math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 13.0s

Alternatives: 24

Speedup: 1.5×

• Could not converge on a ground truth

  1. re = 6.811366093039857e-263
  2. im = 1.2862164107072412e+73

• 49.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{0.5 \cdot \cos re}{\frac{1}{2 \cdot \cosh im}} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (/ (* 0.5 (cos re)) (/ 1.0 (* 2.0 (cosh im)))))

C

double code(double re, double im) {
	return (0.5 * cos(re)) / (1.0 / (2.0 * cosh(im)));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * cos(re)) / (1.0d0 / (2.0d0 * cosh(im)))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.cos(re)) / (1.0 / (2.0 * Math.cosh(im)));
}

Python

def code(re, im):
	return (0.5 * math.cos(re)) / (1.0 / (2.0 * math.cosh(im)))

Julia

function code(re, im)
	return Float64(Float64(0.5 * cos(re)) / Float64(1.0 / Float64(2.0 * cosh(im))))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * cos(re)) / (1.0 / (2.0 * cosh(im)));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[(2.0 * N[Cosh[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. lift-cos.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-neg.f64 N/A

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

   4. lift-exp.f64 N/A

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

   5. lift-exp.f64 N/A

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

   6. flip-+ N/A

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

   7. clear-num N/A

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

   8. un-div-inv N/A

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

   9. lower-/.f64 N/A

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

   10. clear-num N/A

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

   11. flip-+ N/A

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

   12. lift-+.f64 N/A

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

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{\frac{1}{2 \cdot \cosh im}}} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\cosh im \cdot \mathsf{fma}\left(re \cdot re, -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq 0.9999999999971182:\\ \;\;\;\;\cos re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (* (cosh im) (fma (* re re) -0.5 1.0))
     (if (<= t_0 0.9999999999971182)
       (*
        (cos re)
        (fma
         (* im im)
         (fma
          (* im im)
          (fma (* im im) 0.001388888888888889 0.041666666666666664)
          0.5)
         1.0))
       (cosh im)))))

C

double code(double re, double im) {
	double t_0 = (0.5 * cos(re)) * (exp(-im) + exp(im));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = cosh(im) * fma((re * re), -0.5, 1.0);
	} else if (t_0 <= 0.9999999999971182) {
		tmp = cos(re) * fma((im * im), fma((im * im), fma((im * im), 0.001388888888888889, 0.041666666666666664), 0.5), 1.0);
	} else {
		tmp = cosh(im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(cosh(im) * fma(Float64(re * re), -0.5, 1.0));
	elseif (t_0 <= 0.9999999999971182)
		tmp = Float64(cos(re) * fma(Float64(im * im), fma(Float64(im * im), fma(Float64(im * im), 0.001388888888888889, 0.041666666666666664), 0.5), 1.0));
	else
		tmp = cosh(im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[Cosh[im], $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.9999999999971182], N[(N[Cos[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Cosh[im], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. flip-+ N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. clear-num N/A

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

      11. flip-+ N/A

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

      12. lift-+.f64 N/A

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{\frac{1}{2 \cdot \cosh im}}} \]
   5. Step-by-step derivation

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cosh.f64 N/A

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

      4. associate-/r* N/A

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

      5. metadata-eval N/A

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

      6. associate-/r/ N/A

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

      7. div-inv N/A

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

      8. metadata-eval N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. metadata-eval N/A

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

      13. *-rgt-identity N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 100.0

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 100.0

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

   9. Simplified 100.0%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.999999999997118194

   1. Initial program 100.0%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 99.7

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

   7. Simplified 99.7%

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

   if 0.999999999997118194 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Simplified 99.7%

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

      1. lift-cosh.f64 N/A

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

      2. *-lft-identity N/A

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

      3. *-rgt-identity 99.7

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

   9. Applied egg-rr 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\cosh im \cdot \mathsf{fma}\left(re \cdot re, -0.5, 1\right)\\ \mathbf{elif}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.9999999999971182:\\ \;\;\;\;\cos re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024218 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))