math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 9.8s

Alternatives: 17

Speedup: 1.5×

• 50.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.7× speedup

Math

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

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (fma (* 0.5 (cos re)) (exp (- im_m)) (* (* (exp im_m) 0.5) (cos re))))

C

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

Julia

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[Exp[(-im$95$m)], $MachinePrecision] + N[(N[(N[Exp[im$95$m], $MachinePrecision] * 0.5), $MachinePrecision] * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. distribute-lft-in N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := \left(e^{im\_m} + e^{-im\_m}\right) \cdot \left(0.5 \cdot \cos re\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-0.5, re \cdot re, 1\right) \cdot \cosh im\_m\\ \mathbf{elif}\;t\_0 \leq 0.9999992483786312:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im\_m \cdot im\_m, 0.041666666666666664\right), im\_m \cdot im\_m, 0.5\right), im\_m \cdot im\_m, 1\right) \cdot \cos re\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \cosh im\_m\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0 (* (+ (exp im_m) (exp (- im_m))) (* 0.5 (cos re)))))
   (if (<= t_0 (- INFINITY))
     (* (fma -0.5 (* re re) 1.0) (cosh im_m))
     (if (<= t_0 0.9999992483786312)
       (*
        (fma
         (fma
          (fma 0.001388888888888889 (* im_m im_m) 0.041666666666666664)
          (* im_m im_m)
          0.5)
         (* im_m im_m)
         1.0)
        (cos re))
       (* 1.0 (cosh im_m))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double t_0 = (exp(im_m) + exp(-im_m)) * (0.5 * cos(re));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(-0.5, (re * re), 1.0) * cosh(im_m);
	} else if (t_0 <= 0.9999992483786312) {
		tmp = fma(fma(fma(0.001388888888888889, (im_m * im_m), 0.041666666666666664), (im_m * im_m), 0.5), (im_m * im_m), 1.0) * cos(re);
	} else {
		tmp = 1.0 * cosh(im_m);
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	t_0 = Float64(Float64(exp(im_m) + exp(Float64(-im_m))) * Float64(0.5 * cos(re)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(-0.5, Float64(re * re), 1.0) * cosh(im_m));
	elseif (t_0 <= 0.9999992483786312)
		tmp = Float64(fma(fma(fma(0.001388888888888889, Float64(im_m * im_m), 0.041666666666666664), Float64(im_m * im_m), 0.5), Float64(im_m * im_m), 1.0) * cos(re));
	else
		tmp = Float64(1.0 * cosh(im_m));
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := Block[{t$95$0 = N[(N[(N[Exp[im$95$m], $MachinePrecision] + N[Exp[(-im$95$m)], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(-0.5 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * N[Cosh[im$95$m], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.9999992483786312], N[(N[(N[(N[(0.001388888888888889 * N[(im$95$m * im$95$m), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[Cos[re], $MachinePrecision]), $MachinePrecision], N[(1.0 * N[Cosh[im$95$m], $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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. cosh-undef N/A

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

      13. associate-*r* N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.999999248378631189

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. cosh-undef N/A

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

      13. associate-*r* N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   if 0.999999248378631189 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-rgt-in N/A

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

      11. lift-exp.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. lift-exp.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Applied rewrites 99.0%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-0.5, re \cdot re, 1\right) \cdot \cosh im\\ \mathbf{elif}\;\left(e^{im} + e^{-im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq 0.9999992483786312:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \cos re\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \cosh im\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := \left(e^{im\_m} + e^{-im\_m}\right) \cdot \left(0.5 \cdot \cos re\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-0.5, re \cdot re, 1\right) \cdot \cosh im\_m\\ \mathbf{elif}\;t\_0 \leq 0.9999992483786312:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(im\_m \cdot im\_m, 0.041666666666666664, 0.5\right), im\_m \cdot im\_m, 1\right) \cdot \cos re\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \cosh im\_m\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0 (* (+ (exp im_m) (exp (- im_m))) (* 0.5 (cos re)))))
   (if (<= t_0 (- INFINITY))
     (* (fma -0.5 (* re re) 1.0) (cosh im_m))
     (if (<= t_0 0.9999992483786312)
       (*
        (fma (fma (* im_m im_m) 0.041666666666666664 0.5) (* im_m im_m) 1.0)
        (cos re))
       (* 1.0 (cosh im_m))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double t_0 = (exp(im_m) + exp(-im_m)) * (0.5 * cos(re));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(-0.5, (re * re), 1.0) * cosh(im_m);
	} else if (t_0 <= 0.9999992483786312) {
		tmp = fma(fma((im_m * im_m), 0.041666666666666664, 0.5), (im_m * im_m), 1.0) * cos(re);
	} else {
		tmp = 1.0 * cosh(im_m);
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	t_0 = Float64(Float64(exp(im_m) + exp(Float64(-im_m))) * Float64(0.5 * cos(re)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(-0.5, Float64(re * re), 1.0) * cosh(im_m));
	elseif (t_0 <= 0.9999992483786312)
		tmp = Float64(fma(fma(Float64(im_m * im_m), 0.041666666666666664, 0.5), Float64(im_m * im_m), 1.0) * cos(re));
	else
		tmp = Float64(1.0 * cosh(im_m));
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := Block[{t$95$0 = N[(N[(N[Exp[im$95$m], $MachinePrecision] + N[Exp[(-im$95$m)], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(-0.5 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * N[Cosh[im$95$m], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.9999992483786312], N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[Cos[re], $MachinePrecision]), $MachinePrecision], N[(1.0 * N[Cosh[im$95$m], $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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. cosh-undef N/A

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

      13. associate-*r* N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.999999248378631189

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. cosh-undef N/A

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

      13. associate-*r* N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 99.8

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

   7. Applied rewrites 99.8%

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

   if 0.999999248378631189 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-rgt-in N/A

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

      11. lift-exp.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. lift-exp.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Applied rewrites 99.0%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-0.5, re \cdot re, 1\right) \cdot \cosh im\\ \mathbf{elif}\;\left(e^{im} + e^{-im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq 0.9999992483786312:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), im \cdot im, 1\right) \cdot \cos re\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \cosh im\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ t_1 := \left(e^{im\_m} + e^{-im\_m}\right) \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-0.5, re \cdot re, 1\right) \cdot \cosh im\_m\\ \mathbf{elif}\;t\_1 \leq 0.9999992483786312:\\ \;\;\;\;\mathsf{fma}\left(im\_m, im\_m, 2\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \cosh im\_m\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re))) (t_1 (* (+ (exp im_m) (exp (- im_m))) t_0)))
   (if (<= t_1 (- INFINITY))
     (* (fma -0.5 (* re re) 1.0) (cosh im_m))
     (if (<= t_1 0.9999992483786312)
       (* (fma im_m im_m 2.0) t_0)
       (* 1.0 (cosh im_m))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double t_0 = 0.5 * cos(re);
	double t_1 = (exp(im_m) + exp(-im_m)) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(-0.5, (re * re), 1.0) * cosh(im_m);
	} else if (t_1 <= 0.9999992483786312) {
		tmp = fma(im_m, im_m, 2.0) * t_0;
	} else {
		tmp = 1.0 * cosh(im_m);
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	t_0 = Float64(0.5 * cos(re))
	t_1 = Float64(Float64(exp(im_m) + exp(Float64(-im_m))) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(-0.5, Float64(re * re), 1.0) * cosh(im_m));
	elseif (t_1 <= 0.9999992483786312)
		tmp = Float64(fma(im_m, im_m, 2.0) * t_0);
	else
		tmp = Float64(1.0 * cosh(im_m));
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Exp[im$95$m], $MachinePrecision] + N[Exp[(-im$95$m)], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(-0.5 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * N[Cosh[im$95$m], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999992483786312], N[(N[(im$95$m * im$95$m + 2.0), $MachinePrecision] * t$95$0), $MachinePrecision], N[(1.0 * N[Cosh[im$95$m], $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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. cosh-undef N/A

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

      13. associate-*r* N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.999999248378631189

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 98.8

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

   5. Applied rewrites 98.8%

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

   if 0.999999248378631189 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-rgt-in N/A

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

      11. lift-exp.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. lift-exp.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Applied rewrites 99.0%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-0.5, re \cdot re, 1\right) \cdot \cosh im\\ \mathbf{elif}\;\left(e^{im} + e^{-im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq 0.9999992483786312:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \left(0.5 \cdot \cos re\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \cosh im\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ t_1 := \left(e^{im\_m} + e^{-im\_m}\right) \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right), re \cdot re, -0.25\right), re \cdot re, 0.5\right) \cdot \mathsf{fma}\left(im\_m, im\_m, 2\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999992483786312:\\ \;\;\;\;\mathsf{fma}\left(im\_m, im\_m, 2\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \cosh im\_m\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re))) (t_1 (* (+ (exp im_m) (exp (- im_m))) t_0)))
   (if (<= t_1 (- INFINITY))
     (*
      (fma
       (fma
        (fma -0.0006944444444444445 (* re re) 0.020833333333333332)
        (* re re)
        -0.25)
       (* re re)
       0.5)
      (fma im_m im_m 2.0))
     (if (<= t_1 0.9999992483786312)
       (* (fma im_m im_m 2.0) t_0)
       (* 1.0 (cosh im_m))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double t_0 = 0.5 * cos(re);
	double t_1 = (exp(im_m) + exp(-im_m)) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(fma(fma(-0.0006944444444444445, (re * re), 0.020833333333333332), (re * re), -0.25), (re * re), 0.5) * fma(im_m, im_m, 2.0);
	} else if (t_1 <= 0.9999992483786312) {
		tmp = fma(im_m, im_m, 2.0) * t_0;
	} else {
		tmp = 1.0 * cosh(im_m);
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	t_0 = Float64(0.5 * cos(re))
	t_1 = Float64(Float64(exp(im_m) + exp(Float64(-im_m))) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(fma(fma(-0.0006944444444444445, Float64(re * re), 0.020833333333333332), Float64(re * re), -0.25), Float64(re * re), 0.5) * fma(im_m, im_m, 2.0));
	elseif (t_1 <= 0.9999992483786312)
		tmp = Float64(fma(im_m, im_m, 2.0) * t_0);
	else
		tmp = Float64(1.0 * cosh(im_m));
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Exp[im$95$m], $MachinePrecision] + N[Exp[(-im$95$m)], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(-0.0006944444444444445 * N[(re * re), $MachinePrecision] + 0.020833333333333332), $MachinePrecision] * N[(re * re), $MachinePrecision] + -0.25), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im$95$m * im$95$m + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999992483786312], N[(N[(im$95$m * im$95$m + 2.0), $MachinePrecision] * t$95$0), $MachinePrecision], N[(1.0 * N[Cosh[im$95$m], $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

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   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 52.3

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

   5. Applied rewrites 52.3%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 96.7

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

   8. Applied rewrites 96.7%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.999999248378631189

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 98.8

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

   5. Applied rewrites 98.8%

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

   if 0.999999248378631189 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-rgt-in N/A

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

      11. lift-exp.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. lift-exp.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Applied rewrites 99.0%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right), re \cdot re, -0.25\right), re \cdot re, 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;\left(e^{im} + e^{-im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq 0.9999992483786312:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \left(0.5 \cdot \cos re\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \cosh im\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := \left(e^{im\_m} + e^{-im\_m}\right) \cdot \left(0.5 \cdot \cos re\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right), re \cdot re, -0.25\right), re \cdot re, 0.5\right) \cdot \mathsf{fma}\left(im\_m, im\_m, 2\right)\\ \mathbf{elif}\;t\_0 \leq 0.9999992483786312:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \cosh im\_m\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0 (* (+ (exp im_m) (exp (- im_m))) (* 0.5 (cos re)))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma
       (fma
        (fma -0.0006944444444444445 (* re re) 0.020833333333333332)
        (* re re)
        -0.25)
       (* re re)
       0.5)
      (fma im_m im_m 2.0))
     (if (<= t_0 0.9999992483786312) (cos re) (* 1.0 (cosh im_m))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double t_0 = (exp(im_m) + exp(-im_m)) * (0.5 * cos(re));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(fma(fma(-0.0006944444444444445, (re * re), 0.020833333333333332), (re * re), -0.25), (re * re), 0.5) * fma(im_m, im_m, 2.0);
	} else if (t_0 <= 0.9999992483786312) {
		tmp = cos(re);
	} else {
		tmp = 1.0 * cosh(im_m);
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	t_0 = Float64(Float64(exp(im_m) + exp(Float64(-im_m))) * Float64(0.5 * cos(re)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(fma(fma(-0.0006944444444444445, Float64(re * re), 0.020833333333333332), Float64(re * re), -0.25), Float64(re * re), 0.5) * fma(im_m, im_m, 2.0));
	elseif (t_0 <= 0.9999992483786312)
		tmp = cos(re);
	else
		tmp = Float64(1.0 * cosh(im_m));
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := Block[{t$95$0 = N[(N[(N[Exp[im$95$m], $MachinePrecision] + N[Exp[(-im$95$m)], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(-0.0006944444444444445 * N[(re * re), $MachinePrecision] + 0.020833333333333332), $MachinePrecision] * N[(re * re), $MachinePrecision] + -0.25), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im$95$m * im$95$m + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.9999992483786312], N[Cos[re], $MachinePrecision], N[(1.0 * N[Cosh[im$95$m], $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

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   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 52.3

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

   5. Applied rewrites 52.3%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 96.7

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

   8. Applied rewrites 96.7%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.999999248378631189

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0

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

      1. lower-cos.f64 97.2

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

   5. Applied rewrites 97.2%

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

   if 0.999999248378631189 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-rgt-in N/A

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

      11. lift-exp.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. lift-exp.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Applied rewrites 99.0%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right), re \cdot re, -0.25\right), re \cdot re, 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;\left(e^{im} + e^{-im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq 0.9999992483786312:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \cosh im\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := \left(e^{im\_m} + e^{-im\_m}\right) \cdot \left(0.5 \cdot \cos re\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right), re \cdot re, -0.25\right), re \cdot re, 0.5\right) \cdot \mathsf{fma}\left(im\_m, im\_m, 2\right)\\ \mathbf{elif}\;t\_0 \leq 0.9999992483786312:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im\_m \cdot im\_m, 0.001388888888888889, 0.041666666666666664\right), im\_m \cdot im\_m, 0.5\right), im\_m \cdot im\_m, 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0 (* (+ (exp im_m) (exp (- im_m))) (* 0.5 (cos re)))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma
       (fma
        (fma -0.0006944444444444445 (* re re) 0.020833333333333332)
        (* re re)
        -0.25)
       (* re re)
       0.5)
      (fma im_m im_m 2.0))
     (if (<= t_0 0.9999992483786312)
       (cos re)
       (*
        (fma
         (fma
          (fma (* im_m im_m) 0.001388888888888889 0.041666666666666664)
          (* im_m im_m)
          0.5)
         (* im_m im_m)
         1.0)
        1.0)))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double t_0 = (exp(im_m) + exp(-im_m)) * (0.5 * cos(re));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(fma(fma(-0.0006944444444444445, (re * re), 0.020833333333333332), (re * re), -0.25), (re * re), 0.5) * fma(im_m, im_m, 2.0);
	} else if (t_0 <= 0.9999992483786312) {
		tmp = cos(re);
	} else {
		tmp = fma(fma(fma((im_m * im_m), 0.001388888888888889, 0.041666666666666664), (im_m * im_m), 0.5), (im_m * im_m), 1.0) * 1.0;
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	t_0 = Float64(Float64(exp(im_m) + exp(Float64(-im_m))) * Float64(0.5 * cos(re)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(fma(fma(-0.0006944444444444445, Float64(re * re), 0.020833333333333332), Float64(re * re), -0.25), Float64(re * re), 0.5) * fma(im_m, im_m, 2.0));
	elseif (t_0 <= 0.9999992483786312)
		tmp = cos(re);
	else
		tmp = Float64(fma(fma(fma(Float64(im_m * im_m), 0.001388888888888889, 0.041666666666666664), Float64(im_m * im_m), 0.5), Float64(im_m * im_m), 1.0) * 1.0);
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := Block[{t$95$0 = N[(N[(N[Exp[im$95$m], $MachinePrecision] + N[Exp[(-im$95$m)], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(-0.0006944444444444445 * N[(re * re), $MachinePrecision] + 0.020833333333333332), $MachinePrecision] * N[(re * re), $MachinePrecision] + -0.25), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im$95$m * im$95$m + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.9999992483786312], N[Cos[re], $MachinePrecision], N[(N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   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 52.3

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

   5. Applied rewrites 52.3%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 96.7

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

   8. Applied rewrites 96.7%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.999999248378631189

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0

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

      1. lower-cos.f64 97.2

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

   5. Applied rewrites 97.2%

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

   if 0.999999248378631189 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-rgt-in N/A

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

      11. lift-exp.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. lift-exp.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Applied rewrites 99.0%

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

   10. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. +-commutative N/A

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

       8. *-commutative N/A

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

       9. lower-fma.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 N/A

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

       14. unpow2 N/A

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

       15. lower-*.f64 93.2

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

   12. Applied rewrites 93.2%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right), re \cdot re, -0.25\right), re \cdot re, 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;\left(e^{im} + e^{-im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq 0.9999992483786312:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;\left(e^{im\_m} + e^{-im\_m}\right) \cdot \left(0.5 \cdot \cos re\right) \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right), re \cdot re, -0.25\right), re \cdot re, 0.5\right) \cdot \mathsf{fma}\left(im\_m, im\_m, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im\_m \cdot im\_m, 0.001388888888888889, 0.041666666666666664\right), im\_m \cdot im\_m, 0.5\right), im\_m \cdot im\_m, 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (* (+ (exp im_m) (exp (- im_m))) (* 0.5 (cos re))) -0.02)
   (*
    (fma
     (fma
      (fma -0.0006944444444444445 (* re re) 0.020833333333333332)
      (* re re)
      -0.25)
     (* re re)
     0.5)
    (fma im_m im_m 2.0))
   (*
    (fma
     (fma
      (fma (* im_m im_m) 0.001388888888888889 0.041666666666666664)
      (* im_m im_m)
      0.5)
     (* im_m im_m)
     1.0)
    1.0)))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (((exp(im_m) + exp(-im_m)) * (0.5 * cos(re))) <= -0.02) {
		tmp = fma(fma(fma(-0.0006944444444444445, (re * re), 0.020833333333333332), (re * re), -0.25), (re * re), 0.5) * fma(im_m, im_m, 2.0);
	} else {
		tmp = fma(fma(fma((im_m * im_m), 0.001388888888888889, 0.041666666666666664), (im_m * im_m), 0.5), (im_m * im_m), 1.0) * 1.0;
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (Float64(Float64(exp(im_m) + exp(Float64(-im_m))) * Float64(0.5 * cos(re))) <= -0.02)
		tmp = Float64(fma(fma(fma(-0.0006944444444444445, Float64(re * re), 0.020833333333333332), Float64(re * re), -0.25), Float64(re * re), 0.5) * fma(im_m, im_m, 2.0));
	else
		tmp = Float64(fma(fma(fma(Float64(im_m * im_m), 0.001388888888888889, 0.041666666666666664), Float64(im_m * im_m), 0.5), Float64(im_m * im_m), 1.0) * 1.0);
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[(N[(N[Exp[im$95$m], $MachinePrecision] + N[Exp[(-im$95$m)], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.02], N[(N[(N[(N[(-0.0006944444444444445 * N[(re * re), $MachinePrecision] + 0.020833333333333332), $MachinePrecision] * N[(re * re), $MachinePrecision] + -0.25), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im$95$m * im$95$m + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 76.0

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

   5. Applied rewrites 76.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 49.3

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

   8. Applied rewrites 49.3%

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

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

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-rgt-in N/A

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

      11. lift-exp.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. lift-exp.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Applied rewrites 89.0%

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

   10. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. +-commutative N/A

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

       8. *-commutative N/A

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

       9. lower-fma.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 N/A

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

       14. unpow2 N/A

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

       15. lower-*.f64 84.0

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

   12. Applied rewrites 84.0%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right), re \cdot re, -0.25\right), re \cdot re, 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.8% accurate, 0.9× speedup

Math

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

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (* (+ (exp im_m) (exp (- im_m))) (* 0.5 (cos re))) -0.02)
   (*
    (fma (fma (* im_m im_m) 0.041666666666666664 0.5) (* im_m im_m) 1.0)
    (fma -0.5 (* re re) 1.0))
   (*
    (fma
     (fma
      (fma (* im_m im_m) 0.001388888888888889 0.041666666666666664)
      (* im_m im_m)
      0.5)
     (* im_m im_m)
     1.0)
    1.0)))

C

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

Julia

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[(N[(N[Exp[im$95$m], $MachinePrecision] + N[Exp[(-im$95$m)], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.02], N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(-0.5 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. cosh-undef N/A

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

      13. associate-*r* N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 51.3

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

   7. Applied rewrites 51.3%

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

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 46.4

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

   10. Applied rewrites 46.4%

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

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

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-rgt-in N/A

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

      11. lift-exp.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. lift-exp.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Applied rewrites 89.0%

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

   10. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. +-commutative N/A

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

       8. *-commutative N/A

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

       9. lower-fma.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 N/A

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

       14. unpow2 N/A

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

       15. lower-*.f64 84.0

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

   12. Applied rewrites 84.0%

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

4. Final simplification 75.6%

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

Alternative 10: 68.2% accurate, 0.9× speedup

Math

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

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (* (+ (exp im_m) (exp (- im_m))) (* 0.5 (cos re))) -0.02)
   (* (* -0.25 (* re re)) (fma im_m im_m 2.0))
   (*
    (fma (fma (* im_m im_m) 0.041666666666666664 0.5) (* im_m im_m) 1.0)
    1.0)))

C

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

Julia

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[(N[(N[Exp[im$95$m], $MachinePrecision] + N[Exp[(-im$95$m)], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.02], N[(N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(im$95$m * im$95$m + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 76.0

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

   5. Applied rewrites 76.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 0.4

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

   8. Applied rewrites 0.4%

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

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]
   10. 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. lower-fma.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 39.8

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

   11. Applied rewrites 39.8%

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

   12. Taylor expanded in re around inf

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

   14. Applied rewrites 39.8%

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

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

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-rgt-in N/A

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

      11. lift-exp.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. lift-exp.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Applied rewrites 89.0%

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

   10. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 78.7

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

   12. Applied rewrites 78.7%

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

4. Final simplification 70.1%

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

Alternative 11: 59.2% accurate, 0.9× speedup

Math

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

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (* (+ (exp im_m) (exp (- im_m))) (* 0.5 (cos re))) -0.02)
   (* (* -0.25 (* re re)) (fma im_m im_m 2.0))
   (* 0.5 (fma im_m im_m 2.0))))

C

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

Julia

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[(N[(N[Exp[im$95$m], $MachinePrecision] + N[Exp[(-im$95$m)], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.02], N[(N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(im$95$m * im$95$m + 2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im$95$m * im$95$m + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 76.0

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

   5. Applied rewrites 76.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 0.4

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

   8. Applied rewrites 0.4%

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

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]
   10. 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. lower-fma.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 39.8

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

   11. Applied rewrites 39.8%

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

   12. Taylor expanded in re around inf

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

   14. Applied rewrites 39.8%

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

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

   1. Initial program 100.0%

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

   3. 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 77.1

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

   5. Applied rewrites 77.1%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 67.2%

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

4. Final simplification 61.1%

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

Alternative 12: 55.1% accurate, 0.9× speedup

Math

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

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (* (+ (exp im_m) (exp (- im_m))) (* 0.5 (cos re))) -0.02)
   (fma -0.5 (* re re) 1.0)
   (* 0.5 (fma im_m im_m 2.0))))

C

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

Julia

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[(N[(N[Exp[im$95$m], $MachinePrecision] + N[Exp[(-im$95$m)], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.02], N[(-0.5 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision], N[(0.5 * N[(im$95$m * im$95$m + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0

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

      1. lower-cos.f64 50.7

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

   5. Applied rewrites 50.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 23.3%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   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 77.1

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

   5. Applied rewrites 77.1%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 67.2%

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

4. Final simplification 57.4%

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

Alternative 13: 100.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \cosh im\_m \cdot \cos re \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m) :precision binary64 (* (cosh im_m) (cos re)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := N[(N[Cosh[im$95$m], $MachinePrecision] * N[Cos[re], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. distribute-lft-in N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 100.0

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

4. Applied rewrites 100.0%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\cos re \cdot \frac{1}{2}\right)} \cdot e^{-im} + \left(e^{im} \cdot \frac{1}{2}\right) \cdot \cos re \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \cos re\right)} \cdot e^{-im} + \left(e^{im} \cdot \frac{1}{2}\right) \cdot \cos re \]

   4. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \cos re\right)} \cdot e^{-im} + \left(e^{im} \cdot \frac{1}{2}\right) \cdot \cos re \]

   5. *-commutative N/A

      \[\leadsto \color{blue}{e^{-im} \cdot \left(\frac{1}{2} \cdot \cos re\right)} + \left(e^{im} \cdot \frac{1}{2}\right) \cdot \cos re \]

   6. lift-*.f64 N/A

      \[\leadsto e^{-im} \cdot \left(\frac{1}{2} \cdot \cos re\right) + \color{blue}{\left(e^{im} \cdot \frac{1}{2}\right) \cdot \cos re} \]

   7. lift-*.f64 N/A

      \[\leadsto e^{-im} \cdot \left(\frac{1}{2} \cdot \cos re\right) + \color{blue}{\left(e^{im} \cdot \frac{1}{2}\right)} \cdot \cos re \]

   8. associate-*l* N/A

      \[\leadsto e^{-im} \cdot \left(\frac{1}{2} \cdot \cos re\right) + \color{blue}{e^{im} \cdot \left(\frac{1}{2} \cdot \cos re\right)} \]

   9. lift-*.f64 N/A

      \[\leadsto e^{-im} \cdot \left(\frac{1}{2} \cdot \cos re\right) + e^{im} \cdot \color{blue}{\left(\frac{1}{2} \cdot \cos re\right)} \]

   10. distribute-rgt-in N/A

       \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)} \]

   11. lift-exp.f64 N/A

       \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\color{blue}{e^{-im}} + e^{im}\right) \]

   12. lift-neg.f64 N/A

       \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{\color{blue}{\mathsf{neg}\left(im\right)}} + e^{im}\right) \]

   13. lift-exp.f64 N/A

       \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{\mathsf{neg}\left(im\right)} + \color{blue}{e^{im}}\right) \]

   14. *-commutative N/A

       \[\leadsto \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right) \cdot \left(\frac{1}{2} \cdot \cos re\right)} \]

   15. lift-*.f64 N/A

       \[\leadsto \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \cos re\right)} \]

   16. associate-*r* N/A

       \[\leadsto \color{blue}{\left(\left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \cos re} \]

6. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\cosh im \cdot \cos re} \]
7. Add Preprocessing

Alternative 14: 70.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889 \cdot \left(re \cdot re\right), re \cdot re, -0.5\right), re \cdot re, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im\_m \cdot im\_m, 0.001388888888888889, 0.041666666666666664\right), im\_m \cdot im\_m, 0.5\right), im\_m \cdot im\_m, 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (cos re) -0.02)
   (fma (fma (* -0.001388888888888889 (* re re)) (* re re) -0.5) (* re re) 1.0)
   (*
    (fma
     (fma
      (fma (* im_m im_m) 0.001388888888888889 0.041666666666666664)
      (* im_m im_m)
      0.5)
     (* im_m im_m)
     1.0)
    1.0)))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (cos(re) <= -0.02) {
		tmp = fma(fma((-0.001388888888888889 * (re * re)), (re * re), -0.5), (re * re), 1.0);
	} else {
		tmp = fma(fma(fma((im_m * im_m), 0.001388888888888889, 0.041666666666666664), (im_m * im_m), 0.5), (im_m * im_m), 1.0) * 1.0;
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (cos(re) <= -0.02)
		tmp = fma(fma(Float64(-0.001388888888888889 * Float64(re * re)), Float64(re * re), -0.5), Float64(re * re), 1.0);
	else
		tmp = Float64(fma(fma(fma(Float64(im_m * im_m), 0.001388888888888889, 0.041666666666666664), Float64(im_m * im_m), 0.5), Float64(im_m * im_m), 1.0) * 1.0);
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[Cos[re], $MachinePrecision], -0.02], N[(N[(N[(-0.001388888888888889 * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(re * re), $MachinePrecision] + -0.5), $MachinePrecision] * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -0.0200000000000000004

   1. Initial program 99.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re} \]
   4. Step-by-step derivation

      1. lower-cos.f64 50.7

         \[\leadsto \color{blue}{\cos re} \]

   5. Applied rewrites 50.7%

      \[\leadsto \color{blue}{\cos re} \]

   6. Taylor expanded in re around 0

      \[\leadsto 1 + \color{blue}{{re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 44.2%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, -0.5\right), \color{blue}{re \cdot re}, 1\right) \]

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720} \cdot {re}^{2}, re \cdot re, \frac{-1}{2}\right), re \cdot re, 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 44.2%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889 \cdot \left(re \cdot re\right), re \cdot re, -0.5\right), re \cdot re, 1\right) \]

   if -0.0200000000000000004 < (cos.f64 re)

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(e^{-im} + e^{im}\right)} \]

      3. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot e^{-im} + \left(\frac{1}{2} \cdot \cos re\right) \cdot e^{im}} \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot e^{-im} + \color{blue}{e^{im} \cdot \left(\frac{1}{2} \cdot \cos re\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot \cos re, e^{-im}, e^{im} \cdot \left(\frac{1}{2} \cdot \cos re\right)\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{2} \cdot \cos re}, e^{-im}, e^{im} \cdot \left(\frac{1}{2} \cdot \cos re\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\cos re \cdot \frac{1}{2}}, e^{-im}, e^{im} \cdot \left(\frac{1}{2} \cdot \cos re\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\cos re \cdot \frac{1}{2}}, e^{-im}, e^{im} \cdot \left(\frac{1}{2} \cdot \cos re\right)\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\cos re \cdot \frac{1}{2}, e^{-im}, e^{im} \cdot \color{blue}{\left(\frac{1}{2} \cdot \cos re\right)}\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\cos re \cdot \frac{1}{2}, e^{-im}, \color{blue}{\left(e^{im} \cdot \frac{1}{2}\right) \cdot \cos re}\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\cos re \cdot \frac{1}{2}, e^{-im}, \color{blue}{\left(\frac{1}{2} \cdot e^{im}\right)} \cdot \cos re\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\cos re \cdot \frac{1}{2}, e^{-im}, \color{blue}{\left(\frac{1}{2} \cdot e^{im}\right) \cdot \cos re}\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\cos re \cdot \frac{1}{2}, e^{-im}, \color{blue}{\left(e^{im} \cdot \frac{1}{2}\right)} \cdot \cos re\right) \]

      14. lower-*.f64 100.0

          \[\leadsto \mathsf{fma}\left(\cos re \cdot 0.5, e^{-im}, \color{blue}{\left(e^{im} \cdot 0.5\right)} \cdot \cos re\right) \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos re \cdot 0.5, e^{-im}, \left(e^{im} \cdot 0.5\right) \cdot \cos re\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\left(\cos re \cdot \frac{1}{2}\right) \cdot e^{-im} + \left(e^{im} \cdot \frac{1}{2}\right) \cdot \cos re} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\cos re \cdot \frac{1}{2}\right)} \cdot e^{-im} + \left(e^{im} \cdot \frac{1}{2}\right) \cdot \cos re \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \cos re\right)} \cdot e^{-im} + \left(e^{im} \cdot \frac{1}{2}\right) \cdot \cos re \]

      4. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \cos re\right)} \cdot e^{-im} + \left(e^{im} \cdot \frac{1}{2}\right) \cdot \cos re \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{e^{-im} \cdot \left(\frac{1}{2} \cdot \cos re\right)} + \left(e^{im} \cdot \frac{1}{2}\right) \cdot \cos re \]

      6. lift-*.f64 N/A

         \[\leadsto e^{-im} \cdot \left(\frac{1}{2} \cdot \cos re\right) + \color{blue}{\left(e^{im} \cdot \frac{1}{2}\right) \cdot \cos re} \]

      7. lift-*.f64 N/A

         \[\leadsto e^{-im} \cdot \left(\frac{1}{2} \cdot \cos re\right) + \color{blue}{\left(e^{im} \cdot \frac{1}{2}\right)} \cdot \cos re \]

      8. associate-*l* N/A

         \[\leadsto e^{-im} \cdot \left(\frac{1}{2} \cdot \cos re\right) + \color{blue}{e^{im} \cdot \left(\frac{1}{2} \cdot \cos re\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto e^{-im} \cdot \left(\frac{1}{2} \cdot \cos re\right) + e^{im} \cdot \color{blue}{\left(\frac{1}{2} \cdot \cos re\right)} \]

      10. distribute-rgt-in N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)} \]

      11. lift-exp.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\color{blue}{e^{-im}} + e^{im}\right) \]

      12. lift-neg.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{\color{blue}{\mathsf{neg}\left(im\right)}} + e^{im}\right) \]

      13. lift-exp.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{\mathsf{neg}\left(im\right)} + \color{blue}{e^{im}}\right) \]

      14. *-commutative N/A

          \[\leadsto \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right) \cdot \left(\frac{1}{2} \cdot \cos re\right)} \]

      15. lift-*.f64 N/A

          \[\leadsto \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \cos re\right)} \]

      16. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \cos re} \]

   6. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\cosh im \cdot \cos re} \]

   7. Taylor expanded in re around 0

      \[\leadsto \cosh im \cdot \color{blue}{1} \]
   8. Step-by-step derivation

   9. Applied rewrites 89.0%

      \[\leadsto \cosh im \cdot \color{blue}{1} \]

   10. Taylor expanded in im around 0

       \[\leadsto \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \cdot 1 \]
   11. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + 1\right)} \cdot 1 \]

       2. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) \cdot {im}^{2}} + 1\right) \cdot 1 \]

       3. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right), {im}^{2}, 1\right)} \cdot 1 \]

       4. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) + \frac{1}{2}}, {im}^{2}, 1\right) \cdot 1 \]

       5. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) \cdot {im}^{2}} + \frac{1}{2}, {im}^{2}, 1\right) \cdot 1 \]

       6. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}, {im}^{2}, \frac{1}{2}\right)}, {im}^{2}, 1\right) \cdot 1 \]

       7. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{720} \cdot {im}^{2} + \frac{1}{24}}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot 1 \]

       8. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{{im}^{2} \cdot \frac{1}{720}} + \frac{1}{24}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot 1 \]

       9. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{720}, \frac{1}{24}\right)}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot 1 \]

       10. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{720}, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot 1 \]

       11. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{720}, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot 1 \]

       12. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot 1 \]

       13. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot 1 \]

       14. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), \color{blue}{im \cdot im}, 1\right) \cdot 1 \]

       15. lower-*.f64 84.0

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot im, 0.5\right), \color{blue}{im \cdot im}, 1\right) \cdot 1 \]

   12. Applied rewrites 84.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right)} \cdot 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 67.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889 \cdot \left(re \cdot re\right), re \cdot re, -0.5\right), re \cdot re, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(im\_m \cdot im\_m, 0.041666666666666664, 0.5\right), im\_m \cdot im\_m, 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (cos re) -0.02)
   (fma (fma (* -0.001388888888888889 (* re re)) (* re re) -0.5) (* re re) 1.0)
   (*
    (fma (fma (* im_m im_m) 0.041666666666666664 0.5) (* im_m im_m) 1.0)
    1.0)))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (cos(re) <= -0.02) {
		tmp = fma(fma((-0.001388888888888889 * (re * re)), (re * re), -0.5), (re * re), 1.0);
	} else {
		tmp = fma(fma((im_m * im_m), 0.041666666666666664, 0.5), (im_m * im_m), 1.0) * 1.0;
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (cos(re) <= -0.02)
		tmp = fma(fma(Float64(-0.001388888888888889 * Float64(re * re)), Float64(re * re), -0.5), Float64(re * re), 1.0);
	else
		tmp = Float64(fma(fma(Float64(im_m * im_m), 0.041666666666666664, 0.5), Float64(im_m * im_m), 1.0) * 1.0);
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[Cos[re], $MachinePrecision], -0.02], N[(N[(N[(-0.001388888888888889 * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(re * re), $MachinePrecision] + -0.5), $MachinePrecision] * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -0.0200000000000000004

   1. Initial program 99.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re} \]
   4. Step-by-step derivation

      1. lower-cos.f64 50.7

         \[\leadsto \color{blue}{\cos re} \]

   5. Applied rewrites 50.7%

      \[\leadsto \color{blue}{\cos re} \]

   6. Taylor expanded in re around 0

      \[\leadsto 1 + \color{blue}{{re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 44.2%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, -0.5\right), \color{blue}{re \cdot re}, 1\right) \]

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720} \cdot {re}^{2}, re \cdot re, \frac{-1}{2}\right), re \cdot re, 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 44.2%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889 \cdot \left(re \cdot re\right), re \cdot re, -0.5\right), re \cdot re, 1\right) \]

   if -0.0200000000000000004 < (cos.f64 re)

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(e^{-im} + e^{im}\right)} \]

      3. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot e^{-im} + \left(\frac{1}{2} \cdot \cos re\right) \cdot e^{im}} \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot e^{-im} + \color{blue}{e^{im} \cdot \left(\frac{1}{2} \cdot \cos re\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot \cos re, e^{-im}, e^{im} \cdot \left(\frac{1}{2} \cdot \cos re\right)\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{2} \cdot \cos re}, e^{-im}, e^{im} \cdot \left(\frac{1}{2} \cdot \cos re\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\cos re \cdot \frac{1}{2}}, e^{-im}, e^{im} \cdot \left(\frac{1}{2} \cdot \cos re\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\cos re \cdot \frac{1}{2}}, e^{-im}, e^{im} \cdot \left(\frac{1}{2} \cdot \cos re\right)\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\cos re \cdot \frac{1}{2}, e^{-im}, e^{im} \cdot \color{blue}{\left(\frac{1}{2} \cdot \cos re\right)}\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\cos re \cdot \frac{1}{2}, e^{-im}, \color{blue}{\left(e^{im} \cdot \frac{1}{2}\right) \cdot \cos re}\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\cos re \cdot \frac{1}{2}, e^{-im}, \color{blue}{\left(\frac{1}{2} \cdot e^{im}\right)} \cdot \cos re\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\cos re \cdot \frac{1}{2}, e^{-im}, \color{blue}{\left(\frac{1}{2} \cdot e^{im}\right) \cdot \cos re}\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\cos re \cdot \frac{1}{2}, e^{-im}, \color{blue}{\left(e^{im} \cdot \frac{1}{2}\right)} \cdot \cos re\right) \]

      14. lower-*.f64 100.0

          \[\leadsto \mathsf{fma}\left(\cos re \cdot 0.5, e^{-im}, \color{blue}{\left(e^{im} \cdot 0.5\right)} \cdot \cos re\right) \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos re \cdot 0.5, e^{-im}, \left(e^{im} \cdot 0.5\right) \cdot \cos re\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\left(\cos re \cdot \frac{1}{2}\right) \cdot e^{-im} + \left(e^{im} \cdot \frac{1}{2}\right) \cdot \cos re} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\cos re \cdot \frac{1}{2}\right)} \cdot e^{-im} + \left(e^{im} \cdot \frac{1}{2}\right) \cdot \cos re \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \cos re\right)} \cdot e^{-im} + \left(e^{im} \cdot \frac{1}{2}\right) \cdot \cos re \]

      4. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \cos re\right)} \cdot e^{-im} + \left(e^{im} \cdot \frac{1}{2}\right) \cdot \cos re \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{e^{-im} \cdot \left(\frac{1}{2} \cdot \cos re\right)} + \left(e^{im} \cdot \frac{1}{2}\right) \cdot \cos re \]

      6. lift-*.f64 N/A

         \[\leadsto e^{-im} \cdot \left(\frac{1}{2} \cdot \cos re\right) + \color{blue}{\left(e^{im} \cdot \frac{1}{2}\right) \cdot \cos re} \]

      7. lift-*.f64 N/A

         \[\leadsto e^{-im} \cdot \left(\frac{1}{2} \cdot \cos re\right) + \color{blue}{\left(e^{im} \cdot \frac{1}{2}\right)} \cdot \cos re \]

      8. associate-*l* N/A

         \[\leadsto e^{-im} \cdot \left(\frac{1}{2} \cdot \cos re\right) + \color{blue}{e^{im} \cdot \left(\frac{1}{2} \cdot \cos re\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto e^{-im} \cdot \left(\frac{1}{2} \cdot \cos re\right) + e^{im} \cdot \color{blue}{\left(\frac{1}{2} \cdot \cos re\right)} \]

      10. distribute-rgt-in N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)} \]

      11. lift-exp.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\color{blue}{e^{-im}} + e^{im}\right) \]

      12. lift-neg.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{\color{blue}{\mathsf{neg}\left(im\right)}} + e^{im}\right) \]

      13. lift-exp.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{\mathsf{neg}\left(im\right)} + \color{blue}{e^{im}}\right) \]

      14. *-commutative N/A

          \[\leadsto \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right) \cdot \left(\frac{1}{2} \cdot \cos re\right)} \]

      15. lift-*.f64 N/A

          \[\leadsto \left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \cos re\right)} \]

      16. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \cos re} \]

   6. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\cosh im \cdot \cos re} \]

   7. Taylor expanded in re around 0

      \[\leadsto \cosh im \cdot \color{blue}{1} \]
   8. Step-by-step derivation

   9. Applied rewrites 89.0%

      \[\leadsto \cosh im \cdot \color{blue}{1} \]

   10. Taylor expanded in im around 0

       \[\leadsto \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \cdot 1 \]
   11. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) + 1\right)} \cdot 1 \]

       2. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot {im}^{2}} + 1\right) \cdot 1 \]

       3. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}, {im}^{2}, 1\right)} \cdot 1 \]

       4. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}}, {im}^{2}, 1\right) \cdot 1 \]

       5. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{{im}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, {im}^{2}, 1\right) \cdot 1 \]

       6. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{24}, \frac{1}{2}\right)}, {im}^{2}, 1\right) \cdot 1 \]

       7. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{24}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot 1 \]

       8. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{24}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot 1 \]

       9. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{24}, \frac{1}{2}\right), \color{blue}{im \cdot im}, 1\right) \cdot 1 \]

       10. lower-*.f64 78.7

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), \color{blue}{im \cdot im}, 1\right) \cdot 1 \]

   12. Applied rewrites 78.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), im \cdot im, 1\right)} \cdot 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 35.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(-0.5, re \cdot re, 1\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (cos re) -0.02) (fma -0.5 (* re re) 1.0) 1.0))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (cos(re) <= -0.02) {
		tmp = fma(-0.5, (re * re), 1.0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (cos(re) <= -0.02)
		tmp = fma(-0.5, Float64(re * re), 1.0);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[Cos[re], $MachinePrecision], -0.02], N[(-0.5 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -0.0200000000000000004

   1. Initial program 99.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re} \]
   4. Step-by-step derivation

      1. lower-cos.f64 50.7

         \[\leadsto \color{blue}{\cos re} \]

   5. Applied rewrites 50.7%

      \[\leadsto \color{blue}{\cos re} \]

   6. Taylor expanded in re around 0

      \[\leadsto 1 + \color{blue}{\frac{-1}{2} \cdot {re}^{2}} \]
   7. Step-by-step derivation

   8. Applied rewrites 23.3%

      \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{re \cdot re}, 1\right) \]

   if -0.0200000000000000004 < (cos.f64 re)

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re} \]
   4. Step-by-step derivation

      1. lower-cos.f64 43.6

         \[\leadsto \color{blue}{\cos re} \]

   5. Applied rewrites 43.6%

      \[\leadsto \color{blue}{\cos re} \]

   6. Taylor expanded in re around 0

      \[\leadsto 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 33.8%

      \[\leadsto 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 28.5% accurate, 316.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ 1 \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m) :precision binary64 1.0)

C

im_m = fabs(im);
double code(double re, double im_m) {
	return 1.0;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = 1.0d0
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	return 1.0;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	return 1.0

Julia

im_m = abs(im)
function code(re, im_m)
	return 1.0
end

MATLAB

im_m = abs(im);
function tmp = code(re, im_m)
	tmp = 1.0;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := 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 45.2

      \[\leadsto \color{blue}{\cos re} \]

5. Applied rewrites 45.2%

   \[\leadsto \color{blue}{\cos re} \]

6. Taylor expanded in re around 0

   \[\leadsto 1 \]
7. Step-by-step derivation

8. Applied rewrites 26.5%

   \[\leadsto 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024259 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))