math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 10.7s

Alternatives: 22

Speedup: 1.0×

• 24.8% 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: 97.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ t_1 := \cos im \cdot \left(re + 1\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, -0.08333333333333333, 0.16666666666666666\right)\right)\\ \mathbf{elif}\;t\_0 \leq -0.1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;t\_0 \leq 0.995:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))) (t_1 (* (cos im) (+ re 1.0))))
   (if (<= t_0 (- INFINITY))
     (*
      (* re re)
      (* re (fma (* im im) -0.08333333333333333 0.16666666666666666)))
     (if (<= t_0 -0.1)
       t_1
       (if (<= t_0 0.0) (exp re) (if (<= t_0 0.995) t_1 (exp re)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double t_1 = cos(im) * (re + 1.0);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (re * re) * (re * fma((im * im), -0.08333333333333333, 0.16666666666666666));
	} else if (t_0 <= -0.1) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = exp(re);
	} else if (t_0 <= 0.995) {
		tmp = t_1;
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	t_1 = Float64(cos(im) * Float64(re + 1.0))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(re * re) * Float64(re * fma(Float64(im * im), -0.08333333333333333, 0.16666666666666666)));
	elseif (t_0 <= -0.1)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = exp(re);
	elseif (t_0 <= 0.995)
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[Cos[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(re * re), $MachinePrecision] * N[(re * N[(N[(im * im), $MachinePrecision] * -0.08333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.1], t$95$1, If[LessEqual[t$95$0, 0.0], N[Exp[re], $MachinePrecision], If[LessEqual[t$95$0, 0.995], t$95$1, N[Exp[re], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -inf.0

   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 88.9

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

   5. Simplified 88.9%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 100.0

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

   8. Simplified 100.0%

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

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\left({re}^{3} \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right)\right)\right)} \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]
   10. Step-by-step derivation

       1. cube-mult N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. associate-+r+ N/A

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

       5. distribute-lft-in N/A

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

       6. rgt-mult-inverse N/A

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

       7. unpow2 N/A

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

       8. associate-*l* N/A

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

       9. +-commutative N/A

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

       10. distribute-rgt-in N/A

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

       11. associate-*l* N/A

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

       12. lft-mult-inverse N/A

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

       13. metadata-eval N/A

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

   11. Simplified 100.0%

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

   12. Taylor expanded in re around inf

       \[\leadsto \color{blue}{\frac{1}{6} \cdot \left({re}^{3} \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right)} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{1}{6} \cdot \color{blue}{\left(\left(1 + \frac{-1}{2} \cdot {im}^{2}\right) \cdot {re}^{3}\right)} \]

       2. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right) \cdot {re}^{3}} \]

       3. cube-mult N/A

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

       4. unpow2 N/A

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

       5. associate-*r* N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. *-commutative N/A

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

       13. associate-*l* N/A

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

       14. *-commutative N/A

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

       15. lower-*.f64 N/A

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

       16. distribute-rgt-in N/A

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

       17. metadata-eval N/A

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

       18. +-commutative N/A

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

       19. *-commutative N/A

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

       20. associate-*r* N/A

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

       21. *-commutative N/A

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

       22. lower-fma.f64 N/A

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

   14. Simplified 100.0%

       \[\leadsto \color{blue}{\left(re \cdot re\right) \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, -0.08333333333333333, 0.16666666666666666\right)\right)} \]

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.10000000000000001 or 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.994999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 99.1

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

   5. Simplified 99.1%

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

   if -0.10000000000000001 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0 or 0.994999999999999996 < (*.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 100.0

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

   5. Simplified 100.0%

      \[\leadsto \color{blue}{e^{re}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, -0.08333333333333333, 0.16666666666666666\right)\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -0.1:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0.995:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, -0.08333333333333333, 0.16666666666666666\right)\right)\\ \mathbf{elif}\;t\_0 \leq -0.1:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;t\_0 \leq 0.995:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (*
      (* re re)
      (* re (fma (* im im) -0.08333333333333333 0.16666666666666666)))
     (if (<= t_0 -0.1)
       (cos im)
       (if (<= t_0 0.0) (exp re) (if (<= t_0 0.995) (cos im) (exp re)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (re * re) * (re * fma((im * im), -0.08333333333333333, 0.16666666666666666));
	} else if (t_0 <= -0.1) {
		tmp = cos(im);
	} else if (t_0 <= 0.0) {
		tmp = exp(re);
	} else if (t_0 <= 0.995) {
		tmp = cos(im);
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(re * re) * Float64(re * fma(Float64(im * im), -0.08333333333333333, 0.16666666666666666)));
	elseif (t_0 <= -0.1)
		tmp = cos(im);
	elseif (t_0 <= 0.0)
		tmp = exp(re);
	elseif (t_0 <= 0.995)
		tmp = cos(im);
	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, (-Infinity)], N[(N[(re * re), $MachinePrecision] * N[(re * N[(N[(im * im), $MachinePrecision] * -0.08333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.1], N[Cos[im], $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[Exp[re], $MachinePrecision], If[LessEqual[t$95$0, 0.995], N[Cos[im], $MachinePrecision], N[Exp[re], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -inf.0

   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 88.9

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

   5. Simplified 88.9%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 100.0

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

   8. Simplified 100.0%

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

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\left({re}^{3} \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right)\right)\right)} \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]
   10. Step-by-step derivation

       1. cube-mult N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. associate-+r+ N/A

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

       5. distribute-lft-in N/A

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

       6. rgt-mult-inverse N/A

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

       7. unpow2 N/A

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

       8. associate-*l* N/A

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

       9. +-commutative N/A

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

       10. distribute-rgt-in N/A

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

       11. associate-*l* N/A

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

       12. lft-mult-inverse N/A

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

       13. metadata-eval N/A

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

   11. Simplified 100.0%

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

   12. Taylor expanded in re around inf

       \[\leadsto \color{blue}{\frac{1}{6} \cdot \left({re}^{3} \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right)} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{1}{6} \cdot \color{blue}{\left(\left(1 + \frac{-1}{2} \cdot {im}^{2}\right) \cdot {re}^{3}\right)} \]

       2. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right) \cdot {re}^{3}} \]

       3. cube-mult N/A

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

       4. unpow2 N/A

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

       5. associate-*r* N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. *-commutative N/A

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

       13. associate-*l* N/A

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

       14. *-commutative N/A

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

       15. lower-*.f64 N/A

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

       16. distribute-rgt-in N/A

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

       17. metadata-eval N/A

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

       18. +-commutative N/A

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

       19. *-commutative N/A

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

       20. associate-*r* N/A

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

       21. *-commutative N/A

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

       22. lower-fma.f64 N/A

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

   14. Simplified 100.0%

       \[\leadsto \color{blue}{\left(re \cdot re\right) \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, -0.08333333333333333, 0.16666666666666666\right)\right)} \]

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.10000000000000001 or 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.994999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-cos.f64 97.8

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

   5. Simplified 97.8%

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

   if -0.10000000000000001 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0 or 0.994999999999999996 < (*.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 100.0

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

   5. Simplified 100.0%

      \[\leadsto \color{blue}{e^{re}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 95.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ t_1 := \cos im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{if}\;t\_0 \leq -0.1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;t\_0 \leq 0.995:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im)))
        (t_1
         (*
          (cos im)
          (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0))))
   (if (<= t_0 -0.1)
     t_1
     (if (<= t_0 0.0) (exp re) (if (<= t_0 0.995) t_1 (exp re))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double t_1 = cos(im) * fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0);
	double tmp;
	if (t_0 <= -0.1) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = exp(re);
	} else if (t_0 <= 0.995) {
		tmp = t_1;
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	t_1 = Float64(cos(im) * fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0))
	tmp = 0.0
	if (t_0 <= -0.1)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = exp(re);
	elseif (t_0 <= 0.995)
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[Cos[im], $MachinePrecision] * N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.1], t$95$1, If[LessEqual[t$95$0, 0.0], N[Exp[re], $MachinePrecision], If[LessEqual[t$95$0, 0.995], t$95$1, N[Exp[re], $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.10000000000000001 or 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.994999999999999996

   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 97.6

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

   5. Simplified 97.6%

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

   if -0.10000000000000001 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0 or 0.994999999999999996 < (*.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 100.0

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

   5. Simplified 100.0%

      \[\leadsto \color{blue}{e^{re}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -0.1:\\ \;\;\;\;\cos im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0.995:\\ \;\;\;\;\cos im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ t_1 := \cos im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{if}\;t\_0 \leq -0.1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;t\_0 \leq 0.995:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im)))
        (t_1 (* (cos im) (fma re (fma re 0.5 1.0) 1.0))))
   (if (<= t_0 -0.1)
     t_1
     (if (<= t_0 0.0) (exp re) (if (<= t_0 0.995) t_1 (exp re))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double t_1 = cos(im) * fma(re, fma(re, 0.5, 1.0), 1.0);
	double tmp;
	if (t_0 <= -0.1) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = exp(re);
	} else if (t_0 <= 0.995) {
		tmp = t_1;
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	t_1 = Float64(cos(im) * fma(re, fma(re, 0.5, 1.0), 1.0))
	tmp = 0.0
	if (t_0 <= -0.1)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = exp(re);
	elseif (t_0 <= 0.995)
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[Cos[im], $MachinePrecision] * N[(re * N[(re * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.1], t$95$1, If[LessEqual[t$95$0, 0.0], N[Exp[re], $MachinePrecision], If[LessEqual[t$95$0, 0.995], t$95$1, N[Exp[re], $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.10000000000000001 or 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.994999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 95.1

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

   5. Simplified 95.1%

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

   if -0.10000000000000001 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0 or 0.994999999999999996 < (*.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 100.0

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

   5. Simplified 100.0%

      \[\leadsto \color{blue}{e^{re}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -0.1:\\ \;\;\;\;\cos im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0.995:\\ \;\;\;\;\cos im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 56.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -0.1:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right) \cdot \mathsf{fma}\left(im, im \cdot -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(0.16666666666666666, re, 0.5\right), 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 -0.1)
     (*
      (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0)
      (fma im (* im -0.5) 1.0))
     (if (<= t_0 0.0)
       (* 0.041666666666666664 (* (* im im) (* im im)))
       (fma re (fma re (fma 0.16666666666666666 re 0.5) 1.0) 1.0)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -0.1) {
		tmp = fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0) * fma(im, (im * -0.5), 1.0);
	} else if (t_0 <= 0.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else {
		tmp = fma(re, fma(re, fma(0.16666666666666666, re, 0.5), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= -0.1)
		tmp = Float64(fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0) * fma(im, Float64(im * -0.5), 1.0));
	elseif (t_0 <= 0.0)
		tmp = Float64(0.041666666666666664 * Float64(Float64(im * im) * Float64(im * im)));
	else
		tmp = fma(re, fma(re, fma(0.16666666666666666, re, 0.5), 1.0), 1.0);
	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, -0.1], N[(N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im * N[(im * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(0.041666666666666664 * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(re * N[(0.16666666666666666 * re + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.10000000000000001

   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 96.8

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

   5. Simplified 96.8%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 31.7

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

   8. Simplified 31.7%

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

   if -0.10000000000000001 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-cos.f64 3.1

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

   5. Simplified 3.1%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 2.5

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

   8. Simplified 2.5%

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

   9. Taylor expanded in im around inf

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

       1. lower-*.f64 N/A

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 37.4

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

   11. Simplified 37.4%

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

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

   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 88.3

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

   5. Simplified 88.3%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 74.6

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

   8. Simplified 74.6%

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

Alternative 7: 56.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -0.1:\\ \;\;\;\;\mathsf{fma}\left(im, im \cdot -0.5, 1\right) \cdot \left(re \cdot \left(re \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(0.16666666666666666, re, 0.5\right), 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 -0.1)
     (*
      (fma im (* im -0.5) 1.0)
      (* re (* re (fma re 0.16666666666666666 0.5))))
     (if (<= t_0 0.0)
       (* 0.041666666666666664 (* (* im im) (* im im)))
       (fma re (fma re (fma 0.16666666666666666 re 0.5) 1.0) 1.0)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -0.1) {
		tmp = fma(im, (im * -0.5), 1.0) * (re * (re * fma(re, 0.16666666666666666, 0.5)));
	} else if (t_0 <= 0.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else {
		tmp = fma(re, fma(re, fma(0.16666666666666666, re, 0.5), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= -0.1)
		tmp = Float64(fma(im, Float64(im * -0.5), 1.0) * Float64(re * Float64(re * fma(re, 0.16666666666666666, 0.5))));
	elseif (t_0 <= 0.0)
		tmp = Float64(0.041666666666666664 * Float64(Float64(im * im) * Float64(im * im)));
	else
		tmp = fma(re, fma(re, fma(0.16666666666666666, re, 0.5), 1.0), 1.0);
	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, -0.1], N[(N[(im * N[(im * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(0.041666666666666664 * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(re * N[(0.16666666666666666 * re + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.10000000000000001

   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 96.8

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

   5. Simplified 96.8%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 31.7

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

   8. Simplified 31.7%

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

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\left({re}^{3} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)} \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]
   10. Step-by-step derivation

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. unpow2 N/A

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

       5. +-commutative N/A

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

       6. distribute-rgt-in N/A

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

       7. associate-*l* N/A

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

       8. lft-mult-inverse N/A

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

       9. metadata-eval N/A

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

       10. associate-*r* N/A

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

       11. lower-*.f64 N/A

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

       12. lower-*.f64 N/A

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

       13. +-commutative N/A

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

       14. *-commutative N/A

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

       15. lower-fma.f64 31.4

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

   11. Simplified 31.4%

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

   if -0.10000000000000001 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-cos.f64 3.1

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

   5. Simplified 3.1%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 2.5

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

   8. Simplified 2.5%

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

   9. Taylor expanded in im around inf

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

       1. lower-*.f64 N/A

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 37.4

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

   11. Simplified 37.4%

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

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

   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 88.3

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

   5. Simplified 88.3%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 74.6

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

   8. Simplified 74.6%

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

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -0.1:\\ \;\;\;\;\mathsf{fma}\left(im, im \cdot -0.5, 1\right) \cdot \left(re \cdot \left(re \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)\right)\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(0.16666666666666666, re, 0.5\right), 1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -0.1:\\ \;\;\;\;im \cdot \left(im \cdot \left(-0.5 \cdot \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), re\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(0.16666666666666666, re, 0.5\right), 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 -0.1)
     (* im (* im (* -0.5 (fma re (* re (fma re 0.16666666666666666 0.5)) re))))
     (if (<= t_0 0.0)
       (* 0.041666666666666664 (* (* im im) (* im im)))
       (fma re (fma re (fma 0.16666666666666666 re 0.5) 1.0) 1.0)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -0.1) {
		tmp = im * (im * (-0.5 * fma(re, (re * fma(re, 0.16666666666666666, 0.5)), re)));
	} else if (t_0 <= 0.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else {
		tmp = fma(re, fma(re, fma(0.16666666666666666, re, 0.5), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= -0.1)
		tmp = Float64(im * Float64(im * Float64(-0.5 * fma(re, Float64(re * fma(re, 0.16666666666666666, 0.5)), re))));
	elseif (t_0 <= 0.0)
		tmp = Float64(0.041666666666666664 * Float64(Float64(im * im) * Float64(im * im)));
	else
		tmp = fma(re, fma(re, fma(0.16666666666666666, re, 0.5), 1.0), 1.0);
	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, -0.1], N[(im * N[(im * N[(-0.5 * N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(0.041666666666666664 * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(re * N[(0.16666666666666666 * re + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.10000000000000001

   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 96.8

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

   5. Simplified 96.8%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 31.7

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

   8. Simplified 31.7%

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

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\left({re}^{3} \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right)\right)\right)} \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]
   10. Step-by-step derivation

       1. cube-mult N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. associate-+r+ N/A

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

       5. distribute-lft-in N/A

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

       6. rgt-mult-inverse N/A

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

       7. unpow2 N/A

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

       8. associate-*l* N/A

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

       9. +-commutative N/A

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

       10. distribute-rgt-in N/A

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

       11. associate-*l* N/A

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

       12. lft-mult-inverse N/A

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

       13. metadata-eval N/A

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

   11. Simplified 31.2%

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

   12. Taylor expanded in im around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. unpow2 N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. +-commutative N/A

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

       11. unpow2 N/A

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

       12. associate-*r* N/A

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

       13. lower-fma.f64 N/A

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

       14. lower-*.f64 N/A

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

       15. +-commutative N/A

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

       16. *-commutative N/A

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

       17. lower-fma.f64 31.3

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

   14. Simplified 31.3%

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

   if -0.10000000000000001 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-cos.f64 3.1

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

   5. Simplified 3.1%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 2.5

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

   8. Simplified 2.5%

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

   9. Taylor expanded in im around inf

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

       1. lower-*.f64 N/A

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 37.4

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

   11. Simplified 37.4%

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

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

   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 88.3

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

   5. Simplified 88.3%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 74.6

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

   8. Simplified 74.6%

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

Alternative 9: 56.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -0.1:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, -0.08333333333333333, 0.16666666666666666\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(0.16666666666666666, re, 0.5\right), 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 -0.1)
     (*
      (* re re)
      (* re (fma (* im im) -0.08333333333333333 0.16666666666666666)))
     (if (<= t_0 0.0)
       (* 0.041666666666666664 (* (* im im) (* im im)))
       (fma re (fma re (fma 0.16666666666666666 re 0.5) 1.0) 1.0)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -0.1) {
		tmp = (re * re) * (re * fma((im * im), -0.08333333333333333, 0.16666666666666666));
	} else if (t_0 <= 0.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else {
		tmp = fma(re, fma(re, fma(0.16666666666666666, re, 0.5), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= -0.1)
		tmp = Float64(Float64(re * re) * Float64(re * fma(Float64(im * im), -0.08333333333333333, 0.16666666666666666)));
	elseif (t_0 <= 0.0)
		tmp = Float64(0.041666666666666664 * Float64(Float64(im * im) * Float64(im * im)));
	else
		tmp = fma(re, fma(re, fma(0.16666666666666666, re, 0.5), 1.0), 1.0);
	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, -0.1], N[(N[(re * re), $MachinePrecision] * N[(re * N[(N[(im * im), $MachinePrecision] * -0.08333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(0.041666666666666664 * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(re * N[(0.16666666666666666 * re + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.10000000000000001

   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 96.8

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

   5. Simplified 96.8%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 31.7

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

   8. Simplified 31.7%

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

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\left({re}^{3} \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right)\right)\right)} \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]
   10. Step-by-step derivation

       1. cube-mult N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. associate-+r+ N/A

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

       5. distribute-lft-in N/A

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

       6. rgt-mult-inverse N/A

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

       7. unpow2 N/A

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

       8. associate-*l* N/A

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

       9. +-commutative N/A

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

       10. distribute-rgt-in N/A

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

       11. associate-*l* N/A

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

       12. lft-mult-inverse N/A

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

       13. metadata-eval N/A

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

   11. Simplified 31.2%

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

   12. Taylor expanded in re around inf

       \[\leadsto \color{blue}{\frac{1}{6} \cdot \left({re}^{3} \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right)} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{1}{6} \cdot \color{blue}{\left(\left(1 + \frac{-1}{2} \cdot {im}^{2}\right) \cdot {re}^{3}\right)} \]

       2. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right) \cdot {re}^{3}} \]

       3. cube-mult N/A

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

       4. unpow2 N/A

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

       5. associate-*r* N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. *-commutative N/A

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

       13. associate-*l* N/A

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

       14. *-commutative N/A

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

       15. lower-*.f64 N/A

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

       16. distribute-rgt-in N/A

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

       17. metadata-eval N/A

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

       18. +-commutative N/A

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

       19. *-commutative N/A

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

       20. associate-*r* N/A

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

       21. *-commutative N/A

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

       22. lower-fma.f64 N/A

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

   14. Simplified 30.8%

       \[\leadsto \color{blue}{\left(re \cdot re\right) \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, -0.08333333333333333, 0.16666666666666666\right)\right)} \]

   if -0.10000000000000001 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-cos.f64 3.1

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

   5. Simplified 3.1%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 2.5

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

   8. Simplified 2.5%

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

   9. Taylor expanded in im around inf

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

       1. lower-*.f64 N/A

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 37.4

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

   11. Simplified 37.4%

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

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

   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 88.3

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

   5. Simplified 88.3%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 74.6

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

   8. Simplified 74.6%

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

Alternative 10: 56.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -0.1:\\ \;\;\;\;re \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, -0.25, 0.5\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(0.16666666666666666, re, 0.5\right), 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 -0.1)
     (* re (* re (fma (* im im) -0.25 0.5)))
     (if (<= t_0 0.0)
       (* 0.041666666666666664 (* (* im im) (* im im)))
       (fma re (fma re (fma 0.16666666666666666 re 0.5) 1.0) 1.0)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -0.1) {
		tmp = re * (re * fma((im * im), -0.25, 0.5));
	} else if (t_0 <= 0.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else {
		tmp = fma(re, fma(re, fma(0.16666666666666666, re, 0.5), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= -0.1)
		tmp = Float64(re * Float64(re * fma(Float64(im * im), -0.25, 0.5)));
	elseif (t_0 <= 0.0)
		tmp = Float64(0.041666666666666664 * Float64(Float64(im * im) * Float64(im * im)));
	else
		tmp = fma(re, fma(re, fma(0.16666666666666666, re, 0.5), 1.0), 1.0);
	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, -0.1], N[(re * N[(re * N[(N[(im * im), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(0.041666666666666664 * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(re * N[(0.16666666666666666 * re + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.10000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 93.6

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

   5. Simplified 93.6%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 30.2

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

   8. Simplified 30.2%

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

   9. Taylor expanded in re around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. unpow2 N/A

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

       4. associate-*r* N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. *-commutative N/A

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

       10. associate-*r* N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 N/A

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

       13. +-commutative N/A

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

       14. distribute-rgt-in N/A

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

       15. *-commutative N/A

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

       16. associate-*r* N/A

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

       17. *-commutative N/A

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

       18. metadata-eval N/A

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

       19. lower-fma.f64 N/A

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

       20. unpow2 N/A

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

       21. lower-*.f64 N/A

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

       22. metadata-eval 30.4

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

   11. Simplified 30.4%

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

   if -0.10000000000000001 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-cos.f64 3.1

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

   5. Simplified 3.1%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 2.5

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

   8. Simplified 2.5%

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

   9. Taylor expanded in im around inf

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

       1. lower-*.f64 N/A

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 37.4

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

   11. Simplified 37.4%

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

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

   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 88.3

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

   5. Simplified 88.3%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 74.6

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

   8. Simplified 74.6%

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

Alternative 11: 55.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -0.1:\\ \;\;\;\;\left(re + 1\right) \cdot \mathsf{fma}\left(im, im \cdot -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(0.16666666666666666, re, 0.5\right), 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 -0.1)
     (* (+ re 1.0) (fma im (* im -0.5) 1.0))
     (if (<= t_0 0.0)
       (* 0.041666666666666664 (* (* im im) (* im im)))
       (fma re (fma re (fma 0.16666666666666666 re 0.5) 1.0) 1.0)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -0.1) {
		tmp = (re + 1.0) * fma(im, (im * -0.5), 1.0);
	} else if (t_0 <= 0.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else {
		tmp = fma(re, fma(re, fma(0.16666666666666666, re, 0.5), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= -0.1)
		tmp = Float64(Float64(re + 1.0) * fma(im, Float64(im * -0.5), 1.0));
	elseif (t_0 <= 0.0)
		tmp = Float64(0.041666666666666664 * Float64(Float64(im * im) * Float64(im * im)));
	else
		tmp = fma(re, fma(re, fma(0.16666666666666666, re, 0.5), 1.0), 1.0);
	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, -0.1], N[(N[(re + 1.0), $MachinePrecision] * N[(im * N[(im * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(0.041666666666666664 * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(re * N[(0.16666666666666666 * re + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.10000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 73.3

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

   5. Simplified 73.3%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 27.0

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

   8. Simplified 27.0%

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

   if -0.10000000000000001 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-cos.f64 3.1

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

   5. Simplified 3.1%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 2.5

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

   8. Simplified 2.5%

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

   9. Taylor expanded in im around inf

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

       1. lower-*.f64 N/A

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 37.4

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

   11. Simplified 37.4%

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

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

   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 88.3

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

   5. Simplified 88.3%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 74.6

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

   8. Simplified 74.6%

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

Alternative 12: 46.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, re \cdot -0.5, re\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(0.5, re, 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 0.0)
     (fma (* im im) (* re -0.5) re)
     (if (<= t_0 2.0)
       (fma re (fma 0.5 re 1.0) 1.0)
       (fma re (* re (fma re 0.16666666666666666 0.5)) re)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = fma((im * im), (re * -0.5), re);
	} else if (t_0 <= 2.0) {
		tmp = fma(re, fma(0.5, re, 1.0), 1.0);
	} else {
		tmp = fma(re, (re * fma(re, 0.16666666666666666, 0.5)), re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = fma(Float64(im * im), Float64(re * -0.5), re);
	elseif (t_0 <= 2.0)
		tmp = fma(re, fma(0.5, re, 1.0), 1.0);
	else
		tmp = fma(re, Float64(re * fma(re, 0.16666666666666666, 0.5)), 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, 0.0], N[(N[(im * im), $MachinePrecision] * N[(re * -0.5), $MachinePrecision] + re), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(re * N[(0.5 * re + 1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   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 49.3

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

   5. Simplified 49.3%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 16.7

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

   8. Simplified 16.7%

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

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\left({re}^{3} \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right)\right)\right)} \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]
   10. Step-by-step derivation

       1. cube-mult N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. associate-+r+ N/A

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

       5. distribute-lft-in N/A

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

       6. rgt-mult-inverse N/A

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

       7. unpow2 N/A

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

       8. associate-*l* N/A

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

       9. +-commutative N/A

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

       10. distribute-rgt-in N/A

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

       11. associate-*l* N/A

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

       12. lft-mult-inverse N/A

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

       13. metadata-eval N/A

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

   11. Simplified 16.5%

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

   12. Taylor expanded in re around 0

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

       1. +-commutative N/A

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

       2. distribute-rgt-in N/A

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

       3. *-commutative N/A

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

       4. associate-*l* N/A

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

       5. *-lft-identity N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. lower-*.f64 14.2

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

   14. Simplified 14.2%

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

   if 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 2

   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 79.9

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

   5. Simplified 79.9%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 79.9

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

   8. Simplified 79.9%

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

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

   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 62.7

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

   5. Simplified 62.7%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 43.5

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

   8. Simplified 43.5%

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

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\left({re}^{3} \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right)\right)\right)} \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]
   10. Step-by-step derivation

       1. cube-mult N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. associate-+r+ N/A

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

       5. distribute-lft-in N/A

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

       6. rgt-mult-inverse N/A

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

       7. unpow2 N/A

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

       8. associate-*l* N/A

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

       9. +-commutative N/A

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

       10. distribute-rgt-in N/A

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

       11. associate-*l* N/A

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

       12. lft-mult-inverse N/A

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

       13. metadata-eval N/A

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

   11. Simplified 43.5%

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

   12. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. +-commutative N/A

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

       7. *-commutative N/A

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

       8. lower-fma.f64 62.7

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

   14. Simplified 62.7%

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

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, re \cdot -0.5, re\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 2:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(0.5, re, 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 41.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(im, im \cdot -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot 0.5, re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 0.0)
     (fma im (* im -0.5) 1.0)
     (if (<= t_0 2.0) (+ re 1.0) (fma re (* re 0.5) re)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = fma(im, (im * -0.5), 1.0);
	} else if (t_0 <= 2.0) {
		tmp = re + 1.0;
	} else {
		tmp = fma(re, (re * 0.5), re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = fma(im, Float64(im * -0.5), 1.0);
	elseif (t_0 <= 2.0)
		tmp = Float64(re + 1.0);
	else
		tmp = fma(re, Float64(re * 0.5), 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, 0.0], N[(im * N[(im * -0.5), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(re + 1.0), $MachinePrecision], N[(re * N[(re * 0.5), $MachinePrecision] + re), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-cos.f64 37.3

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

   5. Simplified 37.3%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 9.2

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

   8. Simplified 9.2%

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

   if 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 2

   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 79.9

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

   5. Simplified 79.9%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{re + 1} \]

      2. lower-+.f64 79.6

         \[\leadsto \color{blue}{re + 1} \]

   8. Simplified 79.6%

      \[\leadsto \color{blue}{re + 1} \]

   if 2 < (*.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 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 42.6

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

   8. Simplified 42.6%

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

   9. Taylor expanded in re around inf

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

       1. distribute-lft-in N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. rgt-mult-inverse N/A

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

       8. *-rgt-identity N/A

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

       9. lower-fma.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 42.6

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

   11. Simplified 42.6%

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

Alternative 14: 41.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(im, im \cdot -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 0.0)
     (fma im (* im -0.5) 1.0)
     (if (<= t_0 2.0) (+ re 1.0) (* re (* re 0.5))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = fma(im, (im * -0.5), 1.0);
	} else if (t_0 <= 2.0) {
		tmp = re + 1.0;
	} else {
		tmp = re * (re * 0.5);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = fma(im, Float64(im * -0.5), 1.0);
	elseif (t_0 <= 2.0)
		tmp = Float64(re + 1.0);
	else
		tmp = Float64(re * Float64(re * 0.5));
	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, 0.0], N[(im * N[(im * -0.5), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(re + 1.0), $MachinePrecision], N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-cos.f64 37.3

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

   5. Simplified 37.3%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 9.2

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

   8. Simplified 9.2%

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

   if 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 2

   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 79.9

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

   5. Simplified 79.9%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{re + 1} \]

      2. lower-+.f64 79.6

         \[\leadsto \color{blue}{re + 1} \]

   8. Simplified 79.6%

      \[\leadsto \color{blue}{re + 1} \]

   if 2 < (*.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 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 42.6

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

   8. Simplified 42.6%

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

   9. Taylor expanded in re around inf

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

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 42.6

          \[\leadsto re \cdot \color{blue}{\left(re \cdot 0.5\right)} \]

   11. Simplified 42.6%

       \[\leadsto \color{blue}{re \cdot \left(re \cdot 0.5\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 46.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\left(re + 1\right) \cdot \mathsf{fma}\left(im, im \cdot -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(0.16666666666666666, re, 0.5\right), 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (cos im)) 0.0)
   (* (+ re 1.0) (fma im (* im -0.5) 1.0))
   (fma re (fma re (fma 0.16666666666666666 re 0.5) 1.0) 1.0)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * cos(im)) <= 0.0) {
		tmp = (re + 1.0) * fma(im, (im * -0.5), 1.0);
	} else {
		tmp = fma(re, fma(re, fma(0.16666666666666666, re, 0.5), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * cos(im)) <= 0.0)
		tmp = Float64(Float64(re + 1.0) * fma(im, Float64(im * -0.5), 1.0));
	else
		tmp = fma(re, fma(re, fma(0.16666666666666666, re, 0.5), 1.0), 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(re + 1.0), $MachinePrecision] * N[(im * N[(im * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(re * N[(re * N[(0.16666666666666666 * re + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 37.8

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

   5. Simplified 37.8%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 14.5

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

   8. Simplified 14.5%

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

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

   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 88.3

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

   5. Simplified 88.3%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 74.6

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

   8. Simplified 74.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(0.16666666666666666, re, 0.5\right), 1\right), 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 46.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, re \cdot -0.5, re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(0.16666666666666666, re, 0.5\right), 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (cos im)) 0.0)
   (fma (* im im) (* re -0.5) re)
   (fma re (fma re (fma 0.16666666666666666 re 0.5) 1.0) 1.0)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * cos(im)) <= 0.0) {
		tmp = fma((im * im), (re * -0.5), re);
	} else {
		tmp = fma(re, fma(re, fma(0.16666666666666666, re, 0.5), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * cos(im)) <= 0.0)
		tmp = fma(Float64(im * im), Float64(re * -0.5), re);
	else
		tmp = fma(re, fma(re, fma(0.16666666666666666, re, 0.5), 1.0), 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(im * im), $MachinePrecision] * N[(re * -0.5), $MachinePrecision] + re), $MachinePrecision], N[(re * N[(re * N[(0.16666666666666666 * re + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   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 49.3

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

   5. Simplified 49.3%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 16.7

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

   8. Simplified 16.7%

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

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\left({re}^{3} \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right)\right)\right)} \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]
   10. Step-by-step derivation

       1. cube-mult N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. associate-+r+ N/A

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

       5. distribute-lft-in N/A

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

       6. rgt-mult-inverse N/A

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

       7. unpow2 N/A

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

       8. associate-*l* N/A

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

       9. +-commutative N/A

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

       10. distribute-rgt-in N/A

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

       11. associate-*l* N/A

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

       12. lft-mult-inverse N/A

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

       13. metadata-eval N/A

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

   11. Simplified 16.5%

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

   12. Taylor expanded in re around 0

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

       1. +-commutative N/A

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

       2. distribute-rgt-in N/A

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

       3. *-commutative N/A

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

       4. associate-*l* N/A

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

       5. *-lft-identity N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. lower-*.f64 14.2

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

   14. Simplified 14.2%

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

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

   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 88.3

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

   5. Simplified 88.3%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 74.6

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

   8. Simplified 74.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(0.16666666666666666, re, 0.5\right), 1\right), 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.8%

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

Alternative 17: 43.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, re \cdot -0.5, re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(0.5, re, 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (cos im)) 0.0)
   (fma (* im im) (* re -0.5) re)
   (fma re (fma 0.5 re 1.0) 1.0)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * cos(im)) <= 0.0) {
		tmp = fma((im * im), (re * -0.5), re);
	} else {
		tmp = fma(re, fma(0.5, re, 1.0), 1.0);
	}
	return tmp;
}

Julia

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

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(im * im), $MachinePrecision] * N[(re * -0.5), $MachinePrecision] + re), $MachinePrecision], N[(re * N[(0.5 * re + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   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 49.3

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

   5. Simplified 49.3%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 16.7

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

   8. Simplified 16.7%

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

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\left({re}^{3} \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right)\right)\right)} \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]
   10. Step-by-step derivation

       1. cube-mult N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. associate-+r+ N/A

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

       5. distribute-lft-in N/A

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

       6. rgt-mult-inverse N/A

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

       7. unpow2 N/A

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

       8. associate-*l* N/A

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

       9. +-commutative N/A

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

       10. distribute-rgt-in N/A

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

       11. associate-*l* N/A

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

       12. lft-mult-inverse N/A

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

       13. metadata-eval N/A

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

   11. Simplified 16.5%

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

   12. Taylor expanded in re around 0

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

       1. +-commutative N/A

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

       2. distribute-rgt-in N/A

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

       3. *-commutative N/A

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

       4. associate-*l* N/A

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

       5. *-lft-identity N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. lower-*.f64 14.2

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

   14. Simplified 14.2%

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

   if 0.0 < (*.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 86.2

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

   5. Simplified 86.2%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 68.3

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

   8. Simplified 68.3%

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

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, re \cdot -0.5, re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(0.5, re, 1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 42.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\mathsf{fma}\left(im, im \cdot -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(0.5, re, 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (cos im)) 0.0)
   (fma im (* im -0.5) 1.0)
   (fma re (fma 0.5 re 1.0) 1.0)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * cos(im)) <= 0.0) {
		tmp = fma(im, (im * -0.5), 1.0);
	} else {
		tmp = fma(re, fma(0.5, re, 1.0), 1.0);
	}
	return tmp;
}

Julia

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

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], 0.0], N[(im * N[(im * -0.5), $MachinePrecision] + 1.0), $MachinePrecision], N[(re * N[(0.5 * re + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-cos.f64 37.3

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

   5. Simplified 37.3%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\left(im \cdot im\right)} + 1 \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot im\right) \cdot im} + 1 \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{im \cdot \left(\frac{-1}{2} \cdot im\right)} + 1 \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(im, \frac{-1}{2} \cdot im, 1\right)} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(im, \color{blue}{im \cdot \frac{-1}{2}}, 1\right) \]

      7. lower-*.f64 9.2

         \[\leadsto \mathsf{fma}\left(im, \color{blue}{im \cdot -0.5}, 1\right) \]

   8. Simplified 9.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(im, im \cdot -0.5, 1\right)} \]

   if 0.0 < (*.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 86.2

         \[\leadsto \color{blue}{e^{re}} \]

   5. Simplified 86.2%

      \[\leadsto \color{blue}{e^{re}} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{re \cdot \left(1 + \frac{1}{2} \cdot re\right) + 1} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, 1 + \frac{1}{2} \cdot re, 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\frac{1}{2} \cdot re + 1}, 1\right) \]

      4. lower-fma.f64 68.3

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(0.5, re, 1\right)}, 1\right) \]

   8. Simplified 68.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(0.5, re, 1\right), 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 38.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 2:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (cos im)) 2.0) (+ re 1.0) (* re (* re 0.5))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * cos(im)) <= 2.0) {
		tmp = re + 1.0;
	} else {
		tmp = re * (re * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((exp(re) * cos(im)) <= 2.0d0) then
        tmp = re + 1.0d0
    else
        tmp = re * (re * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) * Math.cos(im)) <= 2.0) {
		tmp = re + 1.0;
	} else {
		tmp = re * (re * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) * math.cos(im)) <= 2.0:
		tmp = re + 1.0
	else:
		tmp = re * (re * 0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * cos(im)) <= 2.0)
		tmp = Float64(re + 1.0);
	else
		tmp = Float64(re * Float64(re * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) * cos(im)) <= 2.0)
		tmp = re + 1.0;
	else
		tmp = re * (re * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], 2.0], N[(re + 1.0), $MachinePrecision], N[(re * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < 2

   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 63.6

         \[\leadsto \color{blue}{e^{re}} \]

   5. Simplified 63.6%

      \[\leadsto \color{blue}{e^{re}} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + re} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{re + 1} \]

      2. lower-+.f64 36.5

         \[\leadsto \color{blue}{re + 1} \]

   8. Simplified 36.5%

      \[\leadsto \color{blue}{re + 1} \]

   if 2 < (*.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 100.0

         \[\leadsto \color{blue}{e^{re}} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{e^{re}} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{re \cdot \left(1 + \frac{1}{2} \cdot re\right) + 1} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, 1 + \frac{1}{2} \cdot re, 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\frac{1}{2} \cdot re + 1}, 1\right) \]

      4. lower-fma.f64 42.6

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(0.5, re, 1\right)}, 1\right) \]

   8. Simplified 42.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(0.5, re, 1\right), 1\right)} \]

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot {re}^{2}} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(re \cdot re\right)} \]

       2. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot re\right) \cdot re} \]

       3. *-commutative N/A

          \[\leadsto \color{blue}{re \cdot \left(\frac{1}{2} \cdot re\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{re \cdot \left(\frac{1}{2} \cdot re\right)} \]

       5. *-commutative N/A

          \[\leadsto re \cdot \color{blue}{\left(re \cdot \frac{1}{2}\right)} \]

       6. lower-*.f64 42.6

          \[\leadsto re \cdot \color{blue}{\left(re \cdot 0.5\right)} \]

   11. Simplified 42.6%

       \[\leadsto \color{blue}{re \cdot \left(re \cdot 0.5\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 75.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), re\right)\\ \mathbf{if}\;re \leq -18500000000000:\\ \;\;\;\;0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;re \leq 460000:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;re \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;im \cdot \left(im \cdot \mathsf{fma}\left(-0.5, t\_0, \mathsf{fma}\left(\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), \frac{re \cdot re}{im \cdot im}, \frac{re}{im \cdot im}\right)\right)\right)\\ \mathbf{elif}\;re \leq 6.4 \cdot 10^{+122}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, -0.08333333333333333, 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (fma re (* re (fma re 0.16666666666666666 0.5)) re)))
   (if (<= re -18500000000000.0)
     (* 0.041666666666666664 (* (* im im) (* im im)))
     (if (<= re 460000.0)
       (cos im)
       (if (<= re 1.05e+103)
         (*
          im
          (*
           im
           (fma
            -0.5
            t_0
            (fma
             (fma re 0.16666666666666666 0.5)
             (/ (* re re) (* im im))
             (/ re (* im im))))))
         (if (<= re 6.4e+122)
           t_0
           (*
            (* re re)
            (*
             re
             (fma (* im im) -0.08333333333333333 0.16666666666666666)))))))))

C

double code(double re, double im) {
	double t_0 = fma(re, (re * fma(re, 0.16666666666666666, 0.5)), re);
	double tmp;
	if (re <= -18500000000000.0) {
		tmp = 0.041666666666666664 * ((im * im) * (im * im));
	} else if (re <= 460000.0) {
		tmp = cos(im);
	} else if (re <= 1.05e+103) {
		tmp = im * (im * fma(-0.5, t_0, fma(fma(re, 0.16666666666666666, 0.5), ((re * re) / (im * im)), (re / (im * im)))));
	} else if (re <= 6.4e+122) {
		tmp = t_0;
	} else {
		tmp = (re * re) * (re * fma((im * im), -0.08333333333333333, 0.16666666666666666));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(re, Float64(re * fma(re, 0.16666666666666666, 0.5)), re)
	tmp = 0.0
	if (re <= -18500000000000.0)
		tmp = Float64(0.041666666666666664 * Float64(Float64(im * im) * Float64(im * im)));
	elseif (re <= 460000.0)
		tmp = cos(im);
	elseif (re <= 1.05e+103)
		tmp = Float64(im * Float64(im * fma(-0.5, t_0, fma(fma(re, 0.16666666666666666, 0.5), Float64(Float64(re * re) / Float64(im * im)), Float64(re / Float64(im * im))))));
	elseif (re <= 6.4e+122)
		tmp = t_0;
	else
		tmp = Float64(Float64(re * re) * Float64(re * fma(Float64(im * im), -0.08333333333333333, 0.16666666666666666)));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision]}, If[LessEqual[re, -18500000000000.0], N[(0.041666666666666664 * N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 460000.0], N[Cos[im], $MachinePrecision], If[LessEqual[re, 1.05e+103], N[(im * N[(im * N[(-0.5 * t$95$0 + N[(N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] / N[(im * im), $MachinePrecision]), $MachinePrecision] + N[(re / N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 6.4e+122], t$95$0, N[(N[(re * re), $MachinePrecision] * N[(re * N[(N[(im * im), $MachinePrecision] * -0.08333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if re < -1.85e13

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\cos im} \]
   4. Step-by-step derivation

      1. lower-cos.f64 3.1

         \[\leadsto \color{blue}{\cos im} \]

   5. Simplified 3.1%

      \[\leadsto \color{blue}{\cos im} \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) + 1} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{\left(im \cdot im\right)} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) + 1 \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{im \cdot \left(im \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)} + 1 \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(im, im \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right), 1\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(im, \color{blue}{im \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)}, 1\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(im, im \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}, 1\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(im, im \cdot \left(\color{blue}{{im}^{2} \cdot \frac{1}{24}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), 1\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(im, im \cdot \left({im}^{2} \cdot \frac{1}{24} + \color{blue}{\frac{-1}{2}}\right), 1\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(im, im \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{24}, \frac{-1}{2}\right)}, 1\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{24}, \frac{-1}{2}\right), 1\right) \]

      11. lower-*.f64 2.5

          \[\leadsto \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, 0.041666666666666664, -0.5\right), 1\right) \]

   8. Simplified 2.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, 0.041666666666666664, -0.5\right), 1\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {im}^{4}} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{24} \cdot {im}^{4}} \]

       2. metadata-eval N/A

          \[\leadsto \frac{1}{24} \cdot {im}^{\color{blue}{\left(2 \cdot 2\right)}} \]

       3. pow-sqr N/A

          \[\leadsto \frac{1}{24} \cdot \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{1}{24} \cdot \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)} \]

       5. unpow2 N/A

          \[\leadsto \frac{1}{24} \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot {im}^{2}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{1}{24} \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot {im}^{2}\right) \]

       7. unpow2 N/A

          \[\leadsto \frac{1}{24} \cdot \left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right) \]

       8. lower-*.f64 39.3

          \[\leadsto 0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right) \]

   11. Simplified 39.3%

       \[\leadsto \color{blue}{0.041666666666666664 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)} \]

   if -1.85e13 < re < 4.6e5

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\cos im} \]
   4. Step-by-step derivation

      1. lower-cos.f64 94.6

         \[\leadsto \color{blue}{\cos im} \]

   5. Simplified 94.6%

      \[\leadsto \color{blue}{\cos im} \]

   if 4.6e5 < re < 1.0500000000000001e103

   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 6.0

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \cdot \cos im \]

   5. Simplified 6.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \cdot \cos im \]

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2} + 1\right)} \]

      2. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \left(\frac{-1}{2} \cdot \color{blue}{\left(im \cdot im\right)} + 1\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \left(\color{blue}{\left(\frac{-1}{2} \cdot im\right) \cdot im} + 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \left(\color{blue}{im \cdot \left(\frac{-1}{2} \cdot im\right)} + 1\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \color{blue}{\mathsf{fma}\left(im, \frac{-1}{2} \cdot im, 1\right)} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \mathsf{fma}\left(im, \color{blue}{im \cdot \frac{-1}{2}}, 1\right) \]

      7. lower-*.f64 15.6

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right) \cdot \mathsf{fma}\left(im, \color{blue}{im \cdot -0.5}, 1\right) \]

   8. Simplified 15.6%

      \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right) \cdot \color{blue}{\mathsf{fma}\left(im, im \cdot -0.5, 1\right)} \]

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\left({re}^{3} \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right)\right)\right)} \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]
   10. Step-by-step derivation

       1. cube-mult N/A

          \[\leadsto \left(\color{blue}{\left(re \cdot \left(re \cdot re\right)\right)} \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right)\right)\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]

       2. unpow2 N/A

          \[\leadsto \left(\left(re \cdot \color{blue}{{re}^{2}}\right) \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right)\right)\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{\left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right)\right)\right)\right)} \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]

       4. associate-+r+ N/A

          \[\leadsto \left(re \cdot \left({re}^{2} \cdot \color{blue}{\left(\left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right) + \frac{1}{{re}^{2}}\right)}\right)\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]

       5. distribute-lft-in N/A

          \[\leadsto \left(re \cdot \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right) + {re}^{2} \cdot \frac{1}{{re}^{2}}\right)}\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]

       6. rgt-mult-inverse N/A

          \[\leadsto \left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right) + \color{blue}{1}\right)\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]

       7. unpow2 N/A

          \[\leadsto \left(re \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right) + 1\right)\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]

       8. associate-*l* N/A

          \[\leadsto \left(re \cdot \left(\color{blue}{re \cdot \left(re \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)} + 1\right)\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]

       9. +-commutative N/A

          \[\leadsto \left(re \cdot \left(re \cdot \left(re \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{6}\right)}\right) + 1\right)\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]

       10. distribute-rgt-in N/A

           \[\leadsto \left(re \cdot \left(re \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{1}{re}\right) \cdot re + \frac{1}{6} \cdot re\right)} + 1\right)\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]

       11. associate-*l* N/A

           \[\leadsto \left(re \cdot \left(re \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(\frac{1}{re} \cdot re\right)} + \frac{1}{6} \cdot re\right) + 1\right)\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]

       12. lft-mult-inverse N/A

           \[\leadsto \left(re \cdot \left(re \cdot \left(\frac{1}{2} \cdot \color{blue}{1} + \frac{1}{6} \cdot re\right) + 1\right)\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]

       13. metadata-eval N/A

           \[\leadsto \left(re \cdot \left(re \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{6} \cdot re\right) + 1\right)\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]

   11. Simplified 15.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), re\right)} \cdot \mathsf{fma}\left(im, im \cdot -0.5, 1\right) \]

   12. Taylor expanded in im around inf

       \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{-1}{2} \cdot \left(re + {re}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(\frac{re}{{im}^{2}} + \frac{{re}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)}{{im}^{2}}\right)\right)} \]
   13. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \color{blue}{\left(im \cdot im\right)} \cdot \left(\frac{-1}{2} \cdot \left(re + {re}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(\frac{re}{{im}^{2}} + \frac{{re}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)}{{im}^{2}}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \color{blue}{im \cdot \left(im \cdot \left(\frac{-1}{2} \cdot \left(re + {re}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(\frac{re}{{im}^{2}} + \frac{{re}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)}{{im}^{2}}\right)\right)\right)} \]

       3. *-commutative N/A

          \[\leadsto im \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot \left(re + {re}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(\frac{re}{{im}^{2}} + \frac{{re}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)}{{im}^{2}}\right)\right) \cdot im\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{im \cdot \left(\left(\frac{-1}{2} \cdot \left(re + {re}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(\frac{re}{{im}^{2}} + \frac{{re}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)}{{im}^{2}}\right)\right) \cdot im\right)} \]

       5. *-commutative N/A

          \[\leadsto im \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{2} \cdot \left(re + {re}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(\frac{re}{{im}^{2}} + \frac{{re}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)}{{im}^{2}}\right)\right)\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto im \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{2} \cdot \left(re + {re}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(\frac{re}{{im}^{2}} + \frac{{re}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)}{{im}^{2}}\right)\right)\right)} \]

       7. lower-fma.f64 N/A

          \[\leadsto im \cdot \left(im \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{2}, re + {re}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right), \frac{re}{{im}^{2}} + \frac{{re}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)}{{im}^{2}}\right)}\right) \]

   14. Simplified 64.9%

       \[\leadsto \color{blue}{im \cdot \left(im \cdot \mathsf{fma}\left(-0.5, \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), re\right), \mathsf{fma}\left(\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), \frac{re \cdot re}{im \cdot im}, \frac{re}{im \cdot im}\right)\right)\right)} \]

   if 1.0500000000000001e103 < re < 6.40000000000000024e122

   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 100.0

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \cdot \cos im \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \cdot \cos im \]

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2} + 1\right)} \]

      2. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \left(\frac{-1}{2} \cdot \color{blue}{\left(im \cdot im\right)} + 1\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \left(\color{blue}{\left(\frac{-1}{2} \cdot im\right) \cdot im} + 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \left(\color{blue}{im \cdot \left(\frac{-1}{2} \cdot im\right)} + 1\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \color{blue}{\mathsf{fma}\left(im, \frac{-1}{2} \cdot im, 1\right)} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \mathsf{fma}\left(im, \color{blue}{im \cdot \frac{-1}{2}}, 1\right) \]

      7. lower-*.f64 37.5

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right) \cdot \mathsf{fma}\left(im, \color{blue}{im \cdot -0.5}, 1\right) \]

   8. Simplified 37.5%

      \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right) \cdot \color{blue}{\mathsf{fma}\left(im, im \cdot -0.5, 1\right)} \]

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\left({re}^{3} \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right)\right)\right)} \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]
   10. Step-by-step derivation

       1. cube-mult N/A

          \[\leadsto \left(\color{blue}{\left(re \cdot \left(re \cdot re\right)\right)} \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right)\right)\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]

       2. unpow2 N/A

          \[\leadsto \left(\left(re \cdot \color{blue}{{re}^{2}}\right) \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right)\right)\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{\left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right)\right)\right)\right)} \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]

       4. associate-+r+ N/A

          \[\leadsto \left(re \cdot \left({re}^{2} \cdot \color{blue}{\left(\left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right) + \frac{1}{{re}^{2}}\right)}\right)\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]

       5. distribute-lft-in N/A

          \[\leadsto \left(re \cdot \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right) + {re}^{2} \cdot \frac{1}{{re}^{2}}\right)}\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]

       6. rgt-mult-inverse N/A

          \[\leadsto \left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right) + \color{blue}{1}\right)\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]

       7. unpow2 N/A

          \[\leadsto \left(re \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right) + 1\right)\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]

       8. associate-*l* N/A

          \[\leadsto \left(re \cdot \left(\color{blue}{re \cdot \left(re \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)} + 1\right)\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]

       9. +-commutative N/A

          \[\leadsto \left(re \cdot \left(re \cdot \left(re \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{6}\right)}\right) + 1\right)\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]

       10. distribute-rgt-in N/A

           \[\leadsto \left(re \cdot \left(re \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{1}{re}\right) \cdot re + \frac{1}{6} \cdot re\right)} + 1\right)\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]

       11. associate-*l* N/A

           \[\leadsto \left(re \cdot \left(re \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(\frac{1}{re} \cdot re\right)} + \frac{1}{6} \cdot re\right) + 1\right)\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]

       12. lft-mult-inverse N/A

           \[\leadsto \left(re \cdot \left(re \cdot \left(\frac{1}{2} \cdot \color{blue}{1} + \frac{1}{6} \cdot re\right) + 1\right)\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]

       13. metadata-eval N/A

           \[\leadsto \left(re \cdot \left(re \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{6} \cdot re\right) + 1\right)\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]

   11. Simplified 37.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), re\right)} \cdot \mathsf{fma}\left(im, im \cdot -0.5, 1\right) \]

   12. Taylor expanded in im around 0

       \[\leadsto \color{blue}{re + {re}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)} \]
   13. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{{re}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right) + re} \]

       2. unpow2 N/A

          \[\leadsto \color{blue}{\left(re \cdot re\right)} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right) + re \]

       3. associate-*r* N/A

          \[\leadsto \color{blue}{re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)} + re \]

       4. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(re, re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right), re\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(re, \color{blue}{re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)}, re\right) \]

       6. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(re, re \cdot \color{blue}{\left(\frac{1}{6} \cdot re + \frac{1}{2}\right)}, re\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(re, re \cdot \left(\color{blue}{re \cdot \frac{1}{6}} + \frac{1}{2}\right), re\right) \]

       8. lower-fma.f64 100.0

          \[\leadsto \mathsf{fma}\left(re, re \cdot \color{blue}{\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)}, re\right) \]

   14. Simplified 100.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), re\right)} \]

   if 6.40000000000000024e122 < re

   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 100.0

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \cdot \cos im \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \cdot \cos im \]

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2} + 1\right)} \]

      2. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \left(\frac{-1}{2} \cdot \color{blue}{\left(im \cdot im\right)} + 1\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \left(\color{blue}{\left(\frac{-1}{2} \cdot im\right) \cdot im} + 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \left(\color{blue}{im \cdot \left(\frac{-1}{2} \cdot im\right)} + 1\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \color{blue}{\mathsf{fma}\left(im, \frac{-1}{2} \cdot im, 1\right)} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \mathsf{fma}\left(im, \color{blue}{im \cdot \frac{-1}{2}}, 1\right) \]

      7. lower-*.f64 90.9

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right) \cdot \mathsf{fma}\left(im, \color{blue}{im \cdot -0.5}, 1\right) \]

   8. Simplified 90.9%

      \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right) \cdot \color{blue}{\mathsf{fma}\left(im, im \cdot -0.5, 1\right)} \]

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\left({re}^{3} \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right)\right)\right)} \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]
   10. Step-by-step derivation

       1. cube-mult N/A

          \[\leadsto \left(\color{blue}{\left(re \cdot \left(re \cdot re\right)\right)} \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right)\right)\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]

       2. unpow2 N/A

          \[\leadsto \left(\left(re \cdot \color{blue}{{re}^{2}}\right) \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right)\right)\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{\left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{{re}^{2}}\right)\right)\right)\right)} \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]

       4. associate-+r+ N/A

          \[\leadsto \left(re \cdot \left({re}^{2} \cdot \color{blue}{\left(\left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right) + \frac{1}{{re}^{2}}\right)}\right)\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]

       5. distribute-lft-in N/A

          \[\leadsto \left(re \cdot \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right) + {re}^{2} \cdot \frac{1}{{re}^{2}}\right)}\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]

       6. rgt-mult-inverse N/A

          \[\leadsto \left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right) + \color{blue}{1}\right)\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]

       7. unpow2 N/A

          \[\leadsto \left(re \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right) + 1\right)\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]

       8. associate-*l* N/A

          \[\leadsto \left(re \cdot \left(\color{blue}{re \cdot \left(re \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)} + 1\right)\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]

       9. +-commutative N/A

          \[\leadsto \left(re \cdot \left(re \cdot \left(re \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{re} + \frac{1}{6}\right)}\right) + 1\right)\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]

       10. distribute-rgt-in N/A

           \[\leadsto \left(re \cdot \left(re \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{1}{re}\right) \cdot re + \frac{1}{6} \cdot re\right)} + 1\right)\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]

       11. associate-*l* N/A

           \[\leadsto \left(re \cdot \left(re \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(\frac{1}{re} \cdot re\right)} + \frac{1}{6} \cdot re\right) + 1\right)\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]

       12. lft-mult-inverse N/A

           \[\leadsto \left(re \cdot \left(re \cdot \left(\frac{1}{2} \cdot \color{blue}{1} + \frac{1}{6} \cdot re\right) + 1\right)\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]

       13. metadata-eval N/A

           \[\leadsto \left(re \cdot \left(re \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{6} \cdot re\right) + 1\right)\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{2}, 1\right) \]

   11. Simplified 90.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), re\right)} \cdot \mathsf{fma}\left(im, im \cdot -0.5, 1\right) \]

   12. Taylor expanded in re around inf

       \[\leadsto \color{blue}{\frac{1}{6} \cdot \left({re}^{3} \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right)} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{1}{6} \cdot \color{blue}{\left(\left(1 + \frac{-1}{2} \cdot {im}^{2}\right) \cdot {re}^{3}\right)} \]

       2. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right) \cdot {re}^{3}} \]

       3. cube-mult N/A

          \[\leadsto \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right) \cdot \color{blue}{\left(re \cdot \left(re \cdot re\right)\right)} \]

       4. unpow2 N/A

          \[\leadsto \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right) \cdot \left(re \cdot \color{blue}{{re}^{2}}\right) \]

       5. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(\frac{1}{6} \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right) \cdot re\right) \cdot {re}^{2}} \]

       6. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot \left(\left(1 + \frac{-1}{2} \cdot {im}^{2}\right) \cdot re\right)\right)} \cdot {re}^{2} \]

       7. *-commutative N/A

          \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{\left(re \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right)}\right) \cdot {re}^{2} \]

       8. *-commutative N/A

          \[\leadsto \color{blue}{{re}^{2} \cdot \left(\frac{1}{6} \cdot \left(re \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right)\right)} \]

       9. lower-*.f64 N/A

          \[\leadsto \color{blue}{{re}^{2} \cdot \left(\frac{1}{6} \cdot \left(re \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right)\right)} \]

       10. unpow2 N/A

           \[\leadsto \color{blue}{\left(re \cdot re\right)} \cdot \left(\frac{1}{6} \cdot \left(re \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right)\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \color{blue}{\left(re \cdot re\right)} \cdot \left(\frac{1}{6} \cdot \left(re \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \left(re \cdot re\right) \cdot \color{blue}{\left(\left(re \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right) \cdot \frac{1}{6}\right)} \]

       13. associate-*l* N/A

           \[\leadsto \left(re \cdot re\right) \cdot \color{blue}{\left(re \cdot \left(\left(1 + \frac{-1}{2} \cdot {im}^{2}\right) \cdot \frac{1}{6}\right)\right)} \]

       14. *-commutative N/A

           \[\leadsto \left(re \cdot re\right) \cdot \left(re \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right)}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto \left(re \cdot re\right) \cdot \color{blue}{\left(re \cdot \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right)\right)} \]

       16. distribute-rgt-in N/A

           \[\leadsto \left(re \cdot re\right) \cdot \left(re \cdot \color{blue}{\left(1 \cdot \frac{1}{6} + \left(\frac{-1}{2} \cdot {im}^{2}\right) \cdot \frac{1}{6}\right)}\right) \]

       17. metadata-eval N/A

           \[\leadsto \left(re \cdot re\right) \cdot \left(re \cdot \left(\color{blue}{\frac{1}{6}} + \left(\frac{-1}{2} \cdot {im}^{2}\right) \cdot \frac{1}{6}\right)\right) \]

       18. +-commutative N/A

           \[\leadsto \left(re \cdot re\right) \cdot \left(re \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot {im}^{2}\right) \cdot \frac{1}{6} + \frac{1}{6}\right)}\right) \]

       19. *-commutative N/A

           \[\leadsto \left(re \cdot re\right) \cdot \left(re \cdot \left(\color{blue}{\frac{1}{6} \cdot \left(\frac{-1}{2} \cdot {im}^{2}\right)} + \frac{1}{6}\right)\right) \]

       20. associate-*r* N/A

           \[\leadsto \left(re \cdot re\right) \cdot \left(re \cdot \left(\color{blue}{\left(\frac{1}{6} \cdot \frac{-1}{2}\right) \cdot {im}^{2}} + \frac{1}{6}\right)\right) \]

       21. *-commutative N/A

           \[\leadsto \left(re \cdot re\right) \cdot \left(re \cdot \left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{6} \cdot \frac{-1}{2}\right)} + \frac{1}{6}\right)\right) \]

       22. lower-fma.f64 N/A

           \[\leadsto \left(re \cdot re\right) \cdot \left(re \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{6} \cdot \frac{-1}{2}, \frac{1}{6}\right)}\right) \]

   14. Simplified 90.9%

       \[\leadsto \color{blue}{\left(re \cdot re\right) \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, -0.08333333333333333, 0.16666666666666666\right)\right)} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 21: 29.1% accurate, 51.5× speedup

Math

\[\begin{array}{l} \\ re + 1 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (+ re 1.0))

C

double code(double re, double im) {
	return re + 1.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = re + 1.0d0
end function

Java

public static double code(double re, double im) {
	return re + 1.0;
}

Python

def code(re, im):
	return re + 1.0

Julia

function code(re, im)
	return Float64(re + 1.0)
end

MATLAB

function tmp = code(re, im)
	tmp = re + 1.0;
end

Wolfram

code[re_, im_] := N[(re + 1.0), $MachinePrecision]

Derivation

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 69.8

      \[\leadsto \color{blue}{e^{re}} \]

5. Simplified 69.8%

   \[\leadsto \color{blue}{e^{re}} \]

6. Taylor expanded in re around 0

   \[\leadsto \color{blue}{1 + re} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{re + 1} \]

   2. lower-+.f64 31.2

      \[\leadsto \color{blue}{re + 1} \]

8. Simplified 31.2%

   \[\leadsto \color{blue}{re + 1} \]
9. Add Preprocessing

Alternative 22: 28.6% accurate, 206.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 1.0)

C

double code(double re, double im) {
	return 1.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 1.0d0
end function

Java

public static double code(double re, double im) {
	return 1.0;
}

Python

def code(re, im):
	return 1.0

Julia

function code(re, im)
	return 1.0
end

MATLAB

function tmp = code(re, im)
	tmp = 1.0;
end

Wolfram

code[re_, im_] := 1.0

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \cos im \]
2. Add Preprocessing

3. Taylor expanded in re around 0

   \[\leadsto \color{blue}{\cos im} \]
4. Step-by-step derivation

   1. lower-cos.f64 53.9

      \[\leadsto \color{blue}{\cos im} \]

5. Simplified 53.9%

   \[\leadsto \color{blue}{\cos im} \]

6. Taylor expanded in im around 0

   \[\leadsto \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 30.7%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024215 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))