math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 11.2s

Alternatives: 17

Speedup: 1.5×

• 48.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-cos.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-neg.f64 N/A

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

   4. lift-exp.f64 N/A

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

   5. lift-exp.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*r* N/A

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

   10. lower-*.f64 N/A

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

4. Applied egg-rr 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos re \cdot 0.5\\ t_1 := t\_0 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot \mathsf{fma}\left(re \cdot re, -0.020833333333333332, 0.041666666666666664\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999999999999946:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (cos re) 0.5)) (t_1 (* t_0 (+ (exp (- im)) (exp im)))))
   (if (<= t_1 (- INFINITY))
     (*
      (* im im)
      (* im (* im (fma (* re re) -0.020833333333333332 0.041666666666666664))))
     (if (<= t_1 0.9999999999999946) (* t_0 (fma im im 2.0)) (cosh im)))))

C

double code(double re, double im) {
	double t_0 = cos(re) * 0.5;
	double t_1 = t_0 * (exp(-im) + exp(im));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (im * im) * (im * (im * fma((re * re), -0.020833333333333332, 0.041666666666666664)));
	} else if (t_1 <= 0.9999999999999946) {
		tmp = t_0 * fma(im, im, 2.0);
	} else {
		tmp = cosh(im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(cos(re) * 0.5)
	t_1 = Float64(t_0 * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(im * im) * Float64(im * Float64(im * fma(Float64(re * re), -0.020833333333333332, 0.041666666666666664))));
	elseif (t_1 <= 0.9999999999999946)
		tmp = Float64(t_0 * fma(im, im, 2.0));
	else
		tmp = cosh(im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * N[(N[(re * re), $MachinePrecision] * -0.020833333333333332 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999999946], N[(t$95$0 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[Cosh[im], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 81.0

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

   5. Simplified 81.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 100.0

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

   8. Simplified 100.0%

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

   9. Taylor expanded in im around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. metadata-eval N/A

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

       4. pow-sqr N/A

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

       5. associate-*l* N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. *-commutative N/A

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

       13. associate-*l* N/A

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

       14. unpow2 N/A

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

       15. *-commutative N/A

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

       16. associate-*l* N/A

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

       17. lower-*.f64 N/A

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

       18. lower-*.f64 N/A

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

       19. +-commutative N/A

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

   11. Simplified 100.0%

       \[\leadsto \color{blue}{\left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot \mathsf{fma}\left(re \cdot re, -0.020833333333333332, 0.041666666666666664\right)\right)\right)} \]

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.99999999999999456

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 100.0

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

   5. Simplified 100.0%

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

   if 0.99999999999999456 < (*.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-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Simplified 100.0%

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

      1. lift-cosh.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-rgt-identity 100.0

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

      4. lift-*.f64 N/A

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

      5. *-lft-identity 100.0

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

   9. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot \mathsf{fma}\left(re \cdot re, -0.020833333333333332, 0.041666666666666664\right)\right)\right)\\ \mathbf{elif}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.9999999999999946:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot \mathsf{fma}\left(re \cdot re, -0.020833333333333332, 0.041666666666666664\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0.9999999999999946:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\cosh im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* (cos re) 0.5) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (*
      (* im im)
      (* im (* im (fma (* re re) -0.020833333333333332 0.041666666666666664))))
     (if (<= t_0 0.9999999999999946) (cos re) (cosh im)))))

C

double code(double re, double im) {
	double t_0 = (cos(re) * 0.5) * (exp(-im) + exp(im));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (im * im) * (im * (im * fma((re * re), -0.020833333333333332, 0.041666666666666664)));
	} else if (t_0 <= 0.9999999999999946) {
		tmp = cos(re);
	} else {
		tmp = cosh(im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(cos(re) * 0.5) * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(im * im) * Float64(im * Float64(im * fma(Float64(re * re), -0.020833333333333332, 0.041666666666666664))));
	elseif (t_0 <= 0.9999999999999946)
		tmp = cos(re);
	else
		tmp = cosh(im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * N[(N[(re * re), $MachinePrecision] * -0.020833333333333332 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.9999999999999946], N[Cos[re], $MachinePrecision], N[Cosh[im], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 81.0

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

   5. Simplified 81.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 100.0

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

   8. Simplified 100.0%

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

   9. Taylor expanded in im around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. metadata-eval N/A

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

       4. pow-sqr N/A

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

       5. associate-*l* N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. *-commutative N/A

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

       13. associate-*l* N/A

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

       14. unpow2 N/A

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

       15. *-commutative N/A

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

       16. associate-*l* N/A

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

       17. lower-*.f64 N/A

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

       18. lower-*.f64 N/A

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

       19. +-commutative N/A

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

   11. Simplified 100.0%

       \[\leadsto \color{blue}{\left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot \mathsf{fma}\left(re \cdot re, -0.020833333333333332, 0.041666666666666664\right)\right)\right)} \]

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.99999999999999456

   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 99.9

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

   5. Simplified 99.9%

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

   if 0.99999999999999456 < (*.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-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Simplified 100.0%

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

      1. lift-cosh.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-rgt-identity 100.0

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

      4. lift-*.f64 N/A

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

      5. *-lft-identity 100.0

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

   9. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot \mathsf{fma}\left(re \cdot re, -0.020833333333333332, 0.041666666666666664\right)\right)\right)\\ \mathbf{elif}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.9999999999999946:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\cosh im\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot \mathsf{fma}\left(re \cdot re, -0.020833333333333332, 0.041666666666666664\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0.99995:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, 0.020833333333333332, -0.25\right), 0.5\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot 0.08333333333333333, im\right), 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* (cos re) 0.5) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (*
      (* im im)
      (* im (* im (fma (* re re) -0.020833333333333332 0.041666666666666664))))
     (if (<= t_0 0.99995)
       (cos re)
       (*
        (fma (* re re) (fma (* re re) 0.020833333333333332 -0.25) 0.5)
        (fma im (fma im (* (* im im) 0.08333333333333333) im) 2.0))))))

C

double code(double re, double im) {
	double t_0 = (cos(re) * 0.5) * (exp(-im) + exp(im));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (im * im) * (im * (im * fma((re * re), -0.020833333333333332, 0.041666666666666664)));
	} else if (t_0 <= 0.99995) {
		tmp = cos(re);
	} else {
		tmp = fma((re * re), fma((re * re), 0.020833333333333332, -0.25), 0.5) * fma(im, fma(im, ((im * im) * 0.08333333333333333), im), 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(cos(re) * 0.5) * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(im * im) * Float64(im * Float64(im * fma(Float64(re * re), -0.020833333333333332, 0.041666666666666664))));
	elseif (t_0 <= 0.99995)
		tmp = cos(re);
	else
		tmp = Float64(fma(Float64(re * re), fma(Float64(re * re), 0.020833333333333332, -0.25), 0.5) * fma(im, fma(im, Float64(Float64(im * im) * 0.08333333333333333), im), 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * N[(N[(re * re), $MachinePrecision] * -0.020833333333333332 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.99995], N[Cos[re], $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * 0.020833333333333332 + -0.25), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * 0.08333333333333333), $MachinePrecision] + im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 81.0

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

   5. Simplified 81.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 100.0

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

   8. Simplified 100.0%

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

   9. Taylor expanded in im around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. metadata-eval N/A

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

       4. pow-sqr N/A

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

       5. associate-*l* N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. *-commutative N/A

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

       13. associate-*l* N/A

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

       14. unpow2 N/A

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

       15. *-commutative N/A

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

       16. associate-*l* N/A

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

       17. lower-*.f64 N/A

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

       18. lower-*.f64 N/A

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

       19. +-commutative N/A

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

   11. Simplified 100.0%

       \[\leadsto \color{blue}{\left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot \mathsf{fma}\left(re \cdot re, -0.020833333333333332, 0.041666666666666664\right)\right)\right)} \]

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.999950000000000006

   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 99.9

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

   5. Simplified 99.9%

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

   if 0.999950000000000006 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 85.7

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

   5. Simplified 85.7%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 89.4

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

   8. Simplified 89.4%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot \mathsf{fma}\left(re \cdot re, -0.020833333333333332, 0.041666666666666664\right)\right)\right)\\ \mathbf{elif}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.99995:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, 0.020833333333333332, -0.25\right), 0.5\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot 0.08333333333333333, im\right), 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \mathsf{fma}\left(re, re \cdot -0.25, 0.5\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(0.5, im \cdot im, 1\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* (cos re) 0.5) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 -0.05)
     (* (fma im im 2.0) (fma re (* re -0.25) 0.5))
     (if (<= t_0 2.0)
       (fma 0.5 (* im im) 1.0)
       (* im (* im (fma im (* im 0.041666666666666664) 0.5)))))))

C

double code(double re, double im) {
	double t_0 = (cos(re) * 0.5) * (exp(-im) + exp(im));
	double tmp;
	if (t_0 <= -0.05) {
		tmp = fma(im, im, 2.0) * fma(re, (re * -0.25), 0.5);
	} else if (t_0 <= 2.0) {
		tmp = fma(0.5, (im * im), 1.0);
	} else {
		tmp = im * (im * fma(im, (im * 0.041666666666666664), 0.5));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(cos(re) * 0.5) * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_0 <= -0.05)
		tmp = Float64(fma(im, im, 2.0) * fma(re, Float64(re * -0.25), 0.5));
	elseif (t_0 <= 2.0)
		tmp = fma(0.5, Float64(im * im), 1.0);
	else
		tmp = Float64(im * Float64(im * fma(im, Float64(im * 0.041666666666666664), 0.5)));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.05], N[(N[(im * im + 2.0), $MachinePrecision] * N[(re * N[(re * -0.25), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(0.5 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision], N[(im * N[(im * N[(im * N[(im * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -0.050000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   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 80.2

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

   5. Simplified 80.2%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 40.6

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

   8. Simplified 40.6%

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

   if -0.050000000000000003 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0

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

      1. distribute-lft-in N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 74.0

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

   8. Simplified 74.0%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

   5. Simplified 75.6%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 75.6%

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

   9. Taylor expanded in im around inf

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

       1. distribute-rgt-in N/A

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. associate-*l* N/A

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

       5. associate-*r/ N/A

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

       6. metadata-eval N/A

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

       7. associate-*l/ N/A

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

       8. associate-/l* N/A

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

       9. metadata-eval N/A

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

       10. pow-sqr N/A

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

       11. associate-/l* N/A

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

       12. *-rgt-identity N/A

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

       13. associate-*r/ N/A

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

       14. rgt-mult-inverse N/A

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

       15. *-rgt-identity N/A

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

       16. distribute-rgt-in N/A

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

       17. +-commutative N/A

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

       18. unpow2 N/A

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

   11. Simplified 75.6%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \mathsf{fma}\left(re, re \cdot -0.25, 0.5\right)\\ \mathbf{elif}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 2:\\ \;\;\;\;\mathsf{fma}\left(0.5, im \cdot im, 1\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -0.05:\\ \;\;\;\;im \cdot \left(im \cdot \mathsf{fma}\left(-0.25, re \cdot re, 0.5\right)\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(0.5, im \cdot im, 1\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* (cos re) 0.5) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 -0.05)
     (* im (* im (fma -0.25 (* re re) 0.5)))
     (if (<= t_0 2.0)
       (fma 0.5 (* im im) 1.0)
       (* im (* im (fma im (* im 0.041666666666666664) 0.5)))))))

C

double code(double re, double im) {
	double t_0 = (cos(re) * 0.5) * (exp(-im) + exp(im));
	double tmp;
	if (t_0 <= -0.05) {
		tmp = im * (im * fma(-0.25, (re * re), 0.5));
	} else if (t_0 <= 2.0) {
		tmp = fma(0.5, (im * im), 1.0);
	} else {
		tmp = im * (im * fma(im, (im * 0.041666666666666664), 0.5));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(cos(re) * 0.5) * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_0 <= -0.05)
		tmp = Float64(im * Float64(im * fma(-0.25, Float64(re * re), 0.5)));
	elseif (t_0 <= 2.0)
		tmp = fma(0.5, Float64(im * im), 1.0);
	else
		tmp = Float64(im * Float64(im * fma(im, Float64(im * 0.041666666666666664), 0.5)));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.05], N[(im * N[(im * N[(-0.25 * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(0.5 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision], N[(im * N[(im * N[(im * N[(im * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -0.050000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   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 80.2

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

   5. Simplified 80.2%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 40.6

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

   8. Simplified 40.6%

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

   9. Taylor expanded in im around inf

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. +-commutative N/A

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

       8. lower-fma.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 40.1

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

   11. Simplified 40.1%

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

   if -0.050000000000000003 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0

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

      1. distribute-lft-in N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 74.0

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

   8. Simplified 74.0%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

   5. Simplified 75.6%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 75.6%

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

   9. Taylor expanded in im around inf

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

       1. distribute-rgt-in N/A

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. associate-*l* N/A

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

       5. associate-*r/ N/A

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

       6. metadata-eval N/A

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

       7. associate-*l/ N/A

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

       8. associate-/l* N/A

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

       9. metadata-eval N/A

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

       10. pow-sqr N/A

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

       11. associate-/l* N/A

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

       12. *-rgt-identity N/A

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

       13. associate-*r/ N/A

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

       14. rgt-mult-inverse N/A

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

       15. *-rgt-identity N/A

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

       16. distribute-rgt-in N/A

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

       17. +-commutative N/A

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

       18. unpow2 N/A

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

   11. Simplified 75.6%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.05:\\ \;\;\;\;im \cdot \left(im \cdot \mathsf{fma}\left(-0.25, re \cdot re, 0.5\right)\right)\\ \mathbf{elif}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 2:\\ \;\;\;\;\mathsf{fma}\left(0.5, im \cdot im, 1\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -0.05:\\ \;\;\;\;im \cdot \left(im \cdot \mathsf{fma}\left(-0.25, re \cdot re, 0.5\right)\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(0.5, im \cdot im, 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* (cos re) 0.5) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 -0.05)
     (* im (* im (fma -0.25 (* re re) 0.5)))
     (if (<= t_0 2.0)
       (fma 0.5 (* im im) 1.0)
       (* 0.041666666666666664 (* im (* im (* im im))))))))

C

double code(double re, double im) {
	double t_0 = (cos(re) * 0.5) * (exp(-im) + exp(im));
	double tmp;
	if (t_0 <= -0.05) {
		tmp = im * (im * fma(-0.25, (re * re), 0.5));
	} else if (t_0 <= 2.0) {
		tmp = fma(0.5, (im * im), 1.0);
	} else {
		tmp = 0.041666666666666664 * (im * (im * (im * im)));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(cos(re) * 0.5) * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_0 <= -0.05)
		tmp = Float64(im * Float64(im * fma(-0.25, Float64(re * re), 0.5)));
	elseif (t_0 <= 2.0)
		tmp = fma(0.5, Float64(im * im), 1.0);
	else
		tmp = Float64(0.041666666666666664 * Float64(im * Float64(im * Float64(im * im))));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.05], N[(im * N[(im * N[(-0.25 * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(0.5 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision], N[(0.041666666666666664 * N[(im * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -0.050000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   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 80.2

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

   5. Simplified 80.2%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 40.6

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

   8. Simplified 40.6%

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

   9. Taylor expanded in im around inf

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. +-commutative N/A

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

       8. lower-fma.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 40.1

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

   11. Simplified 40.1%

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

   if -0.050000000000000003 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0

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

      1. distribute-lft-in N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 74.0

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

   8. Simplified 74.0%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

   5. Simplified 75.6%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 75.6%

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

   9. Taylor expanded in im around inf

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

       1. lower-*.f64 N/A

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 75.6

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

   11. Simplified 75.6%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.05:\\ \;\;\;\;im \cdot \left(im \cdot \mathsf{fma}\left(-0.25, re \cdot re, 0.5\right)\right)\\ \mathbf{elif}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 2:\\ \;\;\;\;\mathsf{fma}\left(0.5, im \cdot im, 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(0.5, im \cdot im, 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* (cos re) 0.5) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 -0.05)
     (fma re (* re -0.5) 1.0)
     (if (<= t_0 2.0)
       (fma 0.5 (* im im) 1.0)
       (* 0.041666666666666664 (* im (* im (* im im))))))))

C

double code(double re, double im) {
	double t_0 = (cos(re) * 0.5) * (exp(-im) + exp(im));
	double tmp;
	if (t_0 <= -0.05) {
		tmp = fma(re, (re * -0.5), 1.0);
	} else if (t_0 <= 2.0) {
		tmp = fma(0.5, (im * im), 1.0);
	} else {
		tmp = 0.041666666666666664 * (im * (im * (im * im)));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(cos(re) * 0.5) * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_0 <= -0.05)
		tmp = fma(re, Float64(re * -0.5), 1.0);
	elseif (t_0 <= 2.0)
		tmp = fma(0.5, Float64(im * im), 1.0);
	else
		tmp = Float64(0.041666666666666664 * Float64(im * Float64(im * Float64(im * im))));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.05], N[(re * N[(re * -0.5), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(0.5 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision], N[(0.041666666666666664 * N[(im * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -0.050000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-cos.f64 59.6

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

   5. Simplified 59.6%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 16.7

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

   8. Simplified 16.7%

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

   if -0.050000000000000003 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0

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

      1. distribute-lft-in N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 74.0

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

   8. Simplified 74.0%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

   5. Simplified 75.6%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 75.6%

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

   9. Taylor expanded in im around inf

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

       1. lower-*.f64 N/A

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 75.6

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

   11. Simplified 75.6%

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

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot -0.5, 1\right)\\ \mathbf{elif}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 2:\\ \;\;\;\;\mathsf{fma}\left(0.5, im \cdot im, 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos re \cdot 0.5\\ \mathbf{if}\;t\_0 \cdot \left(e^{-im} + e^{im}\right) \leq 0.9999999999999946:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im\right), 2\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (cos re) 0.5)))
   (if (<= (* t_0 (+ (exp (- im)) (exp im))) 0.9999999999999946)
     (*
      t_0
      (fma
       im
       (fma
        (* im im)
        (* im (fma (* im im) 0.002777777777777778 0.08333333333333333))
        im)
       2.0))
     (cosh im))))

C

double code(double re, double im) {
	double t_0 = cos(re) * 0.5;
	double tmp;
	if ((t_0 * (exp(-im) + exp(im))) <= 0.9999999999999946) {
		tmp = t_0 * fma(im, fma((im * im), (im * fma((im * im), 0.002777777777777778, 0.08333333333333333)), im), 2.0);
	} else {
		tmp = cosh(im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(cos(re) * 0.5)
	tmp = 0.0
	if (Float64(t_0 * Float64(exp(Float64(-im)) + exp(im))) <= 0.9999999999999946)
		tmp = Float64(t_0 * fma(im, fma(Float64(im * im), Float64(im * fma(Float64(im * im), 0.002777777777777778, 0.08333333333333333)), im), 2.0));
	else
		tmp = cosh(im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.9999999999999946], N[(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]), $MachinePrecision], N[Cosh[im], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.99999999999999456

   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. Simplified 96.4%

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

   if 0.99999999999999456 < (*.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-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Simplified 100.0%

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

      1. lift-cosh.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-rgt-identity 100.0

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

      4. lift-*.f64 N/A

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

      5. *-lft-identity 100.0

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

   9. Applied egg-rr 100.0%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.9999999999999946:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im\right), 2\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot 0.08333333333333333, im\right), 2\right) \cdot \mathsf{fma}\left(re, re \cdot -0.25, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (cos re) 0.5) (+ (exp (- im)) (exp im))) -0.05)
   (*
    (fma im (fma im (* (* im im) 0.08333333333333333) im) 2.0)
    (fma re (* re -0.25) 0.5))
   (fma
    im
    (*
     im
     (fma
      (* im im)
      (fma im (* im 0.001388888888888889) 0.041666666666666664)
      0.5))
    1.0)))

C

double code(double re, double im) {
	double tmp;
	if (((cos(re) * 0.5) * (exp(-im) + exp(im))) <= -0.05) {
		tmp = fma(im, fma(im, ((im * im) * 0.08333333333333333), im), 2.0) * fma(re, (re * -0.25), 0.5);
	} else {
		tmp = fma(im, (im * fma((im * im), fma(im, (im * 0.001388888888888889), 0.041666666666666664), 0.5)), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(cos(re) * 0.5) * Float64(exp(Float64(-im)) + exp(im))) <= -0.05)
		tmp = Float64(fma(im, fma(im, Float64(Float64(im * im) * 0.08333333333333333), im), 2.0) * fma(re, Float64(re * -0.25), 0.5));
	else
		tmp = fma(im, Float64(im * fma(Float64(im * im), fma(im, Float64(im * 0.001388888888888889), 0.041666666666666664), 0.5)), 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.05], N[(N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * 0.08333333333333333), $MachinePrecision] + im), $MachinePrecision] + 2.0), $MachinePrecision] * N[(re * N[(re * -0.25), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -0.050000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 92.1

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

   5. Simplified 92.1%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 44.4

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

   8. Simplified 44.4%

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

   if -0.050000000000000003 < (*.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-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Simplified 86.7%

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

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

   10. Simplified 77.4%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot 0.08333333333333333, im\right), 2\right) \cdot \mathsf{fma}\left(re, re \cdot -0.25, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 70.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -0.050000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 92.1

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

   5. Simplified 92.1%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 44.4

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

   8. Simplified 44.4%

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

   9. Taylor expanded in im around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. metadata-eval N/A

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

       4. pow-sqr N/A

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

       5. associate-*l* N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. *-commutative N/A

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

       13. associate-*l* N/A

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

       14. unpow2 N/A

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

       15. *-commutative N/A

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

       16. associate-*l* N/A

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

       17. lower-*.f64 N/A

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

       18. lower-*.f64 N/A

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

       19. +-commutative N/A

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

   11. Simplified 43.4%

       \[\leadsto \color{blue}{\left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot \mathsf{fma}\left(re \cdot re, -0.020833333333333332, 0.041666666666666664\right)\right)\right)} \]

   if -0.050000000000000003 < (*.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-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Simplified 86.7%

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

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

   10. Simplified 77.4%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.05:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot \mathsf{fma}\left(re \cdot re, -0.020833333333333332, 0.041666666666666664\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 67.3% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (cos re) 0.5) (+ (exp (- im)) (exp im))) -0.05)
   (*
    (* im im)
    (* im (* im (fma (* re re) -0.020833333333333332 0.041666666666666664))))
   (fma im (* im (fma im (* im 0.041666666666666664) 0.5)) 1.0)))

C

double code(double re, double im) {
	double tmp;
	if (((cos(re) * 0.5) * (exp(-im) + exp(im))) <= -0.05) {
		tmp = (im * im) * (im * (im * fma((re * re), -0.020833333333333332, 0.041666666666666664)));
	} 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 (Float64(Float64(cos(re) * 0.5) * Float64(exp(Float64(-im)) + exp(im))) <= -0.05)
		tmp = Float64(Float64(im * im) * Float64(im * Float64(im * fma(Float64(re * re), -0.020833333333333332, 0.041666666666666664))));
	else
		tmp = fma(im, Float64(im * fma(im, Float64(im * 0.041666666666666664), 0.5)), 1.0);
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -0.050000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 92.1

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

   5. Simplified 92.1%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 44.4

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

   8. Simplified 44.4%

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

   9. Taylor expanded in im around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. metadata-eval N/A

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

       4. pow-sqr N/A

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

       5. associate-*l* N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. *-commutative N/A

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

       13. associate-*l* N/A

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

       14. unpow2 N/A

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

       15. *-commutative N/A

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

       16. associate-*l* N/A

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

       17. lower-*.f64 N/A

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

       18. lower-*.f64 N/A

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

       19. +-commutative N/A

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

   11. Simplified 43.4%

       \[\leadsto \color{blue}{\left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot \mathsf{fma}\left(re \cdot re, -0.020833333333333332, 0.041666666666666664\right)\right)\right)} \]

   if -0.050000000000000003 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

   5. Simplified 88.1%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 74.8

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

   8. Simplified 74.8%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.05:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot \mathsf{fma}\left(re \cdot re, -0.020833333333333332, 0.041666666666666664\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 47.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-cos.f64 82.3

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

   5. Simplified 82.3%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 42.2%

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

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

   1. Initial program 100.0%

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

   3. 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. Simplified 75.6%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 75.6%

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

   9. Taylor expanded in im around inf

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

       1. distribute-rgt-in N/A

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. associate-*l* N/A

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

       5. associate-*r/ N/A

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

       6. metadata-eval N/A

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

       7. associate-*l/ N/A

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

       8. associate-/l* N/A

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

       9. metadata-eval N/A

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

       10. pow-sqr N/A

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

       11. associate-/l* N/A

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

       12. *-rgt-identity N/A

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

       13. associate-*r/ N/A

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

       14. rgt-mult-inverse N/A

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

       15. *-rgt-identity N/A

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

       16. distribute-rgt-in N/A

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

       17. +-commutative N/A

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

       18. unpow2 N/A

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

   11. Simplified 75.6%

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

   12. Taylor expanded in im around 0

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

   14. Simplified 44.3%

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

4. Final simplification 42.9%

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

Alternative 14: 66.6% accurate, 2.4× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) -0.05)
   (* (fma 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) <= -0.05) {
		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) <= -0.05)
		tmp = Float64(fma(im, im, 2.0) * fma(re, Float64(re * -0.25), 0.5));
	else
		tmp = fma(im, Float64(im * fma(im, Float64(im * 0.041666666666666664), 0.5)), 1.0);
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   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 80.2

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

   5. Simplified 80.2%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 40.6

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

   8. Simplified 40.6%

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

   if -0.050000000000000003 < (cos.f64 re)

   1. Initial program 100.0%

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

   3. 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. Simplified 88.1%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 74.8

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

   8. Simplified 74.8%

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

4. Final simplification 65.2%

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

Alternative 15: 53.4% accurate, 2.7× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= -0.05) {
		tmp = fma(re, (re * -0.5), 1.0);
	} else {
		tmp = fma(0.5, (im * im), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= -0.05)
		tmp = fma(re, Float64(re * -0.5), 1.0);
	else
		tmp = fma(0.5, Float64(im * im), 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], -0.05], N[(re * N[(re * -0.5), $MachinePrecision] + 1.0), $MachinePrecision], N[(0.5 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -0.050000000000000003

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re} \]
   4. Step-by-step derivation

      1. lower-cos.f64 59.6

         \[\leadsto \color{blue}{\cos re} \]

   5. Simplified 59.6%

      \[\leadsto \color{blue}{\cos re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + \frac{-1}{2} \cdot {re}^{2}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot {re}^{2} + 1} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{{re}^{2} \cdot \frac{-1}{2}} + 1 \]

      3. unpow2 N/A

         \[\leadsto \color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{2} + 1 \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{re \cdot \left(re \cdot \frac{-1}{2}\right)} + 1 \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, re \cdot \frac{-1}{2}, 1\right)} \]

      6. lower-*.f64 16.7

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{re \cdot -0.5}, 1\right) \]

   8. Simplified 16.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, re \cdot -0.5, 1\right)} \]

   if -0.050000000000000003 < (cos.f64 re)

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. 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 72.8

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   5. Simplified 72.8%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(2 + {im}^{2}\right)} \]
   7. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot 2 + \frac{1}{2} \cdot {im}^{2}} \]

      2. metadata-eval N/A

         \[\leadsto \color{blue}{1} + \frac{1}{2} \cdot {im}^{2} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot {im}^{2} + 1} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {im}^{2}, 1\right)} \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{im \cdot im}, 1\right) \]

      6. lower-*.f64 59.5

         \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{im \cdot im}, 1\right) \]

   8. Simplified 59.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, im \cdot im, 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 47.1% accurate, 26.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.5, im \cdot im, 1\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (fma 0.5 (* im im) 1.0))

C

double code(double re, double im) {
	return fma(0.5, (im * im), 1.0);
}

Julia

function code(re, im)
	return fma(0.5, Float64(im * im), 1.0)
end

Wolfram

code[re_, im_] := N[(0.5 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
2. Add Preprocessing

3. 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 74.8

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

5. Simplified 74.8%

   \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

6. Taylor expanded in re around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(2 + {im}^{2}\right)} \]
7. Step-by-step derivation

   1. distribute-lft-in N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot 2 + \frac{1}{2} \cdot {im}^{2}} \]

   2. metadata-eval N/A

      \[\leadsto \color{blue}{1} + \frac{1}{2} \cdot {im}^{2} \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot {im}^{2} + 1} \]

   4. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {im}^{2}, 1\right)} \]

   5. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{im \cdot im}, 1\right) \]

   6. lower-*.f64 43.0

      \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{im \cdot im}, 1\right) \]

8. Simplified 43.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, im \cdot im, 1\right)} \]
9. Add Preprocessing

Alternative 17: 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 54.4

      \[\leadsto \color{blue}{\cos re} \]

5. Simplified 54.4%

   \[\leadsto \color{blue}{\cos re} \]

6. Taylor expanded in re around 0

   \[\leadsto \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 28.5%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024207 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))