math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 10.4s

Alternatives: 15

Speedup: 1.0×

• 25.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \cos im \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 99.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+184}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, -0.08333333333333333, -0.25\right), -0.5\right), -0.5\right)\\ \mathbf{elif}\;t\_0 \leq -0.01:\\ \;\;\;\;\cos im \cdot \mathsf{fma}\left(\mathsf{fma}\left(re, re, -1\right), \frac{1}{re + -1}, re \cdot \left(re \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;t\_0 \leq 20:\\ \;\;\;\;\cos im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 -5e+184)
     (*
      (* im im)
      (fma re (fma re (fma re -0.08333333333333333 -0.25) -0.5) -0.5))
     (if (<= t_0 -0.01)
       (*
        (cos im)
        (fma
         (fma re re -1.0)
         (/ 1.0 (+ re -1.0))
         (* re (* re (fma re 0.16666666666666666 0.5)))))
       (if (<= t_0 0.0)
         (exp re)
         (if (<= t_0 20.0)
           (*
            (cos im)
            (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0))
           (exp re)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -5e+184) {
		tmp = (im * im) * fma(re, fma(re, fma(re, -0.08333333333333333, -0.25), -0.5), -0.5);
	} else if (t_0 <= -0.01) {
		tmp = cos(im) * fma(fma(re, re, -1.0), (1.0 / (re + -1.0)), (re * (re * fma(re, 0.16666666666666666, 0.5))));
	} else if (t_0 <= 0.0) {
		tmp = exp(re);
	} else if (t_0 <= 20.0) {
		tmp = cos(im) * fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0);
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= -5e+184)
		tmp = Float64(Float64(im * im) * fma(re, fma(re, fma(re, -0.08333333333333333, -0.25), -0.5), -0.5));
	elseif (t_0 <= -0.01)
		tmp = Float64(cos(im) * fma(fma(re, re, -1.0), Float64(1.0 / Float64(re + -1.0)), Float64(re * Float64(re * fma(re, 0.16666666666666666, 0.5)))));
	elseif (t_0 <= 0.0)
		tmp = exp(re);
	elseif (t_0 <= 20.0)
		tmp = Float64(cos(im) * fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0));
	else
		tmp = exp(re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+184], N[(N[(im * im), $MachinePrecision] * N[(re * N[(re * N[(re * -0.08333333333333333 + -0.25), $MachinePrecision] + -0.5), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.01], N[(N[Cos[im], $MachinePrecision] * N[(N[(re * re + -1.0), $MachinePrecision] * N[(1.0 / N[(re + -1.0), $MachinePrecision]), $MachinePrecision] + N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[Exp[re], $MachinePrecision], If[LessEqual[t$95$0, 20.0], N[(N[Cos[im], $MachinePrecision] * N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Exp[re], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -4.9999999999999999e184

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. *-lft-identity N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 100.0

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

   8. Simplified 100.0%

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

   9. Taylor expanded in re around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

       3. lower-fma.f64 N/A

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

       4. sub-neg N/A

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

       5. metadata-eval N/A

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

       6. lower-fma.f64 N/A

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

       7. sub-neg N/A

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

       8. *-commutative N/A

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

       9. metadata-eval N/A

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

       10. lower-fma.f64 92.6

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

   11. Simplified 92.6%

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

   if -4.9999999999999999e184 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0100000000000000002

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + 1\right)} \cdot \cos im \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, 1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right), 1\right)} \cdot \cos im \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right) + 1}, 1\right) \cdot \cos im \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, \frac{1}{2} + \frac{1}{6} \cdot re, 1\right)}, 1\right) \cdot \cos im \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\frac{1}{6} \cdot re + \frac{1}{2}}, 1\right), 1\right) \cdot \cos im \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{re \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right) \cdot \cos im \]

      7. lower-fma.f64 99.4

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \cdot \cos im \]

   5. Simplified 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \cdot \cos im \]
   6. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \left(re \cdot \left(re \cdot \color{blue}{\mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right)} + 1\right) + 1\right) \cdot \cos im \]

      2. lift-fma.f64 N/A

         \[\leadsto \left(re \cdot \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right)} + 1\right) \cdot \cos im \]

      3. *-rgt-identity N/A

         \[\leadsto \left(\color{blue}{\left(re \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right)\right) \cdot 1} + 1\right) \cdot \cos im \]

      4. *-rgt-identity N/A

         \[\leadsto \left(\color{blue}{re \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right)} + 1\right) \cdot \cos im \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(1 + re \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right)\right)} \cdot \cos im \]

      6. *-commutative N/A

         \[\leadsto \left(1 + \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right) \cdot re}\right) \cdot \cos im \]

      7. lift-fma.f64 N/A

         \[\leadsto \left(1 + \color{blue}{\left(re \cdot \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right) + 1\right)} \cdot re\right) \cdot \cos im \]

      8. distribute-rgt1-in N/A

         \[\leadsto \left(1 + \color{blue}{\left(re + \left(re \cdot \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right)\right) \cdot re\right)}\right) \cdot \cos im \]

      9. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(1 + re\right) + \left(re \cdot \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right)\right) \cdot re\right)} \cdot \cos im \]

      10. +-commutative N/A

          \[\leadsto \left(\color{blue}{\left(re + 1\right)} + \left(re \cdot \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right)\right) \cdot re\right) \cdot \cos im \]

      11. flip-+ N/A

          \[\leadsto \left(\color{blue}{\frac{re \cdot re - 1 \cdot 1}{re - 1}} + \left(re \cdot \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right)\right) \cdot re\right) \cdot \cos im \]

      12. div-inv N/A

          \[\leadsto \left(\color{blue}{\left(re \cdot re - 1 \cdot 1\right) \cdot \frac{1}{re - 1}} + \left(re \cdot \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right)\right) \cdot re\right) \cdot \cos im \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot re - 1 \cdot 1, \frac{1}{re - 1}, \left(re \cdot \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right)\right) \cdot re\right)} \cdot \cos im \]

   7. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(re, re, -1\right), \frac{1}{re + -1}, re \cdot \left(re \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)\right)\right)} \cdot \cos im \]

   if -0.0100000000000000002 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0 or 20 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 99.6

         \[\leadsto \color{blue}{e^{re}} \]

   5. Simplified 99.6%

      \[\leadsto \color{blue}{e^{re}} \]

   if -0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 20

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + 1\right)} \cdot \cos im \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, 1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right), 1\right)} \cdot \cos im \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right) + 1}, 1\right) \cdot \cos im \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, \frac{1}{2} + \frac{1}{6} \cdot re, 1\right)}, 1\right) \cdot \cos im \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\frac{1}{6} \cdot re + \frac{1}{2}}, 1\right), 1\right) \cdot \cos im \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{re \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right) \cdot \cos im \]

      7. lower-fma.f64 99.3

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \cdot \cos im \]

   5. Simplified 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \cdot \cos im \]
3. Recombined 4 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -5 \cdot 10^{+184}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, -0.08333333333333333, -0.25\right), -0.5\right), -0.5\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -0.01:\\ \;\;\;\;\cos im \cdot \mathsf{fma}\left(\mathsf{fma}\left(re, re, -1\right), \frac{1}{re + -1}, re \cdot \left(re \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)\right)\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 20:\\ \;\;\;\;\cos im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
5. Add Preprocessing

Reproduce

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