math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 11.6s

Alternatives: 21

Speedup: 1.5×

• 49.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. distribute-rgt-in N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. associate-*l* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\cosh im \cdot \mathsf{fma}\left(re, re \cdot -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq 0.9999999600086682:\\ \;\;\;\;\cos re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (* (cosh im) (fma re (* re -0.5) 1.0))
     (if (<= t_0 0.9999999600086682)
       (*
        (cos re)
        (fma
         (* im im)
         (fma
          im
          (* im (fma (* im im) 0.001388888888888889 0.041666666666666664))
          0.5)
         1.0))
       (* (cosh im) 1.0)))))

C

double code(double re, double im) {
	double t_0 = (0.5 * cos(re)) * (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 <= 0.9999999600086682) {
		tmp = cos(re) * fma((im * im), fma(im, (im * fma((im * im), 0.001388888888888889, 0.041666666666666664)), 0.5), 1.0);
	} else {
		tmp = cosh(im) * 1.0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * cos(re)) * 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 <= 0.9999999600086682)
		tmp = Float64(cos(re) * fma(Float64(im * im), fma(im, Float64(im * fma(Float64(im * im), 0.001388888888888889, 0.041666666666666664)), 0.5), 1.0));
	else
		tmp = Float64(cosh(im) * 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $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, 0.9999999600086682], N[(N[Cos[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Cosh[im], $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. cosh-undef N/A

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

      13. associate-*r* N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.9999999600086682

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. cosh-undef N/A

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

      13. associate-*r* N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   if 0.9999999600086682 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. cosh-undef N/A

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

      13. associate-*r* N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\cosh im \cdot \mathsf{fma}\left(re, re \cdot -0.5, 1\right)\\ \mathbf{elif}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.9999999600086682:\\ \;\;\;\;\cos re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\cosh im \cdot \mathsf{fma}\left(re, re \cdot -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq 0.9999999600086682:\\ \;\;\;\;\cos re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (* (cosh im) (fma re (* re -0.5) 1.0))
     (if (<= t_0 0.9999999600086682)
       (*
        (cos re)
        (fma (* im im) (fma im (* im 0.041666666666666664) 0.5) 1.0))
       (* (cosh im) 1.0)))))

C

double code(double re, double im) {
	double t_0 = (0.5 * cos(re)) * (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 <= 0.9999999600086682) {
		tmp = cos(re) * fma((im * im), fma(im, (im * 0.041666666666666664), 0.5), 1.0);
	} else {
		tmp = cosh(im) * 1.0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * cos(re)) * 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 <= 0.9999999600086682)
		tmp = Float64(cos(re) * fma(Float64(im * im), fma(im, Float64(im * 0.041666666666666664), 0.5), 1.0));
	else
		tmp = Float64(cosh(im) * 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $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, 0.9999999600086682], N[(N[Cos[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Cosh[im], $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. cosh-undef N/A

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

      13. associate-*r* N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.9999999600086682

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

   5. Applied rewrites 99.8%

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

   if 0.9999999600086682 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. cosh-undef N/A

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

      13. associate-*r* N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\cosh im \cdot \mathsf{fma}\left(re, re \cdot -0.5, 1\right)\\ \mathbf{elif}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.9999999600086682:\\ \;\;\;\;\cos re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ t_1 := t\_0 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\cosh im \cdot \mathsf{fma}\left(re, re \cdot -0.5, 1\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999999600086682:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re))) (t_1 (* t_0 (+ (exp (- im)) (exp im)))))
   (if (<= t_1 (- INFINITY))
     (* (cosh im) (fma re (* re -0.5) 1.0))
     (if (<= t_1 0.9999999600086682)
       (* t_0 (fma im im 2.0))
       (* (cosh im) 1.0)))))

C

double code(double re, double im) {
	double t_0 = 0.5 * cos(re);
	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 <= 0.9999999600086682) {
		tmp = t_0 * fma(im, im, 2.0);
	} else {
		tmp = cosh(im) * 1.0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * cos(re))
	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 <= 0.9999999600086682)
		tmp = Float64(t_0 * fma(im, im, 2.0));
	else
		tmp = Float64(cosh(im) * 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $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, 0.9999999600086682], N[(t$95$0 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[Cosh[im], $MachinePrecision] * 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. cosh-undef N/A

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

      13. associate-*r* N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.9999999600086682

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 99.4

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

   5. Applied rewrites 99.4%

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

   if 0.9999999600086682 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. cosh-undef N/A

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

      13. associate-*r* N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\cosh im \cdot \mathsf{fma}\left(re, re \cdot -0.5, 1\right)\\ \mathbf{elif}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.9999999600086682:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ t_1 := t\_0 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, -0.0006944444444444445, 0.020833333333333332\right), -0.25\right), 0.5\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im\right), 2\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999999600086682:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re))) (t_1 (* t_0 (+ (exp (- im)) (exp im)))))
   (if (<= t_1 (- INFINITY))
     (*
      (fma
       re
       (*
        re
        (fma
         (* re re)
         (fma (* re re) -0.0006944444444444445 0.020833333333333332)
         -0.25))
       0.5)
      (fma
       im
       (fma
        (* im im)
        (* im (fma (* im im) 0.002777777777777778 0.08333333333333333))
        im)
       2.0))
     (if (<= t_1 0.9999999600086682)
       (* t_0 (fma im im 2.0))
       (* (cosh im) 1.0)))))

C

double code(double re, double im) {
	double t_0 = 0.5 * cos(re);
	double t_1 = t_0 * (exp(-im) + exp(im));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(re, (re * fma((re * re), fma((re * re), -0.0006944444444444445, 0.020833333333333332), -0.25)), 0.5) * fma(im, fma((im * im), (im * fma((im * im), 0.002777777777777778, 0.08333333333333333)), im), 2.0);
	} else if (t_1 <= 0.9999999600086682) {
		tmp = t_0 * fma(im, im, 2.0);
	} else {
		tmp = cosh(im) * 1.0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * cos(re))
	t_1 = Float64(t_0 * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(re, Float64(re * fma(Float64(re * re), fma(Float64(re * re), -0.0006944444444444445, 0.020833333333333332), -0.25)), 0.5) * fma(im, fma(Float64(im * im), Float64(im * fma(Float64(im * im), 0.002777777777777778, 0.08333333333333333)), im), 2.0));
	elseif (t_1 <= 0.9999999600086682)
		tmp = Float64(t_0 * fma(im, im, 2.0));
	else
		tmp = Float64(cosh(im) * 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(re * N[(re * N[(N[(re * re), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.0006944444444444445 + 0.020833333333333332), $MachinePrecision] + -0.25), $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$1, 0.9999999600086682], N[(t$95$0 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[Cosh[im], $MachinePrecision] * 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]
   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. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 85.1%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]
   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. lower-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. lower-*.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. lower-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. lower-*.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. lower-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. lower-*.f64 97.0

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

   8. Applied rewrites 97.0%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.9999999600086682

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 99.4

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

   5. Applied rewrites 99.4%

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

   if 0.9999999600086682 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. cosh-undef N/A

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

      13. associate-*r* N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 100.0%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, -0.0006944444444444445, 0.020833333333333332\right), -0.25\right), 0.5\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im\right), 2\right)\\ \mathbf{elif}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.9999999600086682:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, -0.0006944444444444445, 0.020833333333333332\right), -0.25\right), 0.5\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im\right), 2\right)\\ \mathbf{elif}\;t\_0 \leq 0.9999999600086682:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma
       re
       (*
        re
        (fma
         (* re re)
         (fma (* re re) -0.0006944444444444445 0.020833333333333332)
         -0.25))
       0.5)
      (fma
       im
       (fma
        (* im im)
        (* im (fma (* im im) 0.002777777777777778 0.08333333333333333))
        im)
       2.0))
     (if (<= t_0 0.9999999600086682) (cos re) (* (cosh im) 1.0)))))

C

double code(double re, double im) {
	double t_0 = (0.5 * cos(re)) * (exp(-im) + exp(im));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(re, (re * fma((re * re), fma((re * re), -0.0006944444444444445, 0.020833333333333332), -0.25)), 0.5) * fma(im, fma((im * im), (im * fma((im * im), 0.002777777777777778, 0.08333333333333333)), im), 2.0);
	} else if (t_0 <= 0.9999999600086682) {
		tmp = cos(re);
	} else {
		tmp = cosh(im) * 1.0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(re, Float64(re * fma(Float64(re * re), fma(Float64(re * re), -0.0006944444444444445, 0.020833333333333332), -0.25)), 0.5) * fma(im, fma(Float64(im * im), Float64(im * fma(Float64(im * im), 0.002777777777777778, 0.08333333333333333)), im), 2.0));
	elseif (t_0 <= 0.9999999600086682)
		tmp = cos(re);
	else
		tmp = Float64(cosh(im) * 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(re * N[(re * N[(N[(re * re), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.0006944444444444445 + 0.020833333333333332), $MachinePrecision] + -0.25), $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, 0.9999999600086682], N[Cos[re], $MachinePrecision], N[(N[Cosh[im], $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]
   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. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 85.1%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]
   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. lower-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. lower-*.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. lower-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. lower-*.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. lower-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. lower-*.f64 97.0

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

   8. Applied rewrites 97.0%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.9999999600086682

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-cos.f64 98.3

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

   5. Applied rewrites 98.3%

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

   if 0.9999999600086682 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. cosh-undef N/A

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

      13. associate-*r* N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 100.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, -0.0006944444444444445, 0.020833333333333332\right), -0.25\right), 0.5\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im\right), 2\right)\\ \mathbf{elif}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.9999999600086682:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\\ t_1 := \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im\right), 2\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, -0.0006944444444444445, 0.020833333333333332\right), -0.25\right), 0.5\right) \cdot t\_1\\ \mathbf{elif}\;t\_0 \leq 0.9999999600086682:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (cos re)) (+ (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
        (fma
         (* re re)
         (fma (* re re) -0.0006944444444444445 0.020833333333333332)
         -0.25))
       0.5)
      t_1)
     (if (<= t_0 0.9999999600086682) (cos re) (* t_1 0.5)))))

C

double code(double re, double im) {
	double t_0 = (0.5 * cos(re)) * (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 * fma((re * re), fma((re * re), -0.0006944444444444445, 0.020833333333333332), -0.25)), 0.5) * t_1;
	} else if (t_0 <= 0.9999999600086682) {
		tmp = cos(re);
	} else {
		tmp = t_1 * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * cos(re)) * 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 * fma(Float64(re * re), fma(Float64(re * re), -0.0006944444444444445, 0.020833333333333332), -0.25)), 0.5) * t_1);
	elseif (t_0 <= 0.9999999600086682)
		tmp = cos(re);
	else
		tmp = Float64(t_1 * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $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[(N[(re * re), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.0006944444444444445 + 0.020833333333333332), $MachinePrecision] + -0.25), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 0.9999999600086682], N[Cos[re], $MachinePrecision], N[(t$95$1 * 0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. 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. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 85.1%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im\right), 2\right) \]
   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. lower-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. lower-*.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. lower-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. lower-*.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. lower-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. lower-*.f64 97.0

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

   8. Applied rewrites 97.0%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.9999999600086682

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-cos.f64 98.3

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

   5. Applied rewrites 98.3%

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

   if 0.9999999600086682 < (*.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. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 91.5%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 91.5%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, -0.0006944444444444445, 0.020833333333333332\right), -0.25\right), 0.5\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im\right), 2\right)\\ \mathbf{elif}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.9999999600086682:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im\right), 2\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 55.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot -0.5, 1\right)\\ \mathbf{elif}\;\cos re \leq 0.99999998:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, 0.041666666666666664, -0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) -0.02)
   (fma re (* re -0.5) 1.0)
   (if (<= (cos re) 0.99999998)
     (fma re (* re (fma (* re re) 0.041666666666666664 -0.5)) 1.0)
     (* (fma im im 2.0) 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= -0.02) {
		tmp = fma(re, (re * -0.5), 1.0);
	} else if (cos(re) <= 0.99999998) {
		tmp = fma(re, (re * fma((re * re), 0.041666666666666664, -0.5)), 1.0);
	} else {
		tmp = fma(im, im, 2.0) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= -0.02)
		tmp = fma(re, Float64(re * -0.5), 1.0);
	elseif (cos(re) <= 0.99999998)
		tmp = fma(re, Float64(re * fma(Float64(re * re), 0.041666666666666664, -0.5)), 1.0);
	else
		tmp = Float64(fma(im, im, 2.0) * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], -0.02], N[(re * N[(re * -0.5), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[N[Cos[re], $MachinePrecision], 0.99999998], N[(re * N[(re * N[(N[(re * re), $MachinePrecision] * 0.041666666666666664 + -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(im * im + 2.0), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-cos.f64 58.9

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

   5. Applied rewrites 58.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 22.0%

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

   if -0.0200000000000000004 < (cos.f64 re) < 0.999999980000000011

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0

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

      1. lower-cos.f64 45.8

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

   5. Applied rewrites 45.8%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 51.0%

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

   if 0.999999980000000011 < (cos.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 82.3

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

   5. Applied rewrites 82.3%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 82.3%

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

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot -0.5, 1\right)\\ \mathbf{elif}\;\cos re \leq 0.99999998:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, 0.041666666666666664, -0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 100.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-exp.f64 N/A

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

   10. lift-exp.f64 N/A

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

   11. lift-neg.f64 N/A

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

   12. cosh-undef N/A

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

   13. associate-*r* N/A

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

   14. metadata-eval N/A

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

   15. lower-*.f64 N/A

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

   16. lower-cosh.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

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

Alternative 10: 72.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im\right), 2\right)\\ \mathbf{if}\;\cos re \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, -0.0006944444444444445, 0.020833333333333332\right), -0.25\right), 0.5\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (fma
          im
          (fma
           (* im im)
           (* im (fma (* im im) 0.002777777777777778 0.08333333333333333))
           im)
          2.0)))
   (if (<= (cos re) -0.02)
     (*
      (fma
       re
       (*
        re
        (fma
         (* re re)
         (fma (* re re) -0.0006944444444444445 0.020833333333333332)
         -0.25))
       0.5)
      t_0)
     (* t_0 0.5))))

C

double code(double re, double im) {
	double t_0 = fma(im, fma((im * im), (im * fma((im * im), 0.002777777777777778, 0.08333333333333333)), im), 2.0);
	double tmp;
	if (cos(re) <= -0.02) {
		tmp = fma(re, (re * fma((re * re), fma((re * re), -0.0006944444444444445, 0.020833333333333332), -0.25)), 0.5) * t_0;
	} else {
		tmp = t_0 * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(im, fma(Float64(im * im), Float64(im * fma(Float64(im * im), 0.002777777777777778, 0.08333333333333333)), im), 2.0)
	tmp = 0.0
	if (cos(re) <= -0.02)
		tmp = Float64(fma(re, Float64(re * fma(Float64(re * re), fma(Float64(re * re), -0.0006944444444444445, 0.020833333333333332), -0.25)), 0.5) * t_0);
	else
		tmp = Float64(t_0 * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = 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[N[Cos[re], $MachinePrecision], -0.02], N[(N[(re * N[(re * N[(N[(re * re), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.0006944444444444445 + 0.020833333333333332), $MachinePrecision] + -0.25), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] * t$95$0), $MachinePrecision], N[(t$95$0 * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]
   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. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 93.8%

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

   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. lower-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. lower-*.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. lower-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. lower-*.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. lower-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. lower-*.f64 42.5

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

   8. Applied rewrites 42.5%

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

   if -0.0200000000000000004 < (cos.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]
   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. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 92.6%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 82.8%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, -0.0006944444444444445, 0.020833333333333332\right), -0.25\right), 0.5\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im\right), 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im\right), 2\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 71.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right) \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im\right), 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) -0.02)
   (*
    (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
     (fma
      (* im im)
      (* im (fma (* im im) 0.002777777777777778 0.08333333333333333))
      im)
     2.0)
    0.5)))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= -0.02) {
		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, fma((im * im), (im * fma((im * im), 0.002777777777777778, 0.08333333333333333)), im), 2.0) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= -0.02)
		tmp = Float64(fma(Float64(im * im), fma(im, Float64(im * 0.041666666666666664), 0.5), 1.0) * fma(Float64(re * re), fma(re, Float64(re * fma(Float64(re * re), -0.001388888888888889, 0.041666666666666664)), -0.5), 1.0));
	else
		tmp = Float64(fma(im, fma(Float64(im * im), Float64(im * fma(Float64(im * im), 0.002777777777777778, 0.08333333333333333)), im), 2.0) * 0.5);
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

   5. Applied rewrites 91.3%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 42.5%

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

   if -0.0200000000000000004 < (cos.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]
   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. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 92.6%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 82.8%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right) \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im\right), 2\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 71.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, -0.0006944444444444445, 0.020833333333333332\right), -0.25\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im\right), 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) -0.02)
   (*
    (fma im im 2.0)
    (fma
     (* re re)
     (fma
      (* re re)
      (fma (* re re) -0.0006944444444444445 0.020833333333333332)
      -0.25)
     0.5))
   (*
    (fma
     im
     (fma
      (* im im)
      (* im (fma (* im im) 0.002777777777777778 0.08333333333333333))
      im)
     2.0)
    0.5)))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= -0.02) {
		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, fma((im * im), (im * fma((im * im), 0.002777777777777778, 0.08333333333333333)), im), 2.0) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= -0.02)
		tmp = Float64(fma(im, im, 2.0) * fma(Float64(re * re), fma(Float64(re * re), fma(Float64(re * re), -0.0006944444444444445, 0.020833333333333332), -0.25), 0.5));
	else
		tmp = Float64(fma(im, fma(Float64(im * im), Float64(im * fma(Float64(im * im), 0.002777777777777778, 0.08333333333333333)), im), 2.0) * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], -0.02], N[(N[(im * im + 2.0), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.0006944444444444445 + 0.020833333333333332), $MachinePrecision] + -0.25), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], N[(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] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 81.2

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

   5. Applied rewrites 81.2%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 42.5

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

   8. Applied rewrites 42.5%

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

   if -0.0200000000000000004 < (cos.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]
   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. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 92.6%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 82.8%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, -0.0006944444444444445, 0.020833333333333332\right), -0.25\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im\right), 2\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 71.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot -0.5, 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im\right), 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) -0.02)
   (*
    (fma re (* re -0.5) 1.0)
    (fma (* im im) (fma im (* im 0.041666666666666664) 0.5) 1.0))
   (*
    (fma
     im
     (fma
      (* im im)
      (* im (fma (* im im) 0.002777777777777778 0.08333333333333333))
      im)
     2.0)
    0.5)))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= -0.02) {
		tmp = fma(re, (re * -0.5), 1.0) * fma((im * im), fma(im, (im * 0.041666666666666664), 0.5), 1.0);
	} else {
		tmp = fma(im, fma((im * im), (im * fma((im * im), 0.002777777777777778, 0.08333333333333333)), im), 2.0) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= -0.02)
		tmp = Float64(fma(re, Float64(re * -0.5), 1.0) * fma(Float64(im * im), fma(im, Float64(im * 0.041666666666666664), 0.5), 1.0));
	else
		tmp = Float64(fma(im, fma(Float64(im * im), Float64(im * fma(Float64(im * im), 0.002777777777777778, 0.08333333333333333)), im), 2.0) * 0.5);
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

   5. Applied rewrites 91.3%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 39.4%

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

   if -0.0200000000000000004 < (cos.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]
   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. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 92.6%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 82.8%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot -0.5, 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im\right), 2\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 68.5% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

   5. Applied rewrites 91.3%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 39.4%

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

   if -0.0200000000000000004 < (cos.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

   5. Applied rewrites 88.4%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 78.7%

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

Alternative 15: 67.2% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-cos.f64 58.9

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

   5. Applied rewrites 58.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 38.9%

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

   if -0.0200000000000000004 < (cos.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

   5. Applied rewrites 88.4%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 78.7%

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

Alternative 16: 67.8% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]
   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. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 93.8%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 41.8

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

   8. Applied rewrites 41.8%

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

   9. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. unpow2 N/A

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

       3. lower-fma.f64 37.0

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

   11. Applied rewrites 37.0%

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

   if -0.0200000000000000004 < (cos.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

   5. Applied rewrites 88.4%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 78.7%

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

4. Final simplification 66.1%

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

Alternative 17: 63.1% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-cos.f64 58.9

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

   5. Applied rewrites 58.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 22.0%

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

   if -0.0200000000000000004 < (cos.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

   5. Applied rewrites 88.4%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 78.7%

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

Alternative 18: 54.7% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-cos.f64 58.9

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

   5. Applied rewrites 58.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 22.0%

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

   if -0.0200000000000000004 < (cos.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 79.9

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

   5. Applied rewrites 79.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 70.2%

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

4. Final simplification 55.7%

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

Alternative 19: 35.4% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-cos.f64 58.9

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

   5. Applied rewrites 58.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 22.0%

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

   if -0.0200000000000000004 < (cos.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-cos.f64 54.7

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

   5. Applied rewrites 54.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 45.2%

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

Alternative 20: 35.4% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (cos(re) <= (-0.02d0)) then
        tmp = (-0.5d0) * (re * re)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.cos(re) <= -0.02) {
		tmp = -0.5 * (re * re);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.cos(re) <= -0.02:
		tmp = -0.5 * (re * re)
	else:
		tmp = 1.0
	return tmp

Julia

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

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (cos(re) <= -0.02)
		tmp = -0.5 * (re * re);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-cos.f64 58.9

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

   5. Applied rewrites 58.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 22.0%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 22.0%

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

   if -0.0200000000000000004 < (cos.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-cos.f64 54.7

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

   5. Applied rewrites 54.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 45.2%

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

4. Final simplification 38.3%

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

Alternative 21: 28.7% accurate, 316.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return 1.0

Julia

function code(re, im)
	return 1.0
end

MATLAB

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

Wolfram

code[re_, im_] := 1.0

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0

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

   1. lower-cos.f64 56.0

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

5. Applied rewrites 56.0%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 32.0%

   \[\leadsto 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))