math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 13.0s

Alternatives: 20

Speedup: 1.5×

• 49.0% 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. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-lowering-*.f64 N/A

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

   4. *-commutative N/A

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

   5. +-commutative N/A

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

   6. cosh-undef N/A

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

   7. associate-*r* N/A

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

   8. metadata-eval N/A

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

   9. *-lowering-*.f64 N/A

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

   10. cosh-lowering-cosh.f64 N/A

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

   11. cos-lowering-cos.f64 100.0

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

4. Applied egg-rr 100.0%

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

   1. *-lft-identity N/A

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

   2. cosh-lowering-cosh.f64 100.0

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

6. Applied egg-rr 100.0%

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

Alternative 2: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\cosh im \cdot \mathsf{fma}\left(re, re \cdot -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq 0.9999999999999999:\\ \;\;\;\;\cos re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* (cos re) 0.5) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (* (cosh im) (fma re (* re -0.5) 1.0))
     (if (<= t_0 0.9999999999999999)
       (*
        (cos re)
        (fma
         (* im im)
         (fma
          (* im im)
          (fma (* im im) 0.001388888888888889 0.041666666666666664)
          0.5)
         1.0))
       (cosh im)))))

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 = cosh(im) * fma(re, (re * -0.5), 1.0);
	} else if (t_0 <= 0.9999999999999999) {
		tmp = cos(re) * fma((im * im), fma((im * im), fma((im * im), 0.001388888888888889, 0.041666666666666664), 0.5), 1.0);
	} else {
		tmp = cosh(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 <= Float64(-Inf))
		tmp = Float64(cosh(im) * fma(re, Float64(re * -0.5), 1.0));
	elseif (t_0 <= 0.9999999999999999)
		tmp = Float64(cos(re) * fma(Float64(im * im), fma(Float64(im * im), fma(Float64(im * im), 0.001388888888888889, 0.041666666666666664), 0.5), 1.0));
	else
		tmp = cosh(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, (-Infinity)], N[(N[Cosh[im], $MachinePrecision] * N[(re * N[(re * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.9999999999999999], N[(N[Cos[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Cosh[im], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. cosh-undef N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-lowering-*.f64 N/A

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

      10. cosh-lowering-cosh.f64 N/A

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

      11. cos-lowering-cos.f64 100.0

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

   4. Applied egg-rr 100.0%

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

      1. *-lft-identity N/A

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

      2. cosh-lowering-cosh.f64 100.0

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 100.0

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

   9. Simplified 100.0%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.999999999999999889

   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. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. cosh-undef N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-lowering-*.f64 N/A

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

      10. cosh-lowering-cosh.f64 N/A

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

      11. cos-lowering-cos.f64 100.0

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

   4. Applied egg-rr 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. accelerator-lowering-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. *-lowering-*.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. accelerator-lowering-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 99.6

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

   7. Simplified 99.6%

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

   if 0.999999999999999889 < (*.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. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. cosh-undef N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-lowering-*.f64 N/A

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

      10. cosh-lowering-cosh.f64 N/A

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

      11. cos-lowering-cos.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Simplified 99.1%

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

      1. *-lft-identity N/A

         \[\leadsto \color{blue}{\cosh im} \cdot 1 \]

      2. *-rgt-identity N/A

         \[\leadsto \color{blue}{\cosh im} \]

      3. cosh-lowering-cosh.f64 99.1

         \[\leadsto \color{blue}{\cosh im} \]

   9. Applied egg-rr 99.1%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\cosh im \cdot \mathsf{fma}\left(re, re \cdot -0.5, 1\right)\\ \mathbf{elif}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.9999999999999999:\\ \;\;\;\;\cos re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\cosh im \cdot \mathsf{fma}\left(re, re \cdot -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq 0.9999999999999999:\\ \;\;\;\;\cos re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* (cos re) 0.5) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (* (cosh im) (fma re (* re -0.5) 1.0))
     (if (<= t_0 0.9999999999999999)
       (*
        (cos re)
        (fma (* im im) (fma (* im im) 0.041666666666666664 0.5) 1.0))
       (cosh im)))))

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 = cosh(im) * fma(re, (re * -0.5), 1.0);
	} else if (t_0 <= 0.9999999999999999) {
		tmp = cos(re) * fma((im * im), fma((im * im), 0.041666666666666664, 0.5), 1.0);
	} else {
		tmp = cosh(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 <= Float64(-Inf))
		tmp = Float64(cosh(im) * fma(re, Float64(re * -0.5), 1.0));
	elseif (t_0 <= 0.9999999999999999)
		tmp = Float64(cos(re) * fma(Float64(im * im), fma(Float64(im * im), 0.041666666666666664, 0.5), 1.0));
	else
		tmp = cosh(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, (-Infinity)], N[(N[Cosh[im], $MachinePrecision] * N[(re * N[(re * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.9999999999999999], N[(N[Cos[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Cosh[im], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. cosh-undef N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-lowering-*.f64 N/A

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

      10. cosh-lowering-cosh.f64 N/A

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

      11. cos-lowering-cos.f64 100.0

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

   4. Applied egg-rr 100.0%

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

      1. *-lft-identity N/A

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

      2. cosh-lowering-cosh.f64 100.0

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 100.0

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

   9. Simplified 100.0%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.999999999999999889

   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. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-out N/A

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

      11. associate-+r+ N/A

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

      12. +-commutative N/A

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

   5. Simplified 99.3%

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

   if 0.999999999999999889 < (*.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. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. cosh-undef N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-lowering-*.f64 N/A

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

      10. cosh-lowering-cosh.f64 N/A

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

      11. cos-lowering-cos.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Simplified 99.1%

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

      1. *-lft-identity N/A

         \[\leadsto \color{blue}{\cosh im} \cdot 1 \]

      2. *-rgt-identity N/A

         \[\leadsto \color{blue}{\cosh im} \]

      3. cosh-lowering-cosh.f64 99.1

         \[\leadsto \color{blue}{\cosh im} \]

   9. Applied egg-rr 99.1%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\cosh im \cdot \mathsf{fma}\left(re, re \cdot -0.5, 1\right)\\ \mathbf{elif}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.9999999999999999:\\ \;\;\;\;\cos re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos re \cdot 0.5\\ t_1 := t\_0 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\cosh im \cdot \mathsf{fma}\left(re, re \cdot -0.5, 1\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999999999999999:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im\\ \end{array} \end{array} \]

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))
     (* (cosh im) (fma re (* re -0.5) 1.0))
     (if (<= t_1 0.9999999999999999) (* t_0 (fma im im 2.0)) (cosh im)))))

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 = cosh(im) * fma(re, (re * -0.5), 1.0);
	} else if (t_1 <= 0.9999999999999999) {
		tmp = t_0 * fma(im, im, 2.0);
	} else {
		tmp = cosh(im);
	}
	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(cosh(im) * fma(re, Float64(re * -0.5), 1.0));
	elseif (t_1 <= 0.9999999999999999)
		tmp = Float64(t_0 * fma(im, im, 2.0));
	else
		tmp = cosh(im);
	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[Cosh[im], $MachinePrecision] * N[(re * N[(re * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999999999], N[(t$95$0 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[Cosh[im], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. cosh-undef N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-lowering-*.f64 N/A

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

      10. cosh-lowering-cosh.f64 N/A

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

      11. cos-lowering-cos.f64 100.0

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

   4. Applied egg-rr 100.0%

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

      1. *-lft-identity N/A

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

      2. cosh-lowering-cosh.f64 100.0

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 100.0

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

   9. Simplified 100.0%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.999999999999999889

   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. accelerator-lowering-fma.f64 98.9

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

   5. Simplified 98.9%

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

   if 0.999999999999999889 < (*.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. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. cosh-undef N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-lowering-*.f64 N/A

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

      10. cosh-lowering-cosh.f64 N/A

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

      11. cos-lowering-cos.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Simplified 99.1%

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

      1. *-lft-identity N/A

         \[\leadsto \color{blue}{\cosh im} \cdot 1 \]

      2. *-rgt-identity N/A

         \[\leadsto \color{blue}{\cosh im} \]

      3. cosh-lowering-cosh.f64 99.1

         \[\leadsto \color{blue}{\cosh im} \]

   9. Applied egg-rr 99.1%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\cosh im \cdot \mathsf{fma}\left(re, re \cdot -0.5, 1\right)\\ \mathbf{elif}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.9999999999999999:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\cosh im \cdot \mathsf{fma}\left(re, re \cdot -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq 0.9999999999999999:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\cosh im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* (cos re) 0.5) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (* (cosh im) (fma re (* re -0.5) 1.0))
     (if (<= t_0 0.9999999999999999) (cos re) (cosh im)))))

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 = cosh(im) * fma(re, (re * -0.5), 1.0);
	} else if (t_0 <= 0.9999999999999999) {
		tmp = cos(re);
	} else {
		tmp = cosh(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 <= Float64(-Inf))
		tmp = Float64(cosh(im) * fma(re, Float64(re * -0.5), 1.0));
	elseif (t_0 <= 0.9999999999999999)
		tmp = cos(re);
	else
		tmp = cosh(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, (-Infinity)], N[(N[Cosh[im], $MachinePrecision] * N[(re * N[(re * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.9999999999999999], N[Cos[re], $MachinePrecision], N[Cosh[im], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. cosh-undef N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-lowering-*.f64 N/A

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

      10. cosh-lowering-cosh.f64 N/A

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

      11. cos-lowering-cos.f64 100.0

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

   4. Applied egg-rr 100.0%

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

      1. *-lft-identity N/A

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

      2. cosh-lowering-cosh.f64 100.0

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 100.0

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

   9. Simplified 100.0%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.999999999999999889

   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. cos-lowering-cos.f64 97.5

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

   5. Simplified 97.5%

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

   if 0.999999999999999889 < (*.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. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. cosh-undef N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-lowering-*.f64 N/A

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

      10. cosh-lowering-cosh.f64 N/A

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

      11. cos-lowering-cos.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Simplified 99.1%

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

      1. *-lft-identity N/A

         \[\leadsto \color{blue}{\cosh im} \cdot 1 \]

      2. *-rgt-identity N/A

         \[\leadsto \color{blue}{\cosh im} \]

      3. cosh-lowering-cosh.f64 99.1

         \[\leadsto \color{blue}{\cosh im} \]

   9. Applied egg-rr 99.1%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\cosh im \cdot \mathsf{fma}\left(re, re \cdot -0.5, 1\right)\\ \mathbf{elif}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.9999999999999999:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\cosh im\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot \left(re \cdot \left(re \cdot \left(\left(re \cdot re\right) \cdot -0.0006944444444444445\right)\right)\right), 0.5\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im\right), 2\right)\\ \mathbf{elif}\;t\_0 \leq 0.9999999999999999:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\cosh im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* (cos re) 0.5) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma re (* re (* re (* re (* (* re re) -0.0006944444444444445)))) 0.5)
      (fma
       im
       (fma
        (* im im)
        (* im (fma (* im im) 0.002777777777777778 0.08333333333333333))
        im)
       2.0))
     (if (<= t_0 0.9999999999999999) (cos re) (cosh im)))))

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 * (re * (re * ((re * re) * -0.0006944444444444445)))), 0.5) * fma(im, fma((im * im), (im * fma((im * im), 0.002777777777777778, 0.08333333333333333)), im), 2.0);
	} else if (t_0 <= 0.9999999999999999) {
		tmp = cos(re);
	} else {
		tmp = cosh(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 <= Float64(-Inf))
		tmp = Float64(fma(re, Float64(re * Float64(re * Float64(re * Float64(Float64(re * re) * -0.0006944444444444445)))), 0.5) * fma(im, fma(Float64(im * im), Float64(im * fma(Float64(im * im), 0.002777777777777778, 0.08333333333333333)), im), 2.0));
	elseif (t_0 <= 0.9999999999999999)
		tmp = cos(re);
	else
		tmp = cosh(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, (-Infinity)], N[(N[(re * N[(re * N[(re * N[(re * N[(N[(re * re), $MachinePrecision] * -0.0006944444444444445), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * N[(N[(im * im), $MachinePrecision] * N[(im * N[(N[(im * im), $MachinePrecision] * 0.002777777777777778 + 0.08333333333333333), $MachinePrecision]), $MachinePrecision] + im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.9999999999999999], N[Cos[re], $MachinePrecision], N[Cosh[im], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]
   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. accelerator-lowering-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. Simplified 62.5%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) + \frac{1}{2}\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(re \cdot re\right)} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) + \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(\color{blue}{re \cdot \left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right)} + \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, re \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right), \frac{1}{2}\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{re \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(re, re \cdot \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(re, re \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) + \color{blue}{\frac{-1}{4}}\right), \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, re \cdot \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}, \frac{-1}{4}\right)}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}, \frac{-1}{4}\right), \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}, \frac{-1}{4}\right), \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\frac{-1}{1440} \cdot {re}^{2} + \frac{1}{48}}, \frac{-1}{4}\right), \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{{re}^{2} \cdot \frac{-1}{1440}} + \frac{1}{48}, \frac{-1}{4}\right), \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{1440}, \frac{1}{48}\right)}, \frac{-1}{4}\right), \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{1440}, \frac{1}{48}\right), \frac{-1}{4}\right), \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      15. *-lowering-*.f64 94.3

          \[\leadsto \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\color{blue}{re \cdot re}, -0.0006944444444444445, 0.020833333333333332\right), -0.25\right), 0.5\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im\right), 2\right) \]

   8. Simplified 94.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, -0.0006944444444444445, 0.020833333333333332\right), -0.25\right), 0.5\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im\right), 2\right) \]

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{fma}\left(re, re \cdot \color{blue}{\left(\frac{-1}{1440} \cdot {re}^{4}\right)}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(re, re \cdot \left(\frac{-1}{1440} \cdot {re}^{\color{blue}{\left(2 \cdot 2\right)}}\right), \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

       2. pow-sqr N/A

          \[\leadsto \mathsf{fma}\left(re, re \cdot \left(\frac{-1}{1440} \cdot \color{blue}{\left({re}^{2} \cdot {re}^{2}\right)}\right), \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(re, re \cdot \color{blue}{\left(\left(\frac{-1}{1440} \cdot {re}^{2}\right) \cdot {re}^{2}\right)}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(re, re \cdot \color{blue}{\left({re}^{2} \cdot \left(\frac{-1}{1440} \cdot {re}^{2}\right)\right)}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(re, re \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \left(\frac{-1}{1440} \cdot {re}^{2}\right)\right), \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(re, re \cdot \color{blue}{\left(re \cdot \left(re \cdot \left(\frac{-1}{1440} \cdot {re}^{2}\right)\right)\right)}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(re, re \cdot \color{blue}{\left(re \cdot \left(re \cdot \left(\frac{-1}{1440} \cdot {re}^{2}\right)\right)\right)}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(re, re \cdot \left(re \cdot \color{blue}{\left(re \cdot \left(\frac{-1}{1440} \cdot {re}^{2}\right)\right)}\right), \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(re, re \cdot \left(re \cdot \left(re \cdot \color{blue}{\left({re}^{2} \cdot \frac{-1}{1440}\right)}\right)\right), \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(re, re \cdot \left(re \cdot \left(re \cdot \color{blue}{\left({re}^{2} \cdot \frac{-1}{1440}\right)}\right)\right), \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(re, re \cdot \left(re \cdot \left(re \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{1440}\right)\right)\right), \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

       12. *-lowering-*.f64 94.3

           \[\leadsto \mathsf{fma}\left(re, re \cdot \left(re \cdot \left(re \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot -0.0006944444444444445\right)\right)\right), 0.5\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im\right), 2\right) \]

   11. Simplified 94.3%

       \[\leadsto \mathsf{fma}\left(re, re \cdot \color{blue}{\left(re \cdot \left(re \cdot \left(\left(re \cdot re\right) \cdot -0.0006944444444444445\right)\right)\right)}, 0.5\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im\right), 2\right) \]

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.999999999999999889

   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. cos-lowering-cos.f64 97.5

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

   5. Simplified 97.5%

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

   if 0.999999999999999889 < (*.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. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. cosh-undef N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-lowering-*.f64 N/A

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

      10. cosh-lowering-cosh.f64 N/A

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

      11. cos-lowering-cos.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Simplified 99.1%

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

      1. *-lft-identity N/A

         \[\leadsto \color{blue}{\cosh im} \cdot 1 \]

      2. *-rgt-identity N/A

         \[\leadsto \color{blue}{\cosh im} \]

      3. cosh-lowering-cosh.f64 99.1

         \[\leadsto \color{blue}{\cosh im} \]

   9. Applied egg-rr 99.1%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot \left(re \cdot \left(re \cdot \left(\left(re \cdot re\right) \cdot -0.0006944444444444445\right)\right)\right), 0.5\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im\right), 2\right)\\ \mathbf{elif}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.9999999999999999:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\cosh im\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 95.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right)\\ t_1 := \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im\right), 2\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot \left(re \cdot \left(re \cdot \left(\left(re \cdot re\right) \cdot -0.0006944444444444445\right)\right)\right), 0.5\right) \cdot t\_1\\ \mathbf{elif}\;t\_0 \leq 2000:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, 0.020833333333333332, -0.25\right), 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* (cos re) 0.5) (+ (exp (- im)) (exp im))))
        (t_1
         (fma
          im
          (fma
           (* im im)
           (* im (fma (* im im) 0.002777777777777778 0.08333333333333333))
           im)
          2.0)))
   (if (<= t_0 (- INFINITY))
     (*
      (fma re (* re (* re (* re (* (* re re) -0.0006944444444444445)))) 0.5)
      t_1)
     (if (<= t_0 2000.0)
       (cos re)
       (*
        t_1
        (fma re (* re (fma (* re re) 0.020833333333333332 -0.25)) 0.5))))))

C

double code(double re, double im) {
	double t_0 = (cos(re) * 0.5) * (exp(-im) + exp(im));
	double t_1 = fma(im, fma((im * im), (im * fma((im * im), 0.002777777777777778, 0.08333333333333333)), im), 2.0);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(re, (re * (re * (re * ((re * re) * -0.0006944444444444445)))), 0.5) * t_1;
	} else if (t_0 <= 2000.0) {
		tmp = cos(re);
	} else {
		tmp = t_1 * fma(re, (re * fma((re * re), 0.020833333333333332, -0.25)), 0.5);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(cos(re) * 0.5) * Float64(exp(Float64(-im)) + exp(im)))
	t_1 = fma(im, fma(Float64(im * im), Float64(im * fma(Float64(im * im), 0.002777777777777778, 0.08333333333333333)), im), 2.0)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(re, Float64(re * Float64(re * Float64(re * Float64(Float64(re * re) * -0.0006944444444444445)))), 0.5) * t_1);
	elseif (t_0 <= 2000.0)
		tmp = cos(re);
	else
		tmp = Float64(t_1 * fma(re, Float64(re * fma(Float64(re * re), 0.020833333333333332, -0.25)), 0.5));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(im * N[(N[(im * im), $MachinePrecision] * N[(im * N[(N[(im * im), $MachinePrecision] * 0.002777777777777778 + 0.08333333333333333), $MachinePrecision]), $MachinePrecision] + im), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(re * N[(re * N[(re * N[(re * N[(N[(re * re), $MachinePrecision] * -0.0006944444444444445), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 2000.0], N[Cos[re], $MachinePrecision], N[(t$95$1 * N[(re * N[(re * N[(N[(re * re), $MachinePrecision] * 0.020833333333333332 + -0.25), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]
   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. accelerator-lowering-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. Simplified 62.5%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) + \frac{1}{2}\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(re \cdot re\right)} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) + \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(\color{blue}{re \cdot \left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right)} + \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, re \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right), \frac{1}{2}\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{re \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(re, re \cdot \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(re, re \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) + \color{blue}{\frac{-1}{4}}\right), \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, re \cdot \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}, \frac{-1}{4}\right)}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}, \frac{-1}{4}\right), \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}, \frac{-1}{4}\right), \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\frac{-1}{1440} \cdot {re}^{2} + \frac{1}{48}}, \frac{-1}{4}\right), \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{{re}^{2} \cdot \frac{-1}{1440}} + \frac{1}{48}, \frac{-1}{4}\right), \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{1440}, \frac{1}{48}\right)}, \frac{-1}{4}\right), \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{1440}, \frac{1}{48}\right), \frac{-1}{4}\right), \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      15. *-lowering-*.f64 94.3

          \[\leadsto \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\color{blue}{re \cdot re}, -0.0006944444444444445, 0.020833333333333332\right), -0.25\right), 0.5\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im\right), 2\right) \]

   8. Simplified 94.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, -0.0006944444444444445, 0.020833333333333332\right), -0.25\right), 0.5\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im\right), 2\right) \]

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{fma}\left(re, re \cdot \color{blue}{\left(\frac{-1}{1440} \cdot {re}^{4}\right)}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(re, re \cdot \left(\frac{-1}{1440} \cdot {re}^{\color{blue}{\left(2 \cdot 2\right)}}\right), \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

       2. pow-sqr N/A

          \[\leadsto \mathsf{fma}\left(re, re \cdot \left(\frac{-1}{1440} \cdot \color{blue}{\left({re}^{2} \cdot {re}^{2}\right)}\right), \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(re, re \cdot \color{blue}{\left(\left(\frac{-1}{1440} \cdot {re}^{2}\right) \cdot {re}^{2}\right)}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(re, re \cdot \color{blue}{\left({re}^{2} \cdot \left(\frac{-1}{1440} \cdot {re}^{2}\right)\right)}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(re, re \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \left(\frac{-1}{1440} \cdot {re}^{2}\right)\right), \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(re, re \cdot \color{blue}{\left(re \cdot \left(re \cdot \left(\frac{-1}{1440} \cdot {re}^{2}\right)\right)\right)}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(re, re \cdot \color{blue}{\left(re \cdot \left(re \cdot \left(\frac{-1}{1440} \cdot {re}^{2}\right)\right)\right)}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(re, re \cdot \left(re \cdot \color{blue}{\left(re \cdot \left(\frac{-1}{1440} \cdot {re}^{2}\right)\right)}\right), \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(re, re \cdot \left(re \cdot \left(re \cdot \color{blue}{\left({re}^{2} \cdot \frac{-1}{1440}\right)}\right)\right), \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(re, re \cdot \left(re \cdot \left(re \cdot \color{blue}{\left({re}^{2} \cdot \frac{-1}{1440}\right)}\right)\right), \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(re, re \cdot \left(re \cdot \left(re \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{1440}\right)\right)\right), \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

       12. *-lowering-*.f64 94.3

           \[\leadsto \mathsf{fma}\left(re, re \cdot \left(re \cdot \left(re \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot -0.0006944444444444445\right)\right)\right), 0.5\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im\right), 2\right) \]

   11. Simplified 94.3%

       \[\leadsto \mathsf{fma}\left(re, re \cdot \color{blue}{\left(re \cdot \left(re \cdot \left(\left(re \cdot re\right) \cdot -0.0006944444444444445\right)\right)\right)}, 0.5\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im\right), 2\right) \]

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 2e3

   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. cos-lowering-cos.f64 98.1

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

   5. Simplified 98.1%

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

   if 2e3 < (*.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. accelerator-lowering-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. Simplified 90.6%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right)\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right) + \frac{1}{2}\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(re \cdot re\right)} \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right) + \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(\color{blue}{re \cdot \left(re \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right)\right)} + \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, re \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right), \frac{1}{2}\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{re \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right)}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(re, re \cdot \color{blue}{\left(\frac{1}{48} \cdot {re}^{2} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, re \cdot \left(\color{blue}{{re}^{2} \cdot \frac{1}{48}} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right), \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(re, re \cdot \left({re}^{2} \cdot \frac{1}{48} + \color{blue}{\frac{-1}{4}}\right), \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, re \cdot \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{48}, \frac{-1}{4}\right)}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{48}, \frac{-1}{4}\right), \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      11. *-lowering-*.f64 91.6

          \[\leadsto \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, 0.020833333333333332, -0.25\right), 0.5\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im\right), 2\right) \]

   8. Simplified 91.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, 0.020833333333333332, -0.25\right), 0.5\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im\right), 2\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot \left(re \cdot \left(re \cdot \left(\left(re \cdot re\right) \cdot -0.0006944444444444445\right)\right)\right), 0.5\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im\right), 2\right)\\ \mathbf{elif}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 2000:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im\right), 2\right) \cdot \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, 0.020833333333333332, -0.25\right), 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.2:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot \left(re \cdot \left(re \cdot \left(\left(re \cdot re\right) \cdot -0.0006944444444444445\right)\right)\right), 0.5\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im\right), 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot 0.001388888888888889, im, 0.041666666666666664\right), 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (cos re) 0.5) (+ (exp (- im)) (exp im))) -0.2)
   (*
    (fma re (* re (* re (* re (* (* re re) -0.0006944444444444445)))) 0.5)
    (fma
     im
     (fma
      (* im im)
      (* im (fma (* im im) 0.002777777777777778 0.08333333333333333))
      im)
     2.0))
   (fma
    (* im im)
    (fma
     im
     (* im (fma (* im 0.001388888888888889) im 0.041666666666666664))
     0.5)
    1.0)))

C

double code(double re, double im) {
	double tmp;
	if (((cos(re) * 0.5) * (exp(-im) + exp(im))) <= -0.2) {
		tmp = fma(re, (re * (re * (re * ((re * re) * -0.0006944444444444445)))), 0.5) * fma(im, fma((im * im), (im * fma((im * im), 0.002777777777777778, 0.08333333333333333)), im), 2.0);
	} else {
		tmp = fma((im * im), fma(im, (im * fma((im * 0.001388888888888889), im, 0.041666666666666664)), 0.5), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(cos(re) * 0.5) * Float64(exp(Float64(-im)) + exp(im))) <= -0.2)
		tmp = Float64(fma(re, Float64(re * Float64(re * Float64(re * Float64(Float64(re * re) * -0.0006944444444444445)))), 0.5) * fma(im, fma(Float64(im * im), Float64(im * fma(Float64(im * im), 0.002777777777777778, 0.08333333333333333)), im), 2.0));
	else
		tmp = fma(Float64(im * im), fma(im, Float64(im * fma(Float64(im * 0.001388888888888889), im, 0.041666666666666664)), 0.5), 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.2], N[(N[(re * N[(re * N[(re * N[(re * N[(N[(re * re), $MachinePrecision] * -0.0006944444444444445), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * N[(N[(im * im), $MachinePrecision] * N[(im * N[(N[(im * im), $MachinePrecision] * 0.002777777777777778 + 0.08333333333333333), $MachinePrecision]), $MachinePrecision] + im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * N[(N[(im * 0.001388888888888889), $MachinePrecision] * im + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $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))) < -0.20000000000000001

   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. accelerator-lowering-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. Simplified 78.4%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) + \frac{1}{2}\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(re \cdot re\right)} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) + \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(\color{blue}{re \cdot \left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right)} + \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, re \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right), \frac{1}{2}\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{re \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(re, re \cdot \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(re, re \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) + \color{blue}{\frac{-1}{4}}\right), \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, re \cdot \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}, \frac{-1}{4}\right)}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}, \frac{-1}{4}\right), \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}, \frac{-1}{4}\right), \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\frac{-1}{1440} \cdot {re}^{2} + \frac{1}{48}}, \frac{-1}{4}\right), \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{{re}^{2} \cdot \frac{-1}{1440}} + \frac{1}{48}, \frac{-1}{4}\right), \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{1440}, \frac{1}{48}\right)}, \frac{-1}{4}\right), \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{1440}, \frac{1}{48}\right), \frac{-1}{4}\right), \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      15. *-lowering-*.f64 55.1

          \[\leadsto \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\color{blue}{re \cdot re}, -0.0006944444444444445, 0.020833333333333332\right), -0.25\right), 0.5\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im\right), 2\right) \]

   8. Simplified 55.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, -0.0006944444444444445, 0.020833333333333332\right), -0.25\right), 0.5\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im\right), 2\right) \]

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{fma}\left(re, re \cdot \color{blue}{\left(\frac{-1}{1440} \cdot {re}^{4}\right)}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(re, re \cdot \left(\frac{-1}{1440} \cdot {re}^{\color{blue}{\left(2 \cdot 2\right)}}\right), \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

       2. pow-sqr N/A

          \[\leadsto \mathsf{fma}\left(re, re \cdot \left(\frac{-1}{1440} \cdot \color{blue}{\left({re}^{2} \cdot {re}^{2}\right)}\right), \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(re, re \cdot \color{blue}{\left(\left(\frac{-1}{1440} \cdot {re}^{2}\right) \cdot {re}^{2}\right)}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(re, re \cdot \color{blue}{\left({re}^{2} \cdot \left(\frac{-1}{1440} \cdot {re}^{2}\right)\right)}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(re, re \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \left(\frac{-1}{1440} \cdot {re}^{2}\right)\right), \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(re, re \cdot \color{blue}{\left(re \cdot \left(re \cdot \left(\frac{-1}{1440} \cdot {re}^{2}\right)\right)\right)}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(re, re \cdot \color{blue}{\left(re \cdot \left(re \cdot \left(\frac{-1}{1440} \cdot {re}^{2}\right)\right)\right)}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(re, re \cdot \left(re \cdot \color{blue}{\left(re \cdot \left(\frac{-1}{1440} \cdot {re}^{2}\right)\right)}\right), \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(re, re \cdot \left(re \cdot \left(re \cdot \color{blue}{\left({re}^{2} \cdot \frac{-1}{1440}\right)}\right)\right), \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(re, re \cdot \left(re \cdot \left(re \cdot \color{blue}{\left({re}^{2} \cdot \frac{-1}{1440}\right)}\right)\right), \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(re, re \cdot \left(re \cdot \left(re \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{1440}\right)\right)\right), \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

       12. *-lowering-*.f64 55.1

           \[\leadsto \mathsf{fma}\left(re, re \cdot \left(re \cdot \left(re \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot -0.0006944444444444445\right)\right)\right), 0.5\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im\right), 2\right) \]

   11. Simplified 55.1%

       \[\leadsto \mathsf{fma}\left(re, re \cdot \color{blue}{\left(re \cdot \left(re \cdot \left(\left(re \cdot re\right) \cdot -0.0006944444444444445\right)\right)\right)}, 0.5\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im\right), 2\right) \]

   if -0.20000000000000001 < (*.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. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. cosh-undef N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-lowering-*.f64 N/A

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

      10. cosh-lowering-cosh.f64 N/A

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

      11. cos-lowering-cos.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Simplified 87.0%

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

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 82.6

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

   10. Simplified 82.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)} \]
   11. Step-by-step derivation

       1. associate-*l* N/A

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

       2. *-commutative N/A

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

       3. accelerator-lowering-fma.f64 N/A

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

       4. *-lowering-*.f64 82.6

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

   12. Applied egg-rr 82.6%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.2:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot \left(re \cdot \left(re \cdot \left(\left(re \cdot re\right) \cdot -0.0006944444444444445\right)\right)\right), 0.5\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im\right), 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot 0.001388888888888889, im, 0.041666666666666664\right), 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 47.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (((cos(re) * 0.5d0) * (exp(-im) + exp(im))) <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.5d0 * (im * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (((Math.cos(re) * 0.5) * (Math.exp(-im) + Math.exp(im))) <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = 0.5 * (im * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if ((math.cos(re) * 0.5) * (math.exp(-im) + math.exp(im))) <= 2.0:
		tmp = 1.0
	else:
		tmp = 0.5 * (im * im)
	return tmp

Julia

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

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (((cos(re) * 0.5) * (exp(-im) + exp(im))) <= 2.0)
		tmp = 1.0;
	else
		tmp = 0.5 * (im * im);
	end
	tmp_2 = 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], 2.0], 1.0, N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

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

      1. cos-lowering-cos.f64 78.4

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

   5. Simplified 78.4%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 47.4%

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

   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. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. cosh-undef N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-lowering-*.f64 N/A

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

      10. cosh-lowering-cosh.f64 N/A

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

      11. cos-lowering-cos.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Simplified 98.5%

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

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot {im}^{2} + 1} \]

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 54.5

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

   10. Simplified 54.5%

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

   11. Taylor expanded in im around inf

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

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot {im}^{2}} \]

       2. unpow2 N/A

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

       3. *-lowering-*.f64 54.5

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

   13. Simplified 54.5%

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

4. Final simplification 50.2%

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

Alternative 10: 54.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot -0.5, 1\right)\\ \mathbf{elif}\;\cos re \leq 0.965:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) -0.05)
   (fma re (* re -0.5) 1.0)
   (if (<= (cos re) 0.965)
     (* 0.041666666666666664 (* (* re re) (* re re)))
     (fma 0.5 (* im im) 1.0))))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= -0.05) {
		tmp = fma(re, (re * -0.5), 1.0);
	} else if (cos(re) <= 0.965) {
		tmp = 0.041666666666666664 * ((re * re) * (re * re));
	} else {
		tmp = fma(0.5, (im * im), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= -0.05)
		tmp = fma(re, Float64(re * -0.5), 1.0);
	elseif (cos(re) <= 0.965)
		tmp = Float64(0.041666666666666664 * Float64(Float64(re * re) * Float64(re * re)));
	else
		tmp = fma(0.5, Float64(im * im), 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], -0.05], N[(re * N[(re * -0.5), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[N[Cos[re], $MachinePrecision], 0.965], N[(0.041666666666666664 * N[(N[(re * re), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 re) < -0.050000000000000003

   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. cos-lowering-cos.f64 43.7

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

   5. Simplified 43.7%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot {re}^{2} + 1} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{{re}^{2} \cdot \frac{-1}{2}} + 1 \]

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 21.3

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

   8. Simplified 21.3%

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

   if -0.050000000000000003 < (cos.f64 re) < 0.964999999999999969

   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. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-out N/A

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

      11. associate-+r+ N/A

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

      12. +-commutative N/A

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

   5. Simplified 94.7%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 57.3

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

   8. Simplified 57.3%

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

   9. Taylor expanded in re around inf

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

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{24} \cdot {re}^{4}\right) \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \left(\frac{1}{24} \cdot {re}^{4}\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \left(\frac{1}{24} \cdot {re}^{4}\right)} \]

       4. +-commutative N/A

          \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) + 1\right)} \cdot \left(\frac{1}{24} \cdot {re}^{4}\right) \]

       5. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {im}^{2}, 1\right)} \cdot \left(\frac{1}{24} \cdot {re}^{4}\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{2} + \frac{1}{24} \cdot {im}^{2}, 1\right) \cdot \left(\frac{1}{24} \cdot {re}^{4}\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{2} + \frac{1}{24} \cdot {im}^{2}, 1\right) \cdot \left(\frac{1}{24} \cdot {re}^{4}\right) \]

       8. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}}, 1\right) \cdot \left(\frac{1}{24} \cdot {re}^{4}\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right) \cdot \left(\frac{1}{24} \cdot {re}^{4}\right) \]

       10. accelerator-lowering-fma.f64 N/A

           \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{24}, \frac{1}{2}\right)}, 1\right) \cdot \left(\frac{1}{24} \cdot {re}^{4}\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot \left(\frac{1}{24} \cdot {re}^{4}\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot \left(\frac{1}{24} \cdot {re}^{4}\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot \color{blue}{\left(\frac{1}{24} \cdot {re}^{4}\right)} \]

       14. metadata-eval N/A

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

       15. pow-sqr N/A

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

       16. *-lowering-*.f64 N/A

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

       17. unpow2 N/A

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

       18. *-lowering-*.f64 N/A

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

       19. unpow2 N/A

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

       20. *-lowering-*.f64 57.3

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

   11. Simplified 57.3%

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

   12. Taylor expanded in im around 0

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

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{24} \cdot {re}^{4}} \]

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. *-lowering-*.f64 N/A

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 N/A

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

       7. unpow2 N/A

          \[\leadsto \frac{1}{24} \cdot \left(\left(re \cdot re\right) \cdot \color{blue}{\left(re \cdot re\right)}\right) \]

       8. *-lowering-*.f64 45.2

          \[\leadsto 0.041666666666666664 \cdot \left(\left(re \cdot re\right) \cdot \color{blue}{\left(re \cdot re\right)}\right) \]

   14. Simplified 45.2%

       \[\leadsto \color{blue}{0.041666666666666664 \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)} \]

   if 0.964999999999999969 < (cos.f64 re)

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. cosh-undef N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-lowering-*.f64 N/A

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

      10. cosh-lowering-cosh.f64 N/A

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

      11. cos-lowering-cos.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Simplified 95.1%

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

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot {im}^{2} + 1} \]

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 74.7

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

   10. Simplified 74.7%

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

Alternative 11: 70.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), 1\right) \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot 0.001388888888888889, im, 0.041666666666666664\right), 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) -0.05)
   (*
    (fma (* im im) (fma (* im im) 0.041666666666666664 0.5) 1.0)
    (fma
     (* re re)
     (fma
      (* re re)
      (fma (* re re) -0.001388888888888889 0.041666666666666664)
      -0.5)
     1.0))
   (fma
    (* im im)
    (fma
     im
     (* im (fma (* im 0.001388888888888889) im 0.041666666666666664))
     0.5)
    1.0)))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= -0.05) {
		tmp = fma((im * im), fma((im * im), 0.041666666666666664, 0.5), 1.0) * fma((re * re), fma((re * re), fma((re * re), -0.001388888888888889, 0.041666666666666664), -0.5), 1.0);
	} else {
		tmp = fma((im * im), fma(im, (im * fma((im * 0.001388888888888889), im, 0.041666666666666664)), 0.5), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= -0.05)
		tmp = Float64(fma(Float64(im * im), fma(Float64(im * im), 0.041666666666666664, 0.5), 1.0) * fma(Float64(re * re), fma(Float64(re * re), fma(Float64(re * re), -0.001388888888888889, 0.041666666666666664), -0.5), 1.0));
	else
		tmp = fma(Float64(im * im), fma(im, Float64(im * fma(Float64(im * 0.001388888888888889), im, 0.041666666666666664)), 0.5), 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], -0.05], N[(N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * N[(N[(im * 0.001388888888888889), $MachinePrecision] * im + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -0.050000000000000003

   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. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-out N/A

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

      11. associate-+r+ N/A

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

      12. +-commutative N/A

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

   5. Simplified 74.6%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 53.6

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

   8. Simplified 53.6%

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

   if -0.050000000000000003 < (cos.f64 re)

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. cosh-undef N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-lowering-*.f64 N/A

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

      10. cosh-lowering-cosh.f64 N/A

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

      11. cos-lowering-cos.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Simplified 87.0%

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

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 82.6

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

   10. Simplified 82.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)} \]
   11. Step-by-step derivation

       1. associate-*l* N/A

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

       2. *-commutative N/A

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

       3. accelerator-lowering-fma.f64 N/A

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

       4. *-lowering-*.f64 82.6

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

   12. Applied egg-rr 82.6%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), 1\right) \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot 0.001388888888888889, im, 0.041666666666666664\right), 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 70.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, -0.0006944444444444445, 0.020833333333333332\right), -0.25\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot 0.001388888888888889, im, 0.041666666666666664\right), 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) -0.05)
   (*
    (fma im im 2.0)
    (fma
     (* re re)
     (fma
      re
      (* re (fma (* re re) -0.0006944444444444445 0.020833333333333332))
      -0.25)
     0.5))
   (fma
    (* im im)
    (fma
     im
     (* im (fma (* im 0.001388888888888889) im 0.041666666666666664))
     0.5)
    1.0)))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= -0.05) {
		tmp = fma(im, im, 2.0) * fma((re * re), fma(re, (re * fma((re * re), -0.0006944444444444445, 0.020833333333333332)), -0.25), 0.5);
	} else {
		tmp = fma((im * im), fma(im, (im * fma((im * 0.001388888888888889), im, 0.041666666666666664)), 0.5), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= -0.05)
		tmp = Float64(fma(im, im, 2.0) * fma(Float64(re * re), fma(re, Float64(re * fma(Float64(re * re), -0.0006944444444444445, 0.020833333333333332)), -0.25), 0.5));
	else
		tmp = fma(Float64(im * im), fma(im, Float64(im * fma(Float64(im * 0.001388888888888889), im, 0.041666666666666664)), 0.5), 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], -0.05], N[(N[(im * im + 2.0), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * N[(re * N[(re * N[(N[(re * re), $MachinePrecision] * -0.0006944444444444445 + 0.020833333333333332), $MachinePrecision]), $MachinePrecision] + -0.25), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * N[(N[(im * 0.001388888888888889), $MachinePrecision] * im + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -0.050000000000000003

   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. accelerator-lowering-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. Simplified 78.4%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) + \frac{1}{2}\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(re \cdot re\right)} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) + \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(\color{blue}{re \cdot \left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right)} + \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, re \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right), \frac{1}{2}\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{re \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(re, re \cdot \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(re, re \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) + \color{blue}{\frac{-1}{4}}\right), \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, re \cdot \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}, \frac{-1}{4}\right)}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}, \frac{-1}{4}\right), \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}, \frac{-1}{4}\right), \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\frac{-1}{1440} \cdot {re}^{2} + \frac{1}{48}}, \frac{-1}{4}\right), \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{{re}^{2} \cdot \frac{-1}{1440}} + \frac{1}{48}, \frac{-1}{4}\right), \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{1440}, \frac{1}{48}\right)}, \frac{-1}{4}\right), \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{1440}, \frac{1}{48}\right), \frac{-1}{4}\right), \frac{1}{2}\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]

      15. *-lowering-*.f64 55.1

          \[\leadsto \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\color{blue}{re \cdot re}, -0.0006944444444444445, 0.020833333333333332\right), -0.25\right), 0.5\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im\right), 2\right) \]

   8. Simplified 55.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, -0.0006944444444444445, 0.020833333333333332\right), -0.25\right), 0.5\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im\right), 2\right) \]

   9. Taylor expanded in im around 0

      \[\leadsto \color{blue}{2 \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right)} \]
   10. Step-by-step derivation

       1. distribute-rgt-out N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right) \cdot \left(2 + {im}^{2}\right)} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right) \cdot \left(2 + {im}^{2}\right)} \]

   11. Simplified 51.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, -0.0006944444444444445, 0.020833333333333332\right), -0.25\right), 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)} \]

   if -0.050000000000000003 < (cos.f64 re)

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. cosh-undef N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-lowering-*.f64 N/A

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

      10. cosh-lowering-cosh.f64 N/A

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

      11. cos-lowering-cos.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Simplified 87.0%

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

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 82.6

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

   10. Simplified 82.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)} \]
   11. Step-by-step derivation

       1. associate-*l* N/A

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

       2. *-commutative N/A

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

       3. accelerator-lowering-fma.f64 N/A

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

       4. *-lowering-*.f64 82.6

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

   12. Applied egg-rr 82.6%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, -0.0006944444444444445, 0.020833333333333332\right), -0.25\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot 0.001388888888888889, im, 0.041666666666666664\right), 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 69.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, \left(re \cdot re\right) \cdot \left(\left(re \cdot re\right) \cdot -0.001388888888888889\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot 0.001388888888888889, im, 0.041666666666666664\right), 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) -0.05)
   (fma (* re re) (* (* re re) (* (* re re) -0.001388888888888889)) 1.0)
   (fma
    (* im im)
    (fma
     im
     (* im (fma (* im 0.001388888888888889) im 0.041666666666666664))
     0.5)
    1.0)))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= -0.05) {
		tmp = fma((re * re), ((re * re) * ((re * re) * -0.001388888888888889)), 1.0);
	} else {
		tmp = fma((im * im), fma(im, (im * fma((im * 0.001388888888888889), im, 0.041666666666666664)), 0.5), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= -0.05)
		tmp = fma(Float64(re * re), Float64(Float64(re * re) * Float64(Float64(re * re) * -0.001388888888888889)), 1.0);
	else
		tmp = fma(Float64(im * im), fma(im, Float64(im * fma(Float64(im * 0.001388888888888889), im, 0.041666666666666664)), 0.5), 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], -0.05], N[(N[(re * re), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.001388888888888889), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * N[(N[(im * 0.001388888888888889), $MachinePrecision] * im + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -0.050000000000000003

   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. cos-lowering-cos.f64 43.7

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

   5. Simplified 43.7%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 46.8

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

   8. Simplified 46.8%

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

   9. Taylor expanded in re around inf

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

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. *-lowering-*.f64 N/A

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

       6. unpow2 N/A

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

       7. *-lowering-*.f64 N/A

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

       8. *-commutative N/A

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

       9. *-lowering-*.f64 N/A

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

       10. unpow2 N/A

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

       11. *-lowering-*.f64 46.8

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

   11. Simplified 46.8%

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

   if -0.050000000000000003 < (cos.f64 re)

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. cosh-undef N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-lowering-*.f64 N/A

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

      10. cosh-lowering-cosh.f64 N/A

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

      11. cos-lowering-cos.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Simplified 87.0%

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

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 82.6

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

   10. Simplified 82.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)} \]
   11. Step-by-step derivation

       1. associate-*l* N/A

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

       2. *-commutative N/A

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

       3. accelerator-lowering-fma.f64 N/A

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

       4. *-lowering-*.f64 82.6

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

   12. Applied egg-rr 82.6%

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

Alternative 14: 69.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, \left(re \cdot re\right) \cdot \left(\left(re \cdot re\right) \cdot -0.001388888888888889\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right)\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) -0.05)
   (fma (* re re) (* (* re re) (* (* re re) -0.001388888888888889)) 1.0)
   (fma
    (* im im)
    (* im (* im (fma (* im im) 0.001388888888888889 0.041666666666666664)))
    1.0)))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= -0.05) {
		tmp = fma((re * re), ((re * re) * ((re * re) * -0.001388888888888889)), 1.0);
	} else {
		tmp = fma((im * im), (im * (im * fma((im * im), 0.001388888888888889, 0.041666666666666664))), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= -0.05)
		tmp = fma(Float64(re * re), Float64(Float64(re * re) * Float64(Float64(re * re) * -0.001388888888888889)), 1.0);
	else
		tmp = fma(Float64(im * im), Float64(im * Float64(im * fma(Float64(im * im), 0.001388888888888889, 0.041666666666666664))), 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], -0.05], N[(N[(re * re), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.001388888888888889), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -0.050000000000000003

   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. cos-lowering-cos.f64 43.7

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

   5. Simplified 43.7%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 46.8

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

   8. Simplified 46.8%

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

   9. Taylor expanded in re around inf

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

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. *-lowering-*.f64 N/A

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

       6. unpow2 N/A

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

       7. *-lowering-*.f64 N/A

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

       8. *-commutative N/A

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

       9. *-lowering-*.f64 N/A

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

       10. unpow2 N/A

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

       11. *-lowering-*.f64 46.8

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

   11. Simplified 46.8%

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

   if -0.050000000000000003 < (cos.f64 re)

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. cosh-undef N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-lowering-*.f64 N/A

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

      10. cosh-lowering-cosh.f64 N/A

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

      11. cos-lowering-cos.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Simplified 87.0%

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

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 82.6

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

   10. Simplified 82.6%

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

   11. Taylor expanded in im around inf

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

       1. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(im \cdot im, {im}^{4} \cdot \color{blue}{\left(\frac{1}{24} \cdot \frac{1}{{im}^{2}} + \frac{1}{720}\right)}, 1\right) \]

       2. distribute-lft-in N/A

          \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{4} \cdot \left(\frac{1}{24} \cdot \frac{1}{{im}^{2}}\right) + {im}^{4} \cdot \frac{1}{720}}, 1\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{\left(\frac{1}{24} \cdot \frac{1}{{im}^{2}}\right) \cdot {im}^{4}} + {im}^{4} \cdot \frac{1}{720}, 1\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{1}{24} \cdot \left(\frac{1}{{im}^{2}} \cdot {im}^{4}\right)} + {im}^{4} \cdot \frac{1}{720}, 1\right) \]

       5. associate-*l/ N/A

          \[\leadsto \mathsf{fma}\left(im \cdot im, \frac{1}{24} \cdot \color{blue}{\frac{1 \cdot {im}^{4}}{{im}^{2}}} + {im}^{4} \cdot \frac{1}{720}, 1\right) \]

       6. *-lft-identity N/A

          \[\leadsto \mathsf{fma}\left(im \cdot im, \frac{1}{24} \cdot \frac{\color{blue}{{im}^{4}}}{{im}^{2}} + {im}^{4} \cdot \frac{1}{720}, 1\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(im \cdot im, \frac{1}{24} \cdot \frac{{im}^{\color{blue}{\left(2 \cdot 2\right)}}}{{im}^{2}} + {im}^{4} \cdot \frac{1}{720}, 1\right) \]

       8. pow-sqr N/A

          \[\leadsto \mathsf{fma}\left(im \cdot im, \frac{1}{24} \cdot \frac{\color{blue}{{im}^{2} \cdot {im}^{2}}}{{im}^{2}} + {im}^{4} \cdot \frac{1}{720}, 1\right) \]

       9. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(im \cdot im, \frac{1}{24} \cdot \color{blue}{\left({im}^{2} \cdot \frac{{im}^{2}}{{im}^{2}}\right)} + {im}^{4} \cdot \frac{1}{720}, 1\right) \]

       10. *-rgt-identity N/A

           \[\leadsto \mathsf{fma}\left(im \cdot im, \frac{1}{24} \cdot \left({im}^{2} \cdot \frac{\color{blue}{{im}^{2} \cdot 1}}{{im}^{2}}\right) + {im}^{4} \cdot \frac{1}{720}, 1\right) \]

       11. associate-*r/ N/A

           \[\leadsto \mathsf{fma}\left(im \cdot im, \frac{1}{24} \cdot \left({im}^{2} \cdot \color{blue}{\left({im}^{2} \cdot \frac{1}{{im}^{2}}\right)}\right) + {im}^{4} \cdot \frac{1}{720}, 1\right) \]

       12. rgt-mult-inverse N/A

           \[\leadsto \mathsf{fma}\left(im \cdot im, \frac{1}{24} \cdot \left({im}^{2} \cdot \color{blue}{1}\right) + {im}^{4} \cdot \frac{1}{720}, 1\right) \]

       13. *-rgt-identity N/A

           \[\leadsto \mathsf{fma}\left(im \cdot im, \frac{1}{24} \cdot \color{blue}{{im}^{2}} + {im}^{4} \cdot \frac{1}{720}, 1\right) \]

       14. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(im \cdot im, \frac{1}{24} \cdot {im}^{2} + \color{blue}{\frac{1}{720} \cdot {im}^{4}}, 1\right) \]

       15. metadata-eval N/A

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

       16. pow-sqr N/A

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

       17. associate-*l* N/A

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

       18. distribute-rgt-in N/A

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

       19. unpow2 N/A

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

   13. Simplified 82.6%

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

Alternative 15: 69.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, \left(re \cdot re\right) \cdot \left(\left(re \cdot re\right) \cdot -0.001388888888888889\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, im \cdot \left(0.001388888888888889 \cdot \left(im \cdot \left(im \cdot im\right)\right)\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) -0.05)
   (fma (* re re) (* (* re re) (* (* re re) -0.001388888888888889)) 1.0)
   (fma (* im im) (* im (* 0.001388888888888889 (* im (* im im)))) 1.0)))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= -0.05) {
		tmp = fma((re * re), ((re * re) * ((re * re) * -0.001388888888888889)), 1.0);
	} else {
		tmp = fma((im * im), (im * (0.001388888888888889 * (im * (im * im)))), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= -0.05)
		tmp = fma(Float64(re * re), Float64(Float64(re * re) * Float64(Float64(re * re) * -0.001388888888888889)), 1.0);
	else
		tmp = fma(Float64(im * im), Float64(im * Float64(0.001388888888888889 * Float64(im * Float64(im * im)))), 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], -0.05], N[(N[(re * re), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.001388888888888889), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(im * im), $MachinePrecision] * N[(im * N[(0.001388888888888889 * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -0.050000000000000003

   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. cos-lowering-cos.f64 43.7

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

   5. Simplified 43.7%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 46.8

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

   8. Simplified 46.8%

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

   9. Taylor expanded in re around inf

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

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. *-lowering-*.f64 N/A

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

       6. unpow2 N/A

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

       7. *-lowering-*.f64 N/A

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

       8. *-commutative N/A

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

       9. *-lowering-*.f64 N/A

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

       10. unpow2 N/A

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

       11. *-lowering-*.f64 46.8

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

   11. Simplified 46.8%

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

   if -0.050000000000000003 < (cos.f64 re)

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. cosh-undef N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-lowering-*.f64 N/A

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

      10. cosh-lowering-cosh.f64 N/A

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

      11. cos-lowering-cos.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Simplified 87.0%

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

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 82.6

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

   10. Simplified 82.6%

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

   11. Taylor expanded in im around inf

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

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. unpow2 N/A

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

       10. unpow3 N/A

           \[\leadsto \mathsf{fma}\left(im \cdot im, im \cdot \left(\frac{1}{720} \cdot \color{blue}{{im}^{3}}\right), 1\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{im \cdot \left(\frac{1}{720} \cdot {im}^{3}\right)}, 1\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(im \cdot im, im \cdot \color{blue}{\left(\frac{1}{720} \cdot {im}^{3}\right)}, 1\right) \]

       13. cube-mult N/A

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

       14. unpow2 N/A

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

       15. *-lowering-*.f64 N/A

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

       16. unpow2 N/A

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

       17. *-lowering-*.f64 82.6

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

   13. Simplified 82.6%

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

Alternative 16: 65.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, im \cdot \left(0.001388888888888889 \cdot \left(im \cdot \left(im \cdot im\right)\right)\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) -0.05)
   (fma re (* re -0.5) 1.0)
   (fma (* im im) (* im (* 0.001388888888888889 (* im (* im im)))) 1.0)))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= -0.05) {
		tmp = fma(re, (re * -0.5), 1.0);
	} else {
		tmp = fma((im * im), (im * (0.001388888888888889 * (im * (im * im)))), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= -0.05)
		tmp = fma(re, Float64(re * -0.5), 1.0);
	else
		tmp = fma(Float64(im * im), Float64(im * Float64(0.001388888888888889 * Float64(im * Float64(im * im)))), 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], -0.05], N[(re * N[(re * -0.5), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(im * im), $MachinePrecision] * N[(im * N[(0.001388888888888889 * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -0.050000000000000003

   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. cos-lowering-cos.f64 43.7

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

   5. Simplified 43.7%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot {re}^{2} + 1} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{{re}^{2} \cdot \frac{-1}{2}} + 1 \]

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 21.3

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

   8. Simplified 21.3%

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

   if -0.050000000000000003 < (cos.f64 re)

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. cosh-undef N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-lowering-*.f64 N/A

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

      10. cosh-lowering-cosh.f64 N/A

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

      11. cos-lowering-cos.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Simplified 87.0%

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

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 82.6

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

   10. Simplified 82.6%

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

   11. Taylor expanded in im around inf

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

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. unpow2 N/A

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

       10. unpow3 N/A

           \[\leadsto \mathsf{fma}\left(im \cdot im, im \cdot \left(\frac{1}{720} \cdot \color{blue}{{im}^{3}}\right), 1\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{im \cdot \left(\frac{1}{720} \cdot {im}^{3}\right)}, 1\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(im \cdot im, im \cdot \color{blue}{\left(\frac{1}{720} \cdot {im}^{3}\right)}, 1\right) \]

       13. cube-mult N/A

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

       14. unpow2 N/A

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

       15. *-lowering-*.f64 N/A

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

       16. unpow2 N/A

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

       17. *-lowering-*.f64 82.6

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

   13. Simplified 82.6%

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

Alternative 17: 62.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) -0.05)
   (fma re (* re -0.5) 1.0)
   (fma im (* im (fma im (* im 0.041666666666666664) 0.5)) 1.0)))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= -0.05) {
		tmp = fma(re, (re * -0.5), 1.0);
	} else {
		tmp = fma(im, (im * fma(im, (im * 0.041666666666666664), 0.5)), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= -0.05)
		tmp = fma(re, Float64(re * -0.5), 1.0);
	else
		tmp = fma(im, Float64(im * fma(im, Float64(im * 0.041666666666666664), 0.5)), 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], -0.05], N[(re * N[(re * -0.5), $MachinePrecision] + 1.0), $MachinePrecision], N[(im * N[(im * N[(im * N[(im * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -0.050000000000000003

   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. cos-lowering-cos.f64 43.7

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

   5. Simplified 43.7%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot {re}^{2} + 1} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{{re}^{2} \cdot \frac{-1}{2}} + 1 \]

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 21.3

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

   8. Simplified 21.3%

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

   if -0.050000000000000003 < (cos.f64 re)

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. cosh-undef N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-lowering-*.f64 N/A

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

      10. cosh-lowering-cosh.f64 N/A

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

      11. cos-lowering-cos.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Simplified 87.0%

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

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 82.6

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

   10. Simplified 82.6%

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

   11. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. +-commutative N/A

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

       5. distribute-lft-out N/A

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

       6. accelerator-lowering-fma.f64 N/A

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

       7. distribute-lft-out N/A

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

       8. +-commutative N/A

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

       9. *-lowering-*.f64 N/A

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

       10. +-commutative N/A

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

       11. *-commutative N/A

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

       12. unpow2 N/A

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

       13. associate-*l* N/A

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

       14. *-commutative N/A

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

       15. accelerator-lowering-fma.f64 N/A

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

       16. *-commutative N/A

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

       17. *-lowering-*.f64 78.9

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

   13. Simplified 78.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 53.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) -0.05) (fma re (* re -0.5) 1.0) (fma 0.5 (* im im) 1.0)))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= -0.05) {
		tmp = fma(re, (re * -0.5), 1.0);
	} else {
		tmp = fma(0.5, (im * im), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= -0.05)
		tmp = fma(re, Float64(re * -0.5), 1.0);
	else
		tmp = fma(0.5, Float64(im * im), 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], -0.05], N[(re * N[(re * -0.5), $MachinePrecision] + 1.0), $MachinePrecision], N[(0.5 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -0.050000000000000003

   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. cos-lowering-cos.f64 43.7

         \[\leadsto \color{blue}{\cos re} \]

   5. Simplified 43.7%

      \[\leadsto \color{blue}{\cos re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + \frac{-1}{2} \cdot {re}^{2}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot {re}^{2} + 1} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{{re}^{2} \cdot \frac{-1}{2}} + 1 \]

      3. unpow2 N/A

         \[\leadsto \color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{2} + 1 \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{re \cdot \left(re \cdot \frac{-1}{2}\right)} + 1 \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, re \cdot \frac{-1}{2}, 1\right)} \]

      6. *-lowering-*.f64 21.3

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{re \cdot -0.5}, 1\right) \]

   8. Simplified 21.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, re \cdot -0.5, 1\right)} \]

   if -0.050000000000000003 < (cos.f64 re)

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right) \cdot \left(\frac{1}{2} \cdot \cos re\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \cos re} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \cos re} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)} \cdot \cos re \]

      5. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(e^{im} + e^{\mathsf{neg}\left(im\right)}\right)}\right) \cdot \cos re \]

      6. cosh-undef N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(2 \cdot \cosh im\right)}\right) \cdot \cos re \]

      7. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} \cdot 2\right) \cdot \cosh im\right)} \cdot \cos re \]

      8. metadata-eval N/A

         \[\leadsto \left(\color{blue}{1} \cdot \cosh im\right) \cdot \cos re \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right)} \cdot \cos re \]

      10. cosh-lowering-cosh.f64 N/A

          \[\leadsto \left(1 \cdot \color{blue}{\cosh im}\right) \cdot \cos re \]

      11. cos-lowering-cos.f64 100.0

          \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{\cos re} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \cos re} \]

   5. Taylor expanded in re around 0

      \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{1} \]
   6. Step-by-step derivation

   7. Simplified 87.0%

      \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{1} \]

   8. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + \frac{1}{2} \cdot {im}^{2}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot {im}^{2} + 1} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {im}^{2}, 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{im \cdot im}, 1\right) \]

      4. *-lowering-*.f64 64.6

         \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{im \cdot im}, 1\right) \]

   10. Simplified 64.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, im \cdot im, 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 47.2% accurate, 26.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.5, im \cdot im, 1\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (fma 0.5 (* im im) 1.0))

C

double code(double re, double im) {
	return fma(0.5, (im * im), 1.0);
}

Julia

function code(re, im)
	return fma(0.5, Float64(im * im), 1.0)
end

Wolfram

code[re_, im_] := N[(0.5 * N[(im * im), $MachinePrecision] + 1.0), $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. *-commutative N/A

      \[\leadsto \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right) \cdot \left(\frac{1}{2} \cdot \cos re\right)} \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \cos re} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \cos re} \]

   4. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right)\right)} \cdot \cos re \]

   5. +-commutative N/A

      \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(e^{im} + e^{\mathsf{neg}\left(im\right)}\right)}\right) \cdot \cos re \]

   6. cosh-undef N/A

      \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(2 \cdot \cosh im\right)}\right) \cdot \cos re \]

   7. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(\frac{1}{2} \cdot 2\right) \cdot \cosh im\right)} \cdot \cos re \]

   8. metadata-eval N/A

      \[\leadsto \left(\color{blue}{1} \cdot \cosh im\right) \cdot \cos re \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right)} \cdot \cos re \]

   10. cosh-lowering-cosh.f64 N/A

       \[\leadsto \left(1 \cdot \color{blue}{\cosh im}\right) \cdot \cos re \]

   11. cos-lowering-cos.f64 100.0

       \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{\cos re} \]

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \cos re} \]

5. Taylor expanded in re around 0

   \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{1} \]
6. Step-by-step derivation

7. Simplified 67.5%

   \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{1} \]

8. Taylor expanded in im around 0

   \[\leadsto \color{blue}{1 + \frac{1}{2} \cdot {im}^{2}} \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot {im}^{2} + 1} \]

   2. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {im}^{2}, 1\right)} \]

   3. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{im \cdot im}, 1\right) \]

   4. *-lowering-*.f64 50.1

      \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{im \cdot im}, 1\right) \]

10. Simplified 50.1%

    \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, im \cdot im, 1\right)} \]
11. Add Preprocessing

Alternative 20: 28.6% 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. cos-lowering-cos.f64 48.8

      \[\leadsto \color{blue}{\cos re} \]

5. Simplified 48.8%

   \[\leadsto \color{blue}{\cos re} \]

6. Taylor expanded in re around 0

   \[\leadsto \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 30.0%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024199 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))