * t_1);
	elseif (t_0 <= 0.9999999928091802)
		tmp = Float64(cos(re) * t_1);
	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]}, Block[{t$95$1 = N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(re * N[(re * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 0.9999999928091802], N[(N[Cos[re], $MachinePrecision] * t$95$1), $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

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

      1. distribute-lft-in N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. distribute-rgt-out N/A

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

      13. associate-+r+ N/A

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

   5. Applied rewrites 82.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 97.1%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.999999992809180172

   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. distribute-lft-in N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. distribute-rgt-out N/A

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

      13. associate-+r+ N/A

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

   5. Applied rewrites 99.4%

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

   if 0.999999992809180172 < (*.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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot -0.5, 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{elif}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.9999999928091802:\\ \;\;\;\;\cos re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos re \cdot 0.5\\ t_1 := t\_0 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot -0.5, 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999999928091802:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot 1\\ \end{array} \end{array} \]

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 -0.5) 1.0)
      (fma (* im im) (fma (* im im) 0.041666666666666664 0.5) 1.0))
     (if (<= t_1 0.9999999928091802)
       (* 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 * -0.5), 1.0) * fma((im * im), fma((im * im), 0.041666666666666664, 0.5), 1.0);
	} else if (t_1 <= 0.9999999928091802) {
		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 * -0.5), 1.0) * fma(Float64(im * im), fma(Float64(im * im), 0.041666666666666664, 0.5), 1.0));
	elseif (t_1 <= 0.9999999928091802)
		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 * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999928091802], 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

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

      1. distribute-lft-in N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. distribute-rgt-out N/A

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

      13. associate-+r+ N/A

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

   5. Applied rewrites 82.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 97.1%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.999999992809180172

   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 99.2

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

   5. Applied rewrites 99.2%

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

   if 0.999999992809180172 < (*.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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot -0.5, 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{elif}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.9999999928091802:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot -0.5, 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{elif}\;t\_0 \leq 0.9999999928091802:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* (cos re) 0.5) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma re (* re -0.5) 1.0)
      (fma (* im im) (fma (* im im) 0.041666666666666664 0.5) 1.0))
     (if (<= t_0 0.9999999928091802) (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 * -0.5), 1.0) * fma((im * im), fma((im * im), 0.041666666666666664, 0.5), 1.0);
	} else if (t_0 <= 0.9999999928091802) {
		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 * -0.5), 1.0) * fma(Float64(im * im), fma(Float64(im * im), 0.041666666666666664, 0.5), 1.0));
	elseif (t_0 <= 0.9999999928091802)
		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 * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.9999999928091802], 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

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

      1. distribute-lft-in N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. distribute-rgt-out N/A

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

      13. associate-+r+ N/A

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

   5. Applied rewrites 82.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 97.1%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.999999992809180172

   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 98.7

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

   5. Applied rewrites 98.7%

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

   if 0.999999992809180172 < (*.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 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot -0.5, 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{elif}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.9999999928091802:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot -0.5, 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{elif}\;t\_0 \leq 0.99996:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, 0.020833333333333332, -0.25\right), 0.5\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, 0.002777777777777778 \cdot \left(im \cdot \left(im \cdot im\right)\right), im\right), 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* (cos re) 0.5) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma re (* re -0.5) 1.0)
      (fma (* im im) (fma (* im im) 0.041666666666666664 0.5) 1.0))
     (if (<= t_0 0.99996)
       (cos re)
       (*
        (fma (* re re) (fma (* re re) 0.020833333333333332 -0.25) 0.5)
        (fma
         im
         (fma (* im im) (* 0.002777777777777778 (* im (* im im))) im)
         2.0))))))

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 * -0.5), 1.0) * fma((im * im), fma((im * im), 0.041666666666666664, 0.5), 1.0);
	} else if (t_0 <= 0.99996) {
		tmp = cos(re);
	} else {
		tmp = fma((re * re), fma((re * re), 0.020833333333333332, -0.25), 0.5) * fma(im, fma((im * im), (0.002777777777777778 * (im * (im * im))), im), 2.0);
	}
	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 * -0.5), 1.0) * fma(Float64(im * im), fma(Float64(im * im), 0.041666666666666664, 0.5), 1.0));
	elseif (t_0 <= 0.99996)
		tmp = cos(re);
	else
		tmp = Float64(fma(Float64(re * re), fma(Float64(re * re), 0.020833333333333332, -0.25), 0.5) * fma(im, fma(Float64(im * im), Float64(0.002777777777777778 * Float64(im * Float64(im * im))), im), 2.0));
	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 * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.99996], N[Cos[re], $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * 0.020833333333333332 + -0.25), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * N[(N[(im * im), $MachinePrecision] * N[(0.002777777777777778 * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + im), $MachinePrecision] + 2.0), $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

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

      1. distribute-lft-in N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. distribute-rgt-out N/A

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

      13. associate-+r+ N/A

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

   5. Applied rewrites 82.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 97.1%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.99995999999999996

   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 98.7

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

   5. Applied rewrites 98.7%

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

   if 0.99995999999999996 < (*.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

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right)} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\color{blue}{im \cdot \left(im \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} + 2\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right), 2\right)} \]

   5. Applied rewrites 91.4%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im\right), 2\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 91.1%

      \[\leadsto \color{blue}{0.5} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im\right), 2\right) \]

   9. Taylor expanded in im around inf

      \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, \frac{1}{360} \cdot \color{blue}{{im}^{3}}, im\right), 2\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 91.1%

       \[\leadsto 0.5 \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, \left(im \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{0.002777777777777778}, im\right), 2\right) \]

   12. Taylor expanded in re around 0

       \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right)\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, \left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{1}{360}, im\right), 2\right) \]
   13. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right) + \frac{1}{2}\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, \left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{1}{360}, im\right), 2\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{48} \cdot {re}^{2} - \frac{1}{4}, \frac{1}{2}\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, \left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{1}{360}, im\right), 2\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{48} \cdot {re}^{2} - \frac{1}{4}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, \left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{1}{360}, im\right), 2\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{48} \cdot {re}^{2} - \frac{1}{4}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, \left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{1}{360}, im\right), 2\right) \]

       5. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \color{blue}{\frac{1}{48} \cdot {re}^{2} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, \left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{1}{360}, im\right), 2\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \color{blue}{{re}^{2} \cdot \frac{1}{48}} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right), \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, \left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{1}{360}, im\right), 2\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, {re}^{2} \cdot \frac{1}{48} + \color{blue}{\frac{-1}{4}}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, \left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{1}{360}, im\right), 2\right) \]

       8. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{48}, \frac{-1}{4}\right)}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, \left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{1}{360}, im\right), 2\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{48}, \frac{-1}{4}\right), \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, \left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{1}{360}, im\right), 2\right) \]

       10. lower-*.f64 95.3

           \[\leadsto \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\color{blue}{re \cdot re}, 0.020833333333333332, -0.25\right), 0.5\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, \left(im \cdot \left(im \cdot im\right)\right) \cdot 0.002777777777777778, im\right), 2\right) \]

   14. Applied rewrites 95.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, 0.020833333333333332, -0.25\right), 0.5\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, \left(im \cdot \left(im \cdot im\right)\right) \cdot 0.002777777777777778, im\right), 2\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot -0.5, 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{elif}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.99996:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, 0.020833333333333332, -0.25\right), 0.5\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, 0.002777777777777778 \cdot \left(im \cdot \left(im \cdot im\right)\right), im\right), 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot -0.5, 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{elif}\;t\_0 \leq 0.99996:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, 0.020833333333333332, -0.25\right), 0.5\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, 0.002777777777777778 \cdot \left(im \cdot \left(im \cdot im\right)\right), im\right), 2\right)\\ \end{array} \end{array} \]

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)
      (fma (* im im) (fma (* im im) 0.041666666666666664 0.5) 1.0))
     (if (<= t_0 0.99996)
       1.0
       (*
        (fma (* re re) (fma (* re re) 0.020833333333333332 -0.25) 0.5)
        (fma
         im
         (fma (* im im) (* 0.002777777777777778 (* im (* im im))) im)
         2.0))))))

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) * fma((im * im), fma((im * im), 0.041666666666666664, 0.5), 1.0);
	} else if (t_0 <= 0.99996) {
		tmp = 1.0;
	} else {
		tmp = fma((re * re), fma((re * re), 0.020833333333333332, -0.25), 0.5) * fma(im, fma((im * im), (0.002777777777777778 * (im * (im * im))), im), 2.0);
	}
	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 <= -0.02)
		tmp = Float64(fma(re, Float64(re * -0.5), 1.0) * fma(Float64(im * im), fma(Float64(im * im), 0.041666666666666664, 0.5), 1.0));
	elseif (t_0 <= 0.99996)
		tmp = 1.0;
	else
		tmp = Float64(fma(Float64(re * re), fma(Float64(re * re), 0.020833333333333332, -0.25), 0.5) * fma(im, fma(Float64(im * im), Float64(0.002777777777777778 * Float64(im * Float64(im * im))), im), 2.0));
	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, -0.02], N[(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]), $MachinePrecision], If[LessEqual[t$95$0, 0.99996], 1.0, N[(N[(N[(re * re), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * 0.020833333333333332 + -0.25), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * N[(N[(im * im), $MachinePrecision] * N[(0.002777777777777778 * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + im), $MachinePrecision] + 2.0), $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

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

      1. distribute-lft-in N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. distribute-rgt-out N/A

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

      13. associate-+r+ N/A

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

   5. Applied rewrites 92.1%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 44.9%

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

   if -0.0200000000000000004 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.99995999999999996

   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%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 21.5%

      \[\leadsto 1 \]

   if 0.99995999999999996 < (*.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

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right)} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\color{blue}{im \cdot \left(im \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} + 2\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right), 2\right)} \]

   5. Applied rewrites 91.4%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im\right), 2\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 91.1%

      \[\leadsto \color{blue}{0.5} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im\right), 2\right) \]

   9. Taylor expanded in im around inf

      \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, \frac{1}{360} \cdot \color{blue}{{im}^{3}}, im\right), 2\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 91.1%

       \[\leadsto 0.5 \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, \left(im \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{0.002777777777777778}, im\right), 2\right) \]

   12. Taylor expanded in re around 0

       \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right)\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, \left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{1}{360}, im\right), 2\right) \]
   13. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right) + \frac{1}{2}\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, \left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{1}{360}, im\right), 2\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{48} \cdot {re}^{2} - \frac{1}{4}, \frac{1}{2}\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, \left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{1}{360}, im\right), 2\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{48} \cdot {re}^{2} - \frac{1}{4}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, \left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{1}{360}, im\right), 2\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{48} \cdot {re}^{2} - \frac{1}{4}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, \left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{1}{360}, im\right), 2\right) \]

       5. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \color{blue}{\frac{1}{48} \cdot {re}^{2} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, \left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{1}{360}, im\right), 2\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \color{blue}{{re}^{2} \cdot \frac{1}{48}} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right), \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, \left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{1}{360}, im\right), 2\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, {re}^{2} \cdot \frac{1}{48} + \color{blue}{\frac{-1}{4}}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, \left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{1}{360}, im\right), 2\right) \]

       8. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{48}, \frac{-1}{4}\right)}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, \left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{1}{360}, im\right), 2\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{48}, \frac{-1}{4}\right), \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, \left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{1}{360}, im\right), 2\right) \]

       10. lower-*.f64 95.3

           \[\leadsto \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\color{blue}{re \cdot re}, 0.020833333333333332, -0.25\right), 0.5\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, \left(im \cdot \left(im \cdot im\right)\right) \cdot 0.002777777777777778, im\right), 2\right) \]

   14. Applied rewrites 95.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, 0.020833333333333332, -0.25\right), 0.5\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, \left(im \cdot \left(im \cdot im\right)\right) \cdot 0.002777777777777778, im\right), 2\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot -0.5, 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{elif}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.99996:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, 0.020833333333333332, -0.25\right), 0.5\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, 0.002777777777777778 \cdot \left(im \cdot \left(im \cdot im\right)\right), im\right), 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(0.002777777777777778 \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right)\\ \end{array} \end{array} \]

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)
      (fma
       re
       (* re (fma (* re re) -0.001388888888888889 0.041666666666666664))
       -0.5)
      1.0)
     (if (<= t_0 2.0)
       (* (fma im im 2.0) 0.5)
       (*
        0.5
        (* 0.002777777777777778 (* (* im im) (* (* im im) (* im im)))))))))

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), fma(re, (re * fma((re * re), -0.001388888888888889, 0.041666666666666664)), -0.5), 1.0);
	} else if (t_0 <= 2.0) {
		tmp = fma(im, im, 2.0) * 0.5;
	} else {
		tmp = 0.5 * (0.002777777777777778 * ((im * im) * ((im * im) * (im * im))));
	}
	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(Float64(re * re), fma(re, Float64(re * fma(Float64(re * re), -0.001388888888888889, 0.041666666666666664)), -0.5), 1.0);
	elseif (t_0 <= 2.0)
		tmp = Float64(fma(im, im, 2.0) * 0.5);
	else
		tmp = Float64(0.5 * Float64(0.002777777777777778 * Float64(Float64(im * im) * Float64(Float64(im * im) * Float64(im * im)))));
	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[(N[(re * re), $MachinePrecision] * N[(re * N[(re * N[(N[(re * re), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(N[(im * im + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(0.5 * N[(0.002777777777777778 * N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 57.3

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

   5. Applied rewrites 57.3%

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

   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 24.1%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 38.0%

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

   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

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right)} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\color{blue}{im \cdot \left(im \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} + 2\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right), 2\right)} \]

   5. Applied rewrites 99.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im\right), 2\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 69.4%

      \[\leadsto \color{blue}{0.5} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im\right), 2\right) \]

   9. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. unpow2 N/A

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

       3. lower-fma.f64 69.4

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

   11. Applied rewrites 69.4%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right)} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\color{blue}{im \cdot \left(im \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} + 2\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right), 2\right)} \]

   5. Applied rewrites 84.3%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im\right), 2\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 84.3%

      \[\leadsto \color{blue}{0.5} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im\right), 2\right) \]

   9. Taylor expanded in im around inf

      \[\leadsto \frac{1}{2} \cdot \left(\frac{1}{360} \cdot \color{blue}{{im}^{6}}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 84.3%

       \[\leadsto 0.5 \cdot \left(0.002777777777777778 \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)\\ \mathbf{elif}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 2:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(0.002777777777777778 \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.3% accurate, 0.5× speedup

Math

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

FPCore

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

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 = -0.5 * (re * re);
	} else if (t_0 <= 2.0) {
		tmp = fma(im, im, 2.0) * 0.5;
	} else {
		tmp = im * (0.041666666666666664 * (im * (im * im)));
	}
	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 = Float64(-0.5 * Float64(re * re));
	elseif (t_0 <= 2.0)
		tmp = Float64(fma(im, im, 2.0) * 0.5);
	else
		tmp = Float64(im * Float64(0.041666666666666664 * Float64(im * Float64(im * im))));
	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[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(N[(im * im + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(im * N[(0.041666666666666664 * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $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 57.3

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

   5. Applied rewrites 57.3%

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

   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 24.1%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 24.1%

       \[\leadsto \left(re \cdot re\right) \cdot -0.5 \]

   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

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right)} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\color{blue}{im \cdot \left(im \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} + 2\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right), 2\right)} \]

   5. Applied rewrites 99.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im\right), 2\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 69.4%

      \[\leadsto \color{blue}{0.5} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im\right), 2\right) \]

   9. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. unpow2 N/A

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

       3. lower-fma.f64 69.4

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

   11. Applied rewrites 69.4%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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. distribute-lft-in N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. distribute-rgt-out N/A

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

      13. associate-+r+ N/A

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

   5. Applied rewrites 71.7%

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

   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 71.7%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 71.7%

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

4. Final simplification 57.2%

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

Alternative 9: 98.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.9999999928091802:\\ \;\;\;\;\cos re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (cos re) 0.5) (+ (exp (- im)) (exp im))) 0.9999999928091802)
   (*
    (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))) <= 0.9999999928091802) {
		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))) <= 0.9999999928091802)
		tmp = Float64(cos(re) * fma(Float64(im * im), fma(im, Float64(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], 0.9999999928091802], 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] + 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))) < 0.999999992809180172

   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. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 95.5

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

   7. Applied rewrites 95.5%

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

   if 0.999999992809180172 < (*.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 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.9999999928091802:\\ \;\;\;\;\cos re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.9% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (cos re) 0.5) (+ (exp (- im)) (exp im))) -0.02)
   (fma
    (* re re)
    (fma
     re
     (* re (fma (* re re) -0.001388888888888889 0.041666666666666664))
     -0.5)
    1.0)
   (fma (* (* im im) (* im im)) 0.041666666666666664 (fma im (* im 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), fma(re, (re * fma((re * re), -0.001388888888888889, 0.041666666666666664)), -0.5), 1.0);
	} else {
		tmp = fma(((im * im) * (im * im)), 0.041666666666666664, fma(im, (im * 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 = fma(Float64(re * re), fma(re, Float64(re * fma(Float64(re * re), -0.001388888888888889, 0.041666666666666664)), -0.5), 1.0);
	else
		tmp = fma(Float64(Float64(im * im) * Float64(im * im)), 0.041666666666666664, fma(im, Float64(im * 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 * re), $MachinePrecision] * N[(re * N[(re * N[(N[(re * re), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] * 0.041666666666666664 + N[(im * N[(im * 0.5), $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

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

      1. lower-cos.f64 57.3

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

   5. Applied rewrites 57.3%

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

   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 24.1%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 38.0%

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

   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

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

      1. distribute-lft-in N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. distribute-rgt-out N/A

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

      13. associate-+r+ N/A

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

   5. Applied rewrites 87.5%

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

   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 70.4%

      \[\leadsto \mathsf{fma}\left(im, \color{blue}{im \cdot \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right)}, 1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 71.3%

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

4. Final simplification 61.8%

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

Alternative 11: 64.4% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (cos re) 0.5) (+ (exp (- im)) (exp im))) -0.02)
   (* -0.5 (* re re))
   (fma (* (* im im) (* im im)) 0.041666666666666664 (fma im (* im 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 = -0.5 * (re * re);
	} else {
		tmp = fma(((im * im) * (im * im)), 0.041666666666666664, fma(im, (im * 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(-0.5 * Float64(re * re));
	else
		tmp = fma(Float64(Float64(im * im) * Float64(im * im)), 0.041666666666666664, fma(im, Float64(im * 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[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision], N[(N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] * 0.041666666666666664 + N[(im * N[(im * 0.5), $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

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

      1. lower-cos.f64 57.3

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

   5. Applied rewrites 57.3%

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

   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 24.1%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 24.1%

       \[\leadsto \left(re \cdot re\right) \cdot -0.5 \]

   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

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

      1. distribute-lft-in N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. distribute-rgt-out N/A

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

      13. associate-+r+ N/A

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

   5. Applied rewrites 87.5%

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

   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 70.4%

      \[\leadsto \mathsf{fma}\left(im, \color{blue}{im \cdot \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right)}, 1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 71.3%

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

4. Final simplification 57.8%

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

Alternative 12: 64.3% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (cos re) 0.5) (+ (exp (- im)) (exp im))) -0.02)
   (* -0.5 (* re re))
   (fma im (* im (fma (* im 0.041666666666666664) im 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 = -0.5 * (re * re);
	} else {
		tmp = fma(im, (im * fma((im * 0.041666666666666664), im, 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(-0.5 * Float64(re * re));
	else
		tmp = fma(im, Float64(im * fma(Float64(im * 0.041666666666666664), im, 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[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision], N[(im * N[(im * N[(N[(im * 0.041666666666666664), $MachinePrecision] * im + 0.5), $MachinePrecision]), $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))) < -0.0200000000000000004

   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 57.3

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

   5. Applied rewrites 57.3%

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

   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 24.1%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 24.1%

       \[\leadsto \left(re \cdot re\right) \cdot -0.5 \]

   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

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

      1. distribute-lft-in N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. distribute-rgt-out N/A

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

      13. associate-+r+ N/A

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

   5. Applied rewrites 87.5%

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

   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 70.4%

      \[\leadsto \mathsf{fma}\left(im, \color{blue}{im \cdot \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right)}, 1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 70.4%

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

4. Final simplification 57.2%

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

Alternative 13: 64.1% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	double tmp;
	if (((cos(re) * 0.5) * (exp(-im) + exp(im))) <= -0.02) {
		tmp = -0.5 * (re * re);
	} else {
		tmp = fma(im, (im * ((im * im) * 0.041666666666666664)), 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(-0.5 * Float64(re * re));
	else
		tmp = fma(im, Float64(im * Float64(Float64(im * im) * 0.041666666666666664)), 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[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision], N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $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))) < -0.0200000000000000004

   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 57.3

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

   5. Applied rewrites 57.3%

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

   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 24.1%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 24.1%

       \[\leadsto \left(re \cdot re\right) \cdot -0.5 \]

   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

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

      1. distribute-lft-in N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. distribute-rgt-out N/A

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

      13. associate-+r+ N/A

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

   5. Applied rewrites 87.5%

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

   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 70.4%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 70.3%

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

4. Final simplification 57.1%

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

Alternative 14: 55.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	double tmp;
	if (((cos(re) * 0.5) * (exp(-im) + exp(im))) <= -0.02) {
		tmp = -0.5 * (re * re);
	} else {
		tmp = fma(im, im, 2.0) * 0.5;
	}
	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(-0.5 * Float64(re * re));
	else
		tmp = Float64(fma(im, im, 2.0) * 0.5);
	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[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision], N[(N[(im * im + 2.0), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -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 57.3

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

   5. Applied rewrites 57.3%

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

   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 24.1%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 24.1%

       \[\leadsto \left(re \cdot re\right) \cdot -0.5 \]

   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

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right)} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\color{blue}{im \cdot \left(im \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} + 2\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right), 2\right)} \]

   5. Applied rewrites 93.1%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im\right), 2\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 75.9%

      \[\leadsto \color{blue}{0.5} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im\right), 2\right) \]

   9. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. unpow2 N/A

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

       3. lower-fma.f64 59.3

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

   11. Applied rewrites 59.3%

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

4. Final simplification 49.2%

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

Alternative 15: 71.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot -0.5, 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, 0.002777777777777778 \cdot \left(im \cdot \left(im \cdot im\right)\right), im\right), 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) -0.005)
   (*
    (fma re (* re -0.5) 1.0)
    (fma (* im im) (fma (* im im) 0.041666666666666664 0.5) 1.0))
   (*
    (fma im (fma (* im im) (* 0.002777777777777778 (* im (* im im))) im) 2.0)
    0.5)))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= -0.005) {
		tmp = fma(re, (re * -0.5), 1.0) * fma((im * im), fma((im * im), 0.041666666666666664, 0.5), 1.0);
	} else {
		tmp = fma(im, fma((im * im), (0.002777777777777778 * (im * (im * im))), im), 2.0) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= -0.005)
		tmp = Float64(fma(re, Float64(re * -0.5), 1.0) * fma(Float64(im * im), fma(Float64(im * im), 0.041666666666666664, 0.5), 1.0));
	else
		tmp = Float64(fma(im, fma(Float64(im * im), Float64(0.002777777777777778 * Float64(im * Float64(im * im))), im), 2.0) * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], -0.005], N[(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]), $MachinePrecision], N[(N[(im * N[(N[(im * im), $MachinePrecision] * N[(0.002777777777777778 * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + im), $MachinePrecision] + 2.0), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. distribute-lft-in N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. distribute-rgt-out N/A

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

      13. associate-+r+ N/A

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

   5. Applied rewrites 92.1%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 44.9%

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

   if -0.0050000000000000001 < (cos.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right)} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\color{blue}{im \cdot \left(im \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} + 2\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right), 2\right)} \]

   5. Applied rewrites 93.1%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im\right), 2\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 75.9%

      \[\leadsto \color{blue}{0.5} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im\right), 2\right) \]

   9. Taylor expanded in im around inf

      \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, \frac{1}{360} \cdot \color{blue}{{im}^{3}}, im\right), 2\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 75.9%

       \[\leadsto 0.5 \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, \left(im \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{0.002777777777777778}, im\right), 2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot -0.5, 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, 0.002777777777777778 \cdot \left(im \cdot \left(im \cdot im\right)\right), im\right), 2\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 70.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, 0.002777777777777778 \cdot \left(im \cdot \left(im \cdot im\right)\right), im\right), 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) -0.005)
   (fma
    (* re re)
    (fma
     re
     (* re (fma (* re re) -0.001388888888888889 0.041666666666666664))
     -0.5)
    1.0)
   (*
    (fma im (fma (* im im) (* 0.002777777777777778 (* im (* im im))) im) 2.0)
    0.5)))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= -0.005) {
		tmp = fma((re * re), fma(re, (re * fma((re * re), -0.001388888888888889, 0.041666666666666664)), -0.5), 1.0);
	} else {
		tmp = fma(im, fma((im * im), (0.002777777777777778 * (im * (im * im))), im), 2.0) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= -0.005)
		tmp = fma(Float64(re * re), fma(re, Float64(re * fma(Float64(re * re), -0.001388888888888889, 0.041666666666666664)), -0.5), 1.0);
	else
		tmp = Float64(fma(im, fma(Float64(im * im), Float64(0.002777777777777778 * Float64(im * Float64(im * im))), im), 2.0) * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], -0.005], N[(N[(re * re), $MachinePrecision] * N[(re * N[(re * N[(N[(re * re), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(im * N[(N[(im * im), $MachinePrecision] * N[(0.002777777777777778 * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + im), $MachinePrecision] + 2.0), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-cos.f64 57.3

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

   5. Applied rewrites 57.3%

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

   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 24.1%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 38.0%

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

   if -0.0050000000000000001 < (cos.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right)} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\color{blue}{im \cdot \left(im \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} + 2\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right), 2\right)} \]

   5. Applied rewrites 93.1%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im\right), 2\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 75.9%

      \[\leadsto \color{blue}{0.5} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im\right), 2\right) \]

   9. Taylor expanded in im around inf

      \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, \frac{1}{360} \cdot \color{blue}{{im}^{3}}, im\right), 2\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 75.9%

       \[\leadsto 0.5 \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, \left(im \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{0.002777777777777778}, im\right), 2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, 0.002777777777777778 \cdot \left(im \cdot \left(im \cdot im\right)\right), im\right), 2\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 35.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.005:\\ \;\;\;\;-0.5 \cdot \left(re \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) -0.005) (* -0.5 (* re re)) 1.0))

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (cos(re) <= (-0.005d0)) then
        tmp = (-0.5d0) * (re * re)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.cos(re) <= -0.005) {
		tmp = -0.5 * (re * re);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.cos(re) <= -0.005:
		tmp = -0.5 * (re * re)
	else:
		tmp = 1.0
	return tmp

Julia

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

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (cos(re) <= -0.005)
		tmp = -0.5 * (re * re);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-cos.f64 57.3

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

   5. Applied rewrites 57.3%

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

   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 24.1%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 24.1%

       \[\leadsto \left(re \cdot re\right) \cdot -0.5 \]

   if -0.0050000000000000001 < (cos.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-cos.f64 57.6

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

   5. Applied rewrites 57.6%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 40.7%

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

4. Final simplification 35.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq -0.005:\\ \;\;\;\;-0.5 \cdot \left(re \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 29.4% 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 57.5

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

5. Applied rewrites 57.5%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 29.4%

   \[\leadsto 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024227 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))