math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 10.6s

Alternatives: 22

Speedup: 1.0×

• 25.1% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(re, im)
use fmin_fmax_functions
    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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(re, im)
use fmin_fmax_functions
    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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(re, im)
use fmin_fmax_functions
    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: 88.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ t_1 := \left(1 + re\right) \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot re\right) \cdot re\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq -0.04:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;e^{re} \cdot \left(\left(im \cdot im\right) \cdot -0.5\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.027777777777777776, re \cdot re, -0.25\right) \cdot \left(re \cdot re\right)}{0.16666666666666666 \cdot re - 0.5} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))) (t_1 (* (+ 1.0 re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (*
      (* (* (fma 0.16666666666666666 re 0.5) re) re)
      (fma (* im im) -0.5 1.0))
     (if (<= t_0 -0.04)
       t_1
       (if (<= t_0 0.0)
         (* (exp re) (* (* im im) -0.5))
         (if (<= t_0 2.0)
           t_1
           (*
            (/
             (* (fma 0.027777777777777776 (* re re) -0.25) (* re re))
             (- (* 0.16666666666666666 re) 0.5))
            (fma
             (fma 0.041666666666666664 (* im im) -0.5)
             (* im im)
             1.0))))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double t_1 = (1.0 + re) * cos(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = ((fma(0.16666666666666666, re, 0.5) * re) * re) * fma((im * im), -0.5, 1.0);
	} else if (t_0 <= -0.04) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = exp(re) * ((im * im) * -0.5);
	} else if (t_0 <= 2.0) {
		tmp = t_1;
	} else {
		tmp = ((fma(0.027777777777777776, (re * re), -0.25) * (re * re)) / ((0.16666666666666666 * re) - 0.5)) * fma(fma(0.041666666666666664, (im * im), -0.5), (im * im), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	t_1 = Float64(Float64(1.0 + re) * cos(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(fma(0.16666666666666666, re, 0.5) * re) * re) * fma(Float64(im * im), -0.5, 1.0));
	elseif (t_0 <= -0.04)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(exp(re) * Float64(Float64(im * im) * -0.5));
	elseif (t_0 <= 2.0)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(fma(0.027777777777777776, Float64(re * re), -0.25) * Float64(re * re)) / Float64(Float64(0.16666666666666666 * re) - 0.5)) * fma(fma(0.041666666666666664, Float64(im * im), -0.5), Float64(im * im), 1.0));
	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[(1.0 + re), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.04], t$95$1, If[LessEqual[t$95$0, 0.0], N[(N[Exp[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2.0], t$95$1, N[(N[(N[(N[(0.027777777777777776 * N[(re * re), $MachinePrecision] + -0.25), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] / N[(N[(0.16666666666666666 * re), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] * N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision] + -0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 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. fp-cancel-sign-sub-inv N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. metadata-eval N/A

         \[\leadsto \left(\left(re + re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) + \color{blue}{\left(0 + 1\right)}\right) \cdot \cos im \]

      8. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(re + re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) + 0\right) + 1\right)} \cdot \cos im \]

      9. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + re\right)} + 0\right) + 1\right) \cdot \cos im \]

      10. associate-+r+ N/A

          \[\leadsto \left(\color{blue}{\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(re + 0\right)\right)} + 1\right) \cdot \cos im \]

      11. *-rgt-identity N/A

          \[\leadsto \left(\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(\color{blue}{re \cdot 1} + 0\right)\right) + 1\right) \cdot \cos im \]

      12. mul0-rgt N/A

          \[\leadsto \left(\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(re \cdot 1 + \color{blue}{re \cdot 0}\right)\right) + 1\right) \cdot \cos im \]

      13. distribute-lft-in N/A

          \[\leadsto \left(\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \color{blue}{re \cdot \left(1 + 0\right)}\right) + 1\right) \cdot \cos im \]

      14. metadata-eval N/A

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

      15. distribute-lft-in N/A

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

      16. +-commutative N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 83.8%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0400000000000000008 or 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 re around 0

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

      1. lower-+.f64 97.7

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

   5. Applied rewrites 97.7%

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

   if -0.0400000000000000008 < (*.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 im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 73.8

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

   5. Applied rewrites 73.8%

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

   6. Taylor expanded in im around inf

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

   8. Applied rewrites 73.8%

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

   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. fp-cancel-sign-sub-inv N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. metadata-eval N/A

         \[\leadsto \left(\left(re + re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) + \color{blue}{\left(0 + 1\right)}\right) \cdot \cos im \]

      8. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(re + re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) + 0\right) + 1\right)} \cdot \cos im \]

      9. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + re\right)} + 0\right) + 1\right) \cdot \cos im \]

      10. associate-+r+ N/A

          \[\leadsto \left(\color{blue}{\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(re + 0\right)\right)} + 1\right) \cdot \cos im \]

      11. *-rgt-identity N/A

          \[\leadsto \left(\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(\color{blue}{re \cdot 1} + 0\right)\right) + 1\right) \cdot \cos im \]

      12. mul0-rgt N/A

          \[\leadsto \left(\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(re \cdot 1 + \color{blue}{re \cdot 0}\right)\right) + 1\right) \cdot \cos im \]

      13. distribute-lft-in N/A

          \[\leadsto \left(\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \color{blue}{re \cdot \left(1 + 0\right)}\right) + 1\right) \cdot \cos im \]

      14. metadata-eval N/A

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

      15. distribute-lft-in N/A

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

      16. +-commutative N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 64.2%

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

   6. Taylor expanded in im around 0

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

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \color{blue}{\left(1 - \left(\mathsf{neg}\left({im}^{2}\right)\right) \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)} \]

      2. fp-cancel-sub-sign-inv N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({im}^{2}\right)\right)\right)\right) \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({im}^{2}\right)\right)\right)\right) \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) + 1\right)} \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \left(\color{blue}{{im}^{2}} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) + 1\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \left(\color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot {im}^{2}} + 1\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}, {im}^{2}, 1\right)} \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot {im}^{2} - \color{blue}{\frac{1}{2} \cdot 1}, {im}^{2}, 1\right) \]

      8. fp-cancel-sub-sign-inv N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot 1}, {im}^{2}, 1\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\frac{-1}{2}} \cdot 1, {im}^{2}, 1\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\frac{-1}{2}}, {im}^{2}, 1\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {im}^{2}, \frac{-1}{2}\right)}, {im}^{2}, 1\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{-1}{2}\right), {im}^{2}, 1\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{-1}{2}\right), {im}^{2}, 1\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{-1}{2}\right), \color{blue}{im \cdot im}, 1\right) \]

      15. lower-*.f64 78.9

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), \color{blue}{im \cdot im}, 1\right) \]

   8. Applied rewrites 78.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right)} \]

   9. Taylor expanded in re around inf

      \[\leadsto \left({re}^{3} \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{-1}{2}\right), im \cdot im, 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 78.9%

       \[\leadsto \left(\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot re\right) \cdot \color{blue}{re}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 83.0%

       \[\leadsto \frac{\mathsf{fma}\left(0.027777777777777776, re \cdot re, -0.25\right) \cdot \left(re \cdot re\right)}{0.16666666666666666 \cdot re - \color{blue}{0.5}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 3: 79.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ t_1 := \left(1 + re\right) \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot re\right) \cdot re\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq -0.04:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(1 + re\right) \cdot \left(\left(im \cdot im\right) \cdot \left(0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.027777777777777776, re \cdot re, -0.25\right) \cdot \left(re \cdot re\right)}{0.16666666666666666 \cdot re - 0.5} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))) (t_1 (* (+ 1.0 re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (*
      (* (* (fma 0.16666666666666666 re 0.5) re) re)
      (fma (* im im) -0.5 1.0))
     (if (<= t_0 -0.04)
       t_1
       (if (<= t_0 0.0)
         (* (+ 1.0 re) (* (* im im) (* 0.041666666666666664 (* im im))))
         (if (<= t_0 2.0)
           t_1
           (*
            (/
             (* (fma 0.027777777777777776 (* re re) -0.25) (* re re))
             (- (* 0.16666666666666666 re) 0.5))
            (fma
             (fma 0.041666666666666664 (* im im) -0.5)
             (* im im)
             1.0))))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double t_1 = (1.0 + re) * cos(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = ((fma(0.16666666666666666, re, 0.5) * re) * re) * fma((im * im), -0.5, 1.0);
	} else if (t_0 <= -0.04) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = (1.0 + re) * ((im * im) * (0.041666666666666664 * (im * im)));
	} else if (t_0 <= 2.0) {
		tmp = t_1;
	} else {
		tmp = ((fma(0.027777777777777776, (re * re), -0.25) * (re * re)) / ((0.16666666666666666 * re) - 0.5)) * fma(fma(0.041666666666666664, (im * im), -0.5), (im * im), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	t_1 = Float64(Float64(1.0 + re) * cos(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(fma(0.16666666666666666, re, 0.5) * re) * re) * fma(Float64(im * im), -0.5, 1.0));
	elseif (t_0 <= -0.04)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(1.0 + re) * Float64(Float64(im * im) * Float64(0.041666666666666664 * Float64(im * im))));
	elseif (t_0 <= 2.0)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(fma(0.027777777777777776, Float64(re * re), -0.25) * Float64(re * re)) / Float64(Float64(0.16666666666666666 * re) - 0.5)) * fma(fma(0.041666666666666664, Float64(im * im), -0.5), Float64(im * im), 1.0));
	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[(1.0 + re), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.04], t$95$1, If[LessEqual[t$95$0, 0.0], N[(N[(1.0 + re), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2.0], t$95$1, N[(N[(N[(N[(0.027777777777777776 * N[(re * re), $MachinePrecision] + -0.25), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] / N[(N[(0.16666666666666666 * re), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] * N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision] + -0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 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. fp-cancel-sign-sub-inv N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. metadata-eval N/A

         \[\leadsto \left(\left(re + re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) + \color{blue}{\left(0 + 1\right)}\right) \cdot \cos im \]

      8. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(re + re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) + 0\right) + 1\right)} \cdot \cos im \]

      9. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + re\right)} + 0\right) + 1\right) \cdot \cos im \]

      10. associate-+r+ N/A

          \[\leadsto \left(\color{blue}{\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(re + 0\right)\right)} + 1\right) \cdot \cos im \]

      11. *-rgt-identity N/A

          \[\leadsto \left(\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(\color{blue}{re \cdot 1} + 0\right)\right) + 1\right) \cdot \cos im \]

      12. mul0-rgt N/A

          \[\leadsto \left(\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(re \cdot 1 + \color{blue}{re \cdot 0}\right)\right) + 1\right) \cdot \cos im \]

      13. distribute-lft-in N/A

          \[\leadsto \left(\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \color{blue}{re \cdot \left(1 + 0\right)}\right) + 1\right) \cdot \cos im \]

      14. metadata-eval N/A

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

      15. distribute-lft-in N/A

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

      16. +-commutative N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 83.8%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0400000000000000008 or 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 re around 0

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

      1. lower-+.f64 97.7

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

   5. Applied rewrites 97.7%

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

   if -0.0400000000000000008 < (*.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. lower-+.f64 2.3

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

   5. Applied rewrites 2.3%

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

   6. Taylor expanded in im around 0

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. metadata-eval N/A

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

      8. fp-cancel-sub-sign-inv N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 2.0

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

   8. Applied rewrites 2.0%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 45.6%

       \[\leadsto \left(1 + re\right) \cdot \left({im}^{4} \cdot \color{blue}{0.041666666666666664}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 45.6%

       \[\leadsto \left(1 + re\right) \cdot \left(\left(im \cdot im\right) \cdot \left(0.041666666666666664 \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \]

   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. fp-cancel-sign-sub-inv N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. metadata-eval N/A

         \[\leadsto \left(\left(re + re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) + \color{blue}{\left(0 + 1\right)}\right) \cdot \cos im \]

      8. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(re + re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) + 0\right) + 1\right)} \cdot \cos im \]

      9. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + re\right)} + 0\right) + 1\right) \cdot \cos im \]

      10. associate-+r+ N/A

          \[\leadsto \left(\color{blue}{\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(re + 0\right)\right)} + 1\right) \cdot \cos im \]

      11. *-rgt-identity N/A

          \[\leadsto \left(\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(\color{blue}{re \cdot 1} + 0\right)\right) + 1\right) \cdot \cos im \]

      12. mul0-rgt N/A

          \[\leadsto \left(\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(re \cdot 1 + \color{blue}{re \cdot 0}\right)\right) + 1\right) \cdot \cos im \]

      13. distribute-lft-in N/A

          \[\leadsto \left(\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \color{blue}{re \cdot \left(1 + 0\right)}\right) + 1\right) \cdot \cos im \]

      14. metadata-eval N/A

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

      15. distribute-lft-in N/A

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

      16. +-commutative N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 64.2%

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

   6. Taylor expanded in im around 0

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

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \color{blue}{\left(1 - \left(\mathsf{neg}\left({im}^{2}\right)\right) \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)} \]

      2. fp-cancel-sub-sign-inv N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({im}^{2}\right)\right)\right)\right) \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({im}^{2}\right)\right)\right)\right) \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) + 1\right)} \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \left(\color{blue}{{im}^{2}} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) + 1\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \left(\color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot {im}^{2}} + 1\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}, {im}^{2}, 1\right)} \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot {im}^{2} - \color{blue}{\frac{1}{2} \cdot 1}, {im}^{2}, 1\right) \]

      8. fp-cancel-sub-sign-inv N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot 1}, {im}^{2}, 1\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\frac{-1}{2}} \cdot 1, {im}^{2}, 1\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\frac{-1}{2}}, {im}^{2}, 1\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {im}^{2}, \frac{-1}{2}\right)}, {im}^{2}, 1\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{-1}{2}\right), {im}^{2}, 1\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{-1}{2}\right), {im}^{2}, 1\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{-1}{2}\right), \color{blue}{im \cdot im}, 1\right) \]

      15. lower-*.f64 78.9

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), \color{blue}{im \cdot im}, 1\right) \]

   8. Applied rewrites 78.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right)} \]

   9. Taylor expanded in re around inf

      \[\leadsto \left({re}^{3} \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{-1}{2}\right), im \cdot im, 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 78.9%

       \[\leadsto \left(\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot re\right) \cdot \color{blue}{re}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 83.0%

       \[\leadsto \frac{\mathsf{fma}\left(0.027777777777777776, re \cdot re, -0.25\right) \cdot \left(re \cdot re\right)}{0.16666666666666666 \cdot re - \color{blue}{0.5}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 4: 79.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot re\right) \cdot re\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq -0.04:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(1 + re\right) \cdot \left(\left(im \cdot im\right) \cdot \left(0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.027777777777777776, re \cdot re, -0.25\right) \cdot \left(re \cdot re\right)}{0.16666666666666666 \cdot re - 0.5} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (*
      (* (* (fma 0.16666666666666666 re 0.5) re) re)
      (fma (* im im) -0.5 1.0))
     (if (<= t_0 -0.04)
       (cos im)
       (if (<= t_0 0.0)
         (* (+ 1.0 re) (* (* im im) (* 0.041666666666666664 (* im im))))
         (if (<= t_0 2.0)
           (cos im)
           (*
            (/
             (* (fma 0.027777777777777776 (* re re) -0.25) (* re re))
             (- (* 0.16666666666666666 re) 0.5))
            (fma
             (fma 0.041666666666666664 (* im im) -0.5)
             (* im im)
             1.0))))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = ((fma(0.16666666666666666, re, 0.5) * re) * re) * fma((im * im), -0.5, 1.0);
	} else if (t_0 <= -0.04) {
		tmp = cos(im);
	} else if (t_0 <= 0.0) {
		tmp = (1.0 + re) * ((im * im) * (0.041666666666666664 * (im * im)));
	} else if (t_0 <= 2.0) {
		tmp = cos(im);
	} else {
		tmp = ((fma(0.027777777777777776, (re * re), -0.25) * (re * re)) / ((0.16666666666666666 * re) - 0.5)) * fma(fma(0.041666666666666664, (im * im), -0.5), (im * im), 1.0);
	}
	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(Float64(fma(0.16666666666666666, re, 0.5) * re) * re) * fma(Float64(im * im), -0.5, 1.0));
	elseif (t_0 <= -0.04)
		tmp = cos(im);
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(1.0 + re) * Float64(Float64(im * im) * Float64(0.041666666666666664 * Float64(im * im))));
	elseif (t_0 <= 2.0)
		tmp = cos(im);
	else
		tmp = Float64(Float64(Float64(fma(0.027777777777777776, Float64(re * re), -0.25) * Float64(re * re)) / Float64(Float64(0.16666666666666666 * re) - 0.5)) * fma(fma(0.041666666666666664, Float64(im * im), -0.5), Float64(im * im), 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, (-Infinity)], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.04], N[Cos[im], $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(1.0 + re), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[Cos[im], $MachinePrecision], N[(N[(N[(N[(0.027777777777777776 * N[(re * re), $MachinePrecision] + -0.25), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] / N[(N[(0.16666666666666666 * re), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] * N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision] + -0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 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. fp-cancel-sign-sub-inv N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. metadata-eval N/A

         \[\leadsto \left(\left(re + re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) + \color{blue}{\left(0 + 1\right)}\right) \cdot \cos im \]

      8. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(re + re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) + 0\right) + 1\right)} \cdot \cos im \]

      9. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + re\right)} + 0\right) + 1\right) \cdot \cos im \]

      10. associate-+r+ N/A

          \[\leadsto \left(\color{blue}{\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(re + 0\right)\right)} + 1\right) \cdot \cos im \]

      11. *-rgt-identity N/A

          \[\leadsto \left(\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(\color{blue}{re \cdot 1} + 0\right)\right) + 1\right) \cdot \cos im \]

      12. mul0-rgt N/A

          \[\leadsto \left(\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(re \cdot 1 + \color{blue}{re \cdot 0}\right)\right) + 1\right) \cdot \cos im \]

      13. distribute-lft-in N/A

          \[\leadsto \left(\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \color{blue}{re \cdot \left(1 + 0\right)}\right) + 1\right) \cdot \cos im \]

      14. metadata-eval N/A

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

      15. distribute-lft-in N/A

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

      16. +-commutative N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 83.8%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0400000000000000008 or 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 re around 0

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

      1. lower-cos.f64 96.5

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

   5. Applied rewrites 96.5%

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

   if -0.0400000000000000008 < (*.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. lower-+.f64 2.3

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

   5. Applied rewrites 2.3%

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

   6. Taylor expanded in im around 0

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. metadata-eval N/A

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

      8. fp-cancel-sub-sign-inv N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 2.0

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

   8. Applied rewrites 2.0%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 45.6%

       \[\leadsto \left(1 + re\right) \cdot \left({im}^{4} \cdot \color{blue}{0.041666666666666664}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 45.6%

       \[\leadsto \left(1 + re\right) \cdot \left(\left(im \cdot im\right) \cdot \left(0.041666666666666664 \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \]

   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. fp-cancel-sign-sub-inv N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. metadata-eval N/A

         \[\leadsto \left(\left(re + re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) + \color{blue}{\left(0 + 1\right)}\right) \cdot \cos im \]

      8. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(re + re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) + 0\right) + 1\right)} \cdot \cos im \]

      9. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + re\right)} + 0\right) + 1\right) \cdot \cos im \]

      10. associate-+r+ N/A

          \[\leadsto \left(\color{blue}{\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(re + 0\right)\right)} + 1\right) \cdot \cos im \]

      11. *-rgt-identity N/A

          \[\leadsto \left(\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(\color{blue}{re \cdot 1} + 0\right)\right) + 1\right) \cdot \cos im \]

      12. mul0-rgt N/A

          \[\leadsto \left(\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(re \cdot 1 + \color{blue}{re \cdot 0}\right)\right) + 1\right) \cdot \cos im \]

      13. distribute-lft-in N/A

          \[\leadsto \left(\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \color{blue}{re \cdot \left(1 + 0\right)}\right) + 1\right) \cdot \cos im \]

      14. metadata-eval N/A

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

      15. distribute-lft-in N/A

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

      16. +-commutative N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 64.2%

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

   6. Taylor expanded in im around 0

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

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \color{blue}{\left(1 - \left(\mathsf{neg}\left({im}^{2}\right)\right) \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)} \]

      2. fp-cancel-sub-sign-inv N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({im}^{2}\right)\right)\right)\right) \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({im}^{2}\right)\right)\right)\right) \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) + 1\right)} \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \left(\color{blue}{{im}^{2}} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) + 1\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \left(\color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot {im}^{2}} + 1\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}, {im}^{2}, 1\right)} \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot {im}^{2} - \color{blue}{\frac{1}{2} \cdot 1}, {im}^{2}, 1\right) \]

      8. fp-cancel-sub-sign-inv N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot 1}, {im}^{2}, 1\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\frac{-1}{2}} \cdot 1, {im}^{2}, 1\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\frac{-1}{2}}, {im}^{2}, 1\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {im}^{2}, \frac{-1}{2}\right)}, {im}^{2}, 1\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{-1}{2}\right), {im}^{2}, 1\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{-1}{2}\right), {im}^{2}, 1\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{-1}{2}\right), \color{blue}{im \cdot im}, 1\right) \]

      15. lower-*.f64 78.9

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), \color{blue}{im \cdot im}, 1\right) \]

   8. Applied rewrites 78.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right)} \]

   9. Taylor expanded in re around inf

      \[\leadsto \left({re}^{3} \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{-1}{2}\right), im \cdot im, 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 78.9%

       \[\leadsto \left(\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot re\right) \cdot \color{blue}{re}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 83.0%

       \[\leadsto \frac{\mathsf{fma}\left(0.027777777777777776, re \cdot re, -0.25\right) \cdot \left(re \cdot re\right)}{0.16666666666666666 \cdot re - \color{blue}{0.5}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 5: 58.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -0.04:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot re\right) \cdot re\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(1 + re\right) \cdot \left(\left(im \cdot im\right) \cdot \left(0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;{im}^{-1} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.027777777777777776, re \cdot re, -0.25\right) \cdot \left(re \cdot re\right)}{0.16666666666666666 \cdot re - 0.5} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 -0.04)
     (*
      (* (* (fma 0.16666666666666666 re 0.5) re) re)
      (fma (* im im) -0.5 1.0))
     (if (<= t_0 0.0)
       (* (+ 1.0 re) (* (* im im) (* 0.041666666666666664 (* im im))))
       (if (<= t_0 2.0)
         (* (pow im -1.0) im)
         (*
          (/
           (* (fma 0.027777777777777776 (* re re) -0.25) (* re re))
           (- (* 0.16666666666666666 re) 0.5))
          (fma (fma 0.041666666666666664 (* im im) -0.5) (* im im) 1.0)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -0.04) {
		tmp = ((fma(0.16666666666666666, re, 0.5) * re) * re) * fma((im * im), -0.5, 1.0);
	} else if (t_0 <= 0.0) {
		tmp = (1.0 + re) * ((im * im) * (0.041666666666666664 * (im * im)));
	} else if (t_0 <= 2.0) {
		tmp = pow(im, -1.0) * im;
	} else {
		tmp = ((fma(0.027777777777777776, (re * re), -0.25) * (re * re)) / ((0.16666666666666666 * re) - 0.5)) * fma(fma(0.041666666666666664, (im * im), -0.5), (im * im), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= -0.04)
		tmp = Float64(Float64(Float64(fma(0.16666666666666666, re, 0.5) * re) * re) * fma(Float64(im * im), -0.5, 1.0));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(1.0 + re) * Float64(Float64(im * im) * Float64(0.041666666666666664 * Float64(im * im))));
	elseif (t_0 <= 2.0)
		tmp = Float64((im ^ -1.0) * im);
	else
		tmp = Float64(Float64(Float64(fma(0.027777777777777776, Float64(re * re), -0.25) * Float64(re * re)) / Float64(Float64(0.16666666666666666 * re) - 0.5)) * fma(fma(0.041666666666666664, Float64(im * im), -0.5), Float64(im * im), 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, -0.04], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(1.0 + re), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(N[Power[im, -1.0], $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(N[(0.027777777777777776 * N[(re * re), $MachinePrecision] + -0.25), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] / N[(N[(0.16666666666666666 * re), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] * N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision] + -0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0400000000000000008

   1. Initial program 99.9%

      \[e^{re} \cdot \cos im \]
   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. fp-cancel-sign-sub-inv N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. metadata-eval N/A

         \[\leadsto \left(\left(re + re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) + \color{blue}{\left(0 + 1\right)}\right) \cdot \cos im \]

      8. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(re + re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) + 0\right) + 1\right)} \cdot \cos im \]

      9. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + re\right)} + 0\right) + 1\right) \cdot \cos im \]

      10. associate-+r+ N/A

          \[\leadsto \left(\color{blue}{\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(re + 0\right)\right)} + 1\right) \cdot \cos im \]

      11. *-rgt-identity N/A

          \[\leadsto \left(\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(\color{blue}{re \cdot 1} + 0\right)\right) + 1\right) \cdot \cos im \]

      12. mul0-rgt N/A

          \[\leadsto \left(\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(re \cdot 1 + \color{blue}{re \cdot 0}\right)\right) + 1\right) \cdot \cos im \]

      13. distribute-lft-in N/A

          \[\leadsto \left(\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \color{blue}{re \cdot \left(1 + 0\right)}\right) + 1\right) \cdot \cos im \]

      14. metadata-eval N/A

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

      15. distribute-lft-in N/A

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

      16. +-commutative N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 92.9%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 36.3

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

   8. Applied rewrites 36.3%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 36.3%

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

   if -0.0400000000000000008 < (*.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. lower-+.f64 2.3

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

   5. Applied rewrites 2.3%

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

   6. Taylor expanded in im around 0

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. metadata-eval N/A

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

      8. fp-cancel-sub-sign-inv N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 2.0

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

   8. Applied rewrites 2.0%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 45.6%

       \[\leadsto \left(1 + re\right) \cdot \left({im}^{4} \cdot \color{blue}{0.041666666666666664}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 45.6%

       \[\leadsto \left(1 + re\right) \cdot \left(\left(im \cdot im\right) \cdot \left(0.041666666666666664 \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \]

   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 re around 0

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

      1. lower-cos.f64 97.0

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

   5. Applied rewrites 97.0%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 64.4%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 40.2%

       \[\leadsto \left(\left(\frac{1}{im \cdot im} - 0.5\right) \cdot im\right) \cdot im \]

   12. Taylor expanded in im around 0

       \[\leadsto \frac{1}{im} \cdot im \]
   13. Step-by-step derivation

   14. Applied rewrites 70.5%

       \[\leadsto \frac{1}{im} \cdot im \]

   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. fp-cancel-sign-sub-inv N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. metadata-eval N/A

         \[\leadsto \left(\left(re + re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) + \color{blue}{\left(0 + 1\right)}\right) \cdot \cos im \]

      8. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(re + re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) + 0\right) + 1\right)} \cdot \cos im \]

      9. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + re\right)} + 0\right) + 1\right) \cdot \cos im \]

      10. associate-+r+ N/A

          \[\leadsto \left(\color{blue}{\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(re + 0\right)\right)} + 1\right) \cdot \cos im \]

      11. *-rgt-identity N/A

          \[\leadsto \left(\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(\color{blue}{re \cdot 1} + 0\right)\right) + 1\right) \cdot \cos im \]

      12. mul0-rgt N/A

          \[\leadsto \left(\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(re \cdot 1 + \color{blue}{re \cdot 0}\right)\right) + 1\right) \cdot \cos im \]

      13. distribute-lft-in N/A

          \[\leadsto \left(\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \color{blue}{re \cdot \left(1 + 0\right)}\right) + 1\right) \cdot \cos im \]

      14. metadata-eval N/A

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

      15. distribute-lft-in N/A

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

      16. +-commutative N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 64.2%

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

   6. Taylor expanded in im around 0

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

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \color{blue}{\left(1 - \left(\mathsf{neg}\left({im}^{2}\right)\right) \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)} \]

      2. fp-cancel-sub-sign-inv N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({im}^{2}\right)\right)\right)\right) \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({im}^{2}\right)\right)\right)\right) \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) + 1\right)} \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \left(\color{blue}{{im}^{2}} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) + 1\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \left(\color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot {im}^{2}} + 1\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}, {im}^{2}, 1\right)} \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot {im}^{2} - \color{blue}{\frac{1}{2} \cdot 1}, {im}^{2}, 1\right) \]

      8. fp-cancel-sub-sign-inv N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot 1}, {im}^{2}, 1\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\frac{-1}{2}} \cdot 1, {im}^{2}, 1\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\frac{-1}{2}}, {im}^{2}, 1\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {im}^{2}, \frac{-1}{2}\right)}, {im}^{2}, 1\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{-1}{2}\right), {im}^{2}, 1\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{-1}{2}\right), {im}^{2}, 1\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{-1}{2}\right), \color{blue}{im \cdot im}, 1\right) \]

      15. lower-*.f64 78.9

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), \color{blue}{im \cdot im}, 1\right) \]

   8. Applied rewrites 78.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right)} \]

   9. Taylor expanded in re around inf

      \[\leadsto \left({re}^{3} \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{-1}{2}\right), im \cdot im, 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 78.9%

       \[\leadsto \left(\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot re\right) \cdot \color{blue}{re}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 83.0%

       \[\leadsto \frac{\mathsf{fma}\left(0.027777777777777776, re \cdot re, -0.25\right) \cdot \left(re \cdot re\right)}{0.16666666666666666 \cdot re - \color{blue}{0.5}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -0.04:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot re\right) \cdot re\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\left(1 + re\right) \cdot \left(\left(im \cdot im\right) \cdot \left(0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 2:\\ \;\;\;\;{im}^{-1} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.027777777777777776, re \cdot re, -0.25\right) \cdot \left(re \cdot re\right)}{0.16666666666666666 \cdot re - 0.5} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 57.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -0.04:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot re\right) \cdot re\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(1 + re\right) \cdot \left(\left(im \cdot im\right) \cdot \left(0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0.95:\\ \;\;\;\;{im}^{-1} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 -0.04)
     (*
      (* (* (fma 0.16666666666666666 re 0.5) re) re)
      (fma (* im im) -0.5 1.0))
     (if (<= t_0 0.0)
       (* (+ 1.0 re) (* (* im im) (* 0.041666666666666664 (* im im))))
       (if (<= t_0 0.95)
         (* (pow im -1.0) im)
         (*
          (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)
          (fma (fma 0.041666666666666664 (* im im) -0.5) (* im im) 1.0)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -0.04) {
		tmp = ((fma(0.16666666666666666, re, 0.5) * re) * re) * fma((im * im), -0.5, 1.0);
	} else if (t_0 <= 0.0) {
		tmp = (1.0 + re) * ((im * im) * (0.041666666666666664 * (im * im)));
	} else if (t_0 <= 0.95) {
		tmp = pow(im, -1.0) * im;
	} else {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * fma(fma(0.041666666666666664, (im * im), -0.5), (im * im), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= -0.04)
		tmp = Float64(Float64(Float64(fma(0.16666666666666666, re, 0.5) * re) * re) * fma(Float64(im * im), -0.5, 1.0));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(1.0 + re) * Float64(Float64(im * im) * Float64(0.041666666666666664 * Float64(im * im))));
	elseif (t_0 <= 0.95)
		tmp = Float64((im ^ -1.0) * im);
	else
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * fma(fma(0.041666666666666664, Float64(im * im), -0.5), Float64(im * im), 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, -0.04], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(1.0 + re), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.95], N[(N[Power[im, -1.0], $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision] + -0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0400000000000000008

   1. Initial program 99.9%

      \[e^{re} \cdot \cos im \]
   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. fp-cancel-sign-sub-inv N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. metadata-eval N/A

         \[\leadsto \left(\left(re + re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) + \color{blue}{\left(0 + 1\right)}\right) \cdot \cos im \]

      8. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(re + re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) + 0\right) + 1\right)} \cdot \cos im \]

      9. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + re\right)} + 0\right) + 1\right) \cdot \cos im \]

      10. associate-+r+ N/A

          \[\leadsto \left(\color{blue}{\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(re + 0\right)\right)} + 1\right) \cdot \cos im \]

      11. *-rgt-identity N/A

          \[\leadsto \left(\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(\color{blue}{re \cdot 1} + 0\right)\right) + 1\right) \cdot \cos im \]

      12. mul0-rgt N/A

          \[\leadsto \left(\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(re \cdot 1 + \color{blue}{re \cdot 0}\right)\right) + 1\right) \cdot \cos im \]

      13. distribute-lft-in N/A

          \[\leadsto \left(\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \color{blue}{re \cdot \left(1 + 0\right)}\right) + 1\right) \cdot \cos im \]

      14. metadata-eval N/A

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

      15. distribute-lft-in N/A

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

      16. +-commutative N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 92.9%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 36.3

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

   8. Applied rewrites 36.3%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 36.3%

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

   if -0.0400000000000000008 < (*.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. lower-+.f64 2.3

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

   5. Applied rewrites 2.3%

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

   6. Taylor expanded in im around 0

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. metadata-eval N/A

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

      8. fp-cancel-sub-sign-inv N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 2.0

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

   8. Applied rewrites 2.0%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 45.6%

       \[\leadsto \left(1 + re\right) \cdot \left({im}^{4} \cdot \color{blue}{0.041666666666666664}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 45.6%

       \[\leadsto \left(1 + re\right) \cdot \left(\left(im \cdot im\right) \cdot \left(0.041666666666666664 \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \]

   if 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.94999999999999996

   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 97.7

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

   5. Applied rewrites 97.7%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 0.9%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 0.9%

       \[\leadsto \left(\left(\frac{1}{im \cdot im} - 0.5\right) \cdot im\right) \cdot im \]

   12. Taylor expanded in im around 0

       \[\leadsto \frac{1}{im} \cdot im \]
   13. Step-by-step derivation

   14. Applied rewrites 19.4%

       \[\leadsto \frac{1}{im} \cdot im \]

   if 0.94999999999999996 < (*.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. fp-cancel-sign-sub-inv N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. metadata-eval N/A

         \[\leadsto \left(\left(re + re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) + \color{blue}{\left(0 + 1\right)}\right) \cdot \cos im \]

      8. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(re + re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) + 0\right) + 1\right)} \cdot \cos im \]

      9. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + re\right)} + 0\right) + 1\right) \cdot \cos im \]

      10. associate-+r+ N/A

          \[\leadsto \left(\color{blue}{\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(re + 0\right)\right)} + 1\right) \cdot \cos im \]

      11. *-rgt-identity N/A

          \[\leadsto \left(\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(\color{blue}{re \cdot 1} + 0\right)\right) + 1\right) \cdot \cos im \]

      12. mul0-rgt N/A

          \[\leadsto \left(\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(re \cdot 1 + \color{blue}{re \cdot 0}\right)\right) + 1\right) \cdot \cos im \]

      13. distribute-lft-in N/A

          \[\leadsto \left(\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \color{blue}{re \cdot \left(1 + 0\right)}\right) + 1\right) \cdot \cos im \]

      14. metadata-eval N/A

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

      15. distribute-lft-in N/A

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

      16. +-commutative N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 84.6%

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

   6. Taylor expanded in im around 0

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

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \color{blue}{\left(1 - \left(\mathsf{neg}\left({im}^{2}\right)\right) \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)} \]

      2. fp-cancel-sub-sign-inv N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({im}^{2}\right)\right)\right)\right) \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({im}^{2}\right)\right)\right)\right) \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) + 1\right)} \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \left(\color{blue}{{im}^{2}} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) + 1\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \left(\color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot {im}^{2}} + 1\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}, {im}^{2}, 1\right)} \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot {im}^{2} - \color{blue}{\frac{1}{2} \cdot 1}, {im}^{2}, 1\right) \]

      8. fp-cancel-sub-sign-inv N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot 1}, {im}^{2}, 1\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\frac{-1}{2}} \cdot 1, {im}^{2}, 1\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\frac{-1}{2}}, {im}^{2}, 1\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {im}^{2}, \frac{-1}{2}\right)}, {im}^{2}, 1\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{-1}{2}\right), {im}^{2}, 1\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{-1}{2}\right), {im}^{2}, 1\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{-1}{2}\right), \color{blue}{im \cdot im}, 1\right) \]

      15. lower-*.f64 89.8

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), \color{blue}{im \cdot im}, 1\right) \]

   8. Applied rewrites 89.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -0.04:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot re\right) \cdot re\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\left(1 + re\right) \cdot \left(\left(im \cdot im\right) \cdot \left(0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0.95:\\ \;\;\;\;{im}^{-1} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -0.04:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot re\right) \cdot re\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(1 + re\right) \cdot \left(\left(im \cdot im\right) \cdot \left(0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;{im}^{-1} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(0.16666666666666666 \cdot re\right) \cdot re\right) \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 -0.04)
     (*
      (* (* (fma 0.16666666666666666 re 0.5) re) re)
      (fma (* im im) -0.5 1.0))
     (if (<= t_0 0.0)
       (* (+ 1.0 re) (* (* im im) (* 0.041666666666666664 (* im im))))
       (if (<= t_0 2.0)
         (* (pow im -1.0) im)
         (*
          (* (* (* 0.16666666666666666 re) re) re)
          (fma (fma 0.041666666666666664 (* im im) -0.5) (* im im) 1.0)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -0.04) {
		tmp = ((fma(0.16666666666666666, re, 0.5) * re) * re) * fma((im * im), -0.5, 1.0);
	} else if (t_0 <= 0.0) {
		tmp = (1.0 + re) * ((im * im) * (0.041666666666666664 * (im * im)));
	} else if (t_0 <= 2.0) {
		tmp = pow(im, -1.0) * im;
	} else {
		tmp = (((0.16666666666666666 * re) * re) * re) * fma(fma(0.041666666666666664, (im * im), -0.5), (im * im), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= -0.04)
		tmp = Float64(Float64(Float64(fma(0.16666666666666666, re, 0.5) * re) * re) * fma(Float64(im * im), -0.5, 1.0));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(1.0 + re) * Float64(Float64(im * im) * Float64(0.041666666666666664 * Float64(im * im))));
	elseif (t_0 <= 2.0)
		tmp = Float64((im ^ -1.0) * im);
	else
		tmp = Float64(Float64(Float64(Float64(0.16666666666666666 * re) * re) * re) * fma(fma(0.041666666666666664, Float64(im * im), -0.5), Float64(im * im), 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, -0.04], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(1.0 + re), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(N[Power[im, -1.0], $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision] * N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision] + -0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0400000000000000008

   1. Initial program 99.9%

      \[e^{re} \cdot \cos im \]
   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. fp-cancel-sign-sub-inv N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. metadata-eval N/A

         \[\leadsto \left(\left(re + re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) + \color{blue}{\left(0 + 1\right)}\right) \cdot \cos im \]

      8. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(re + re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) + 0\right) + 1\right)} \cdot \cos im \]

      9. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + re\right)} + 0\right) + 1\right) \cdot \cos im \]

      10. associate-+r+ N/A

          \[\leadsto \left(\color{blue}{\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(re + 0\right)\right)} + 1\right) \cdot \cos im \]

      11. *-rgt-identity N/A

          \[\leadsto \left(\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(\color{blue}{re \cdot 1} + 0\right)\right) + 1\right) \cdot \cos im \]

      12. mul0-rgt N/A

          \[\leadsto \left(\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(re \cdot 1 + \color{blue}{re \cdot 0}\right)\right) + 1\right) \cdot \cos im \]

      13. distribute-lft-in N/A

          \[\leadsto \left(\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \color{blue}{re \cdot \left(1 + 0\right)}\right) + 1\right) \cdot \cos im \]

      14. metadata-eval N/A

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

      15. distribute-lft-in N/A

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

      16. +-commutative N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 92.9%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 36.3

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

   8. Applied rewrites 36.3%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 36.3%

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

   if -0.0400000000000000008 < (*.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. lower-+.f64 2.3

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

   5. Applied rewrites 2.3%

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

   6. Taylor expanded in im around 0

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. metadata-eval N/A

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

      8. fp-cancel-sub-sign-inv N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 2.0

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

   8. Applied rewrites 2.0%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 45.6%

       \[\leadsto \left(1 + re\right) \cdot \left({im}^{4} \cdot \color{blue}{0.041666666666666664}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 45.6%

       \[\leadsto \left(1 + re\right) \cdot \left(\left(im \cdot im\right) \cdot \left(0.041666666666666664 \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \]

   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 re around 0

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

      1. lower-cos.f64 97.0

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

   5. Applied rewrites 97.0%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 64.4%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 40.2%

       \[\leadsto \left(\left(\frac{1}{im \cdot im} - 0.5\right) \cdot im\right) \cdot im \]

   12. Taylor expanded in im around 0

       \[\leadsto \frac{1}{im} \cdot im \]
   13. Step-by-step derivation

   14. Applied rewrites 70.5%

       \[\leadsto \frac{1}{im} \cdot im \]

   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. fp-cancel-sign-sub-inv N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. metadata-eval N/A

         \[\leadsto \left(\left(re + re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) + \color{blue}{\left(0 + 1\right)}\right) \cdot \cos im \]

      8. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(re + re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) + 0\right) + 1\right)} \cdot \cos im \]

      9. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + re\right)} + 0\right) + 1\right) \cdot \cos im \]

      10. associate-+r+ N/A

          \[\leadsto \left(\color{blue}{\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(re + 0\right)\right)} + 1\right) \cdot \cos im \]

      11. *-rgt-identity N/A

          \[\leadsto \left(\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(\color{blue}{re \cdot 1} + 0\right)\right) + 1\right) \cdot \cos im \]

      12. mul0-rgt N/A

          \[\leadsto \left(\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(re \cdot 1 + \color{blue}{re \cdot 0}\right)\right) + 1\right) \cdot \cos im \]

      13. distribute-lft-in N/A

          \[\leadsto \left(\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \color{blue}{re \cdot \left(1 + 0\right)}\right) + 1\right) \cdot \cos im \]

      14. metadata-eval N/A

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

      15. distribute-lft-in N/A

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

      16. +-commutative N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 64.2%

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

   6. Taylor expanded in im around 0

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

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \color{blue}{\left(1 - \left(\mathsf{neg}\left({im}^{2}\right)\right) \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)} \]

      2. fp-cancel-sub-sign-inv N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({im}^{2}\right)\right)\right)\right) \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({im}^{2}\right)\right)\right)\right) \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) + 1\right)} \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \left(\color{blue}{{im}^{2}} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) + 1\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \left(\color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot {im}^{2}} + 1\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}, {im}^{2}, 1\right)} \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot {im}^{2} - \color{blue}{\frac{1}{2} \cdot 1}, {im}^{2}, 1\right) \]

      8. fp-cancel-sub-sign-inv N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot 1}, {im}^{2}, 1\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\frac{-1}{2}} \cdot 1, {im}^{2}, 1\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\frac{-1}{2}}, {im}^{2}, 1\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {im}^{2}, \frac{-1}{2}\right)}, {im}^{2}, 1\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{-1}{2}\right), {im}^{2}, 1\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{-1}{2}\right), {im}^{2}, 1\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{-1}{2}\right), \color{blue}{im \cdot im}, 1\right) \]

      15. lower-*.f64 78.9

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), \color{blue}{im \cdot im}, 1\right) \]

   8. Applied rewrites 78.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right)} \]

   9. Taylor expanded in re around inf

      \[\leadsto \left({re}^{3} \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{-1}{2}\right), im \cdot im, 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 78.9%

       \[\leadsto \left(\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot re\right) \cdot \color{blue}{re}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right) \]

   12. Taylor expanded in re around inf

       \[\leadsto \left(\left(\left(\frac{1}{6} \cdot re\right) \cdot re\right) \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{-1}{2}\right), im \cdot im, 1\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 78.9%

       \[\leadsto \left(\left(\left(0.16666666666666666 \cdot re\right) \cdot re\right) \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -0.04:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot re\right) \cdot re\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\left(1 + re\right) \cdot \left(\left(im \cdot im\right) \cdot \left(0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 2:\\ \;\;\;\;{im}^{-1} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(0.16666666666666666 \cdot re\right) \cdot re\right) \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 55.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -0.04:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot re\right) \cdot re\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(1 + re\right) \cdot \left(\left(im \cdot im\right) \cdot \left(0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0.999:\\ \;\;\;\;{im}^{-1} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 -0.04)
     (*
      (* (* (fma 0.16666666666666666 re 0.5) re) re)
      (fma (* im im) -0.5 1.0))
     (if (<= t_0 0.0)
       (* (+ 1.0 re) (* (* im im) (* 0.041666666666666664 (* im im))))
       (if (<= t_0 0.999)
         (* (pow im -1.0) im)
         (*
          (fma (fma 0.5 re 1.0) re 1.0)
          (fma (fma 0.041666666666666664 (* im im) -0.5) (* im im) 1.0)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -0.04) {
		tmp = ((fma(0.16666666666666666, re, 0.5) * re) * re) * fma((im * im), -0.5, 1.0);
	} else if (t_0 <= 0.0) {
		tmp = (1.0 + re) * ((im * im) * (0.041666666666666664 * (im * im)));
	} else if (t_0 <= 0.999) {
		tmp = pow(im, -1.0) * im;
	} else {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * fma(fma(0.041666666666666664, (im * im), -0.5), (im * im), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= -0.04)
		tmp = Float64(Float64(Float64(fma(0.16666666666666666, re, 0.5) * re) * re) * fma(Float64(im * im), -0.5, 1.0));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(1.0 + re) * Float64(Float64(im * im) * Float64(0.041666666666666664 * Float64(im * im))));
	elseif (t_0 <= 0.999)
		tmp = Float64((im ^ -1.0) * im);
	else
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * fma(fma(0.041666666666666664, Float64(im * im), -0.5), Float64(im * im), 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, -0.04], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(1.0 + re), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.999], N[(N[Power[im, -1.0], $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision] + -0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0400000000000000008

   1. Initial program 99.9%

      \[e^{re} \cdot \cos im \]
   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. fp-cancel-sign-sub-inv N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. metadata-eval N/A

         \[\leadsto \left(\left(re + re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) + \color{blue}{\left(0 + 1\right)}\right) \cdot \cos im \]

      8. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(re + re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) + 0\right) + 1\right)} \cdot \cos im \]

      9. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + re\right)} + 0\right) + 1\right) \cdot \cos im \]

      10. associate-+r+ N/A

          \[\leadsto \left(\color{blue}{\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(re + 0\right)\right)} + 1\right) \cdot \cos im \]

      11. *-rgt-identity N/A

          \[\leadsto \left(\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(\color{blue}{re \cdot 1} + 0\right)\right) + 1\right) \cdot \cos im \]

      12. mul0-rgt N/A

          \[\leadsto \left(\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(re \cdot 1 + \color{blue}{re \cdot 0}\right)\right) + 1\right) \cdot \cos im \]

      13. distribute-lft-in N/A

          \[\leadsto \left(\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \color{blue}{re \cdot \left(1 + 0\right)}\right) + 1\right) \cdot \cos im \]

      14. metadata-eval N/A

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

      15. distribute-lft-in N/A

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

      16. +-commutative N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 92.9%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 36.3

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

   8. Applied rewrites 36.3%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 36.3%

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

   if -0.0400000000000000008 < (*.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. lower-+.f64 2.3

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

   5. Applied rewrites 2.3%

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

   6. Taylor expanded in im around 0

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. metadata-eval N/A

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

      8. fp-cancel-sub-sign-inv N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 2.0

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

   8. Applied rewrites 2.0%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 45.6%

       \[\leadsto \left(1 + re\right) \cdot \left({im}^{4} \cdot \color{blue}{0.041666666666666664}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 45.6%

       \[\leadsto \left(1 + re\right) \cdot \left(\left(im \cdot im\right) \cdot \left(0.041666666666666664 \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \]

   if 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.998999999999999999

   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 95.6

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

   5. Applied rewrites 95.6%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 1.7%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 1.0%

       \[\leadsto \left(\left(\frac{1}{im \cdot im} - 0.5\right) \cdot im\right) \cdot im \]

   12. Taylor expanded in im around 0

       \[\leadsto \frac{1}{im} \cdot im \]
   13. Step-by-step derivation

   14. Applied rewrites 20.0%

       \[\leadsto \frac{1}{im} \cdot im \]

   if 0.998999999999999999 < (*.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

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-neg-out N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. remove-double-neg N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt1-in N/A

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

      10. *-commutative N/A

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

      11. metadata-eval N/A

          \[\leadsto \left(\left(re + re \cdot \left(\frac{1}{2} \cdot re\right)\right) + \color{blue}{\left(0 + 1\right)}\right) \cdot \cos im \]

      12. associate-+r+ N/A

          \[\leadsto \color{blue}{\left(\left(\left(re + re \cdot \left(\frac{1}{2} \cdot re\right)\right) + 0\right) + 1\right)} \cdot \cos im \]

   5. Applied rewrites 79.1%

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

   6. Taylor expanded in im around 0

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. metadata-eval N/A

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

      8. fp-cancel-sub-sign-inv N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 84.5

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

   8. Applied rewrites 84.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -0.04:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot re\right) \cdot re\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\left(1 + re\right) \cdot \left(\left(im \cdot im\right) \cdot \left(0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0.999:\\ \;\;\;\;{im}^{-1} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 54.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -0.04:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot re\right) \cdot re\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(1 + re\right) \cdot \left(\left(im \cdot im\right) \cdot \left(0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;{im}^{-1} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.5 \cdot re\right) \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 -0.04)
     (*
      (* (* (fma 0.16666666666666666 re 0.5) re) re)
      (fma (* im im) -0.5 1.0))
     (if (<= t_0 0.0)
       (* (+ 1.0 re) (* (* im im) (* 0.041666666666666664 (* im im))))
       (if (<= t_0 2.0)
         (* (pow im -1.0) im)
         (*
          (* (* 0.5 re) re)
          (fma (fma 0.041666666666666664 (* im im) -0.5) (* im im) 1.0)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -0.04) {
		tmp = ((fma(0.16666666666666666, re, 0.5) * re) * re) * fma((im * im), -0.5, 1.0);
	} else if (t_0 <= 0.0) {
		tmp = (1.0 + re) * ((im * im) * (0.041666666666666664 * (im * im)));
	} else if (t_0 <= 2.0) {
		tmp = pow(im, -1.0) * im;
	} else {
		tmp = ((0.5 * re) * re) * fma(fma(0.041666666666666664, (im * im), -0.5), (im * im), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= -0.04)
		tmp = Float64(Float64(Float64(fma(0.16666666666666666, re, 0.5) * re) * re) * fma(Float64(im * im), -0.5, 1.0));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(1.0 + re) * Float64(Float64(im * im) * Float64(0.041666666666666664 * Float64(im * im))));
	elseif (t_0 <= 2.0)
		tmp = Float64((im ^ -1.0) * im);
	else
		tmp = Float64(Float64(Float64(0.5 * re) * re) * fma(fma(0.041666666666666664, Float64(im * im), -0.5), Float64(im * im), 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, -0.04], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(1.0 + re), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(N[Power[im, -1.0], $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(0.5 * re), $MachinePrecision] * re), $MachinePrecision] * N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision] + -0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0400000000000000008

   1. Initial program 99.9%

      \[e^{re} \cdot \cos im \]
   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. fp-cancel-sign-sub-inv N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. metadata-eval N/A

         \[\leadsto \left(\left(re + re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) + \color{blue}{\left(0 + 1\right)}\right) \cdot \cos im \]

      8. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(re + re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) + 0\right) + 1\right)} \cdot \cos im \]

      9. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + re\right)} + 0\right) + 1\right) \cdot \cos im \]

      10. associate-+r+ N/A

          \[\leadsto \left(\color{blue}{\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(re + 0\right)\right)} + 1\right) \cdot \cos im \]

      11. *-rgt-identity N/A

          \[\leadsto \left(\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(\color{blue}{re \cdot 1} + 0\right)\right) + 1\right) \cdot \cos im \]

      12. mul0-rgt N/A

          \[\leadsto \left(\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(re \cdot 1 + \color{blue}{re \cdot 0}\right)\right) + 1\right) \cdot \cos im \]

      13. distribute-lft-in N/A

          \[\leadsto \left(\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \color{blue}{re \cdot \left(1 + 0\right)}\right) + 1\right) \cdot \cos im \]

      14. metadata-eval N/A

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

      15. distribute-lft-in N/A

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

      16. +-commutative N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 92.9%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 36.3

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

   8. Applied rewrites 36.3%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 36.3%

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

   if -0.0400000000000000008 < (*.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. lower-+.f64 2.3

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

   5. Applied rewrites 2.3%

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

   6. Taylor expanded in im around 0

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. metadata-eval N/A

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

      8. fp-cancel-sub-sign-inv N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 2.0

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

   8. Applied rewrites 2.0%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 45.6%

       \[\leadsto \left(1 + re\right) \cdot \left({im}^{4} \cdot \color{blue}{0.041666666666666664}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 45.6%

       \[\leadsto \left(1 + re\right) \cdot \left(\left(im \cdot im\right) \cdot \left(0.041666666666666664 \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \]

   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 re around 0

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

      1. lower-cos.f64 97.0

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

   5. Applied rewrites 97.0%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 64.4%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 40.2%

       \[\leadsto \left(\left(\frac{1}{im \cdot im} - 0.5\right) \cdot im\right) \cdot im \]

   12. Taylor expanded in im around 0

       \[\leadsto \frac{1}{im} \cdot im \]
   13. Step-by-step derivation

   14. Applied rewrites 70.5%

       \[\leadsto \frac{1}{im} \cdot im \]

   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. fp-cancel-sign-sub-inv N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. metadata-eval N/A

         \[\leadsto \left(\left(re + re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) + \color{blue}{\left(0 + 1\right)}\right) \cdot \cos im \]

      8. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(re + re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) + 0\right) + 1\right)} \cdot \cos im \]

      9. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + re\right)} + 0\right) + 1\right) \cdot \cos im \]

      10. associate-+r+ N/A

          \[\leadsto \left(\color{blue}{\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(re + 0\right)\right)} + 1\right) \cdot \cos im \]

      11. *-rgt-identity N/A

          \[\leadsto \left(\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(\color{blue}{re \cdot 1} + 0\right)\right) + 1\right) \cdot \cos im \]

      12. mul0-rgt N/A

          \[\leadsto \left(\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(re \cdot 1 + \color{blue}{re \cdot 0}\right)\right) + 1\right) \cdot \cos im \]

      13. distribute-lft-in N/A

          \[\leadsto \left(\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \color{blue}{re \cdot \left(1 + 0\right)}\right) + 1\right) \cdot \cos im \]

      14. metadata-eval N/A

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

      15. distribute-lft-in N/A

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

      16. +-commutative N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 64.2%

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

   6. Taylor expanded in im around 0

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

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \color{blue}{\left(1 - \left(\mathsf{neg}\left({im}^{2}\right)\right) \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)} \]

      2. fp-cancel-sub-sign-inv N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({im}^{2}\right)\right)\right)\right) \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({im}^{2}\right)\right)\right)\right) \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) + 1\right)} \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \left(\color{blue}{{im}^{2}} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) + 1\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \left(\color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot {im}^{2}} + 1\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}, {im}^{2}, 1\right)} \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot {im}^{2} - \color{blue}{\frac{1}{2} \cdot 1}, {im}^{2}, 1\right) \]

      8. fp-cancel-sub-sign-inv N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot 1}, {im}^{2}, 1\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\frac{-1}{2}} \cdot 1, {im}^{2}, 1\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\frac{-1}{2}}, {im}^{2}, 1\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {im}^{2}, \frac{-1}{2}\right)}, {im}^{2}, 1\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{-1}{2}\right), {im}^{2}, 1\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{-1}{2}\right), {im}^{2}, 1\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{-1}{2}\right), \color{blue}{im \cdot im}, 1\right) \]

      15. lower-*.f64 78.9

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), \color{blue}{im \cdot im}, 1\right) \]

   8. Applied rewrites 78.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right)} \]

   9. Taylor expanded in re around inf

      \[\leadsto \left({re}^{3} \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{-1}{2}\right), im \cdot im, 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 78.9%

       \[\leadsto \left(\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot re\right) \cdot \color{blue}{re}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right) \]

   12. Taylor expanded in re around 0

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

   14. Applied rewrites 64.3%

       \[\leadsto \left(\left(0.5 \cdot re\right) \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -0.04:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot re\right) \cdot re\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\left(1 + re\right) \cdot \left(\left(im \cdot im\right) \cdot \left(0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 2:\\ \;\;\;\;{im}^{-1} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.5 \cdot re\right) \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 43.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -100:\\ \;\;\;\;\left(1 + re\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(-0.5 \cdot im\right) \cdot im\\ \mathbf{elif}\;t\_0 \leq 0.999:\\ \;\;\;\;{im}^{-1} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 -100.0)
     (* (+ 1.0 re) (fma (* im im) -0.5 1.0))
     (if (<= t_0 0.0)
       (* (* -0.5 im) im)
       (if (<= t_0 0.999)
         (* (pow im -1.0) im)
         (fma (fma 0.041666666666666664 (* im im) -0.5) (* im im) 1.0))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -100.0) {
		tmp = (1.0 + re) * fma((im * im), -0.5, 1.0);
	} else if (t_0 <= 0.0) {
		tmp = (-0.5 * im) * im;
	} else if (t_0 <= 0.999) {
		tmp = pow(im, -1.0) * im;
	} else {
		tmp = fma(fma(0.041666666666666664, (im * im), -0.5), (im * im), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= -100.0)
		tmp = Float64(Float64(1.0 + re) * fma(Float64(im * im), -0.5, 1.0));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(-0.5 * im) * im);
	elseif (t_0 <= 0.999)
		tmp = Float64((im ^ -1.0) * im);
	else
		tmp = fma(fma(0.041666666666666664, Float64(im * im), -0.5), Float64(im * im), 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, -100.0], N[(N[(1.0 + re), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(-0.5 * im), $MachinePrecision] * im), $MachinePrecision], If[LessEqual[t$95$0, 0.999], N[(N[Power[im, -1.0], $MachinePrecision] * im), $MachinePrecision], N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision] + -0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -100

   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. lower-+.f64 6.8

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

   5. Applied rewrites 6.8%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 68.4

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

   8. Applied rewrites 68.4%

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

   if -100 < (*.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 35.0

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

   5. Applied rewrites 35.0%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 3.2%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 21.3%

       \[\leadsto \left(im \cdot im\right) \cdot -0.5 \]
   12. Step-by-step derivation

   13. Applied rewrites 21.3%

       \[\leadsto \left(-0.5 \cdot im\right) \cdot im \]

   if 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.998999999999999999

   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 95.6

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

   5. Applied rewrites 95.6%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 1.7%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 1.0%

       \[\leadsto \left(\left(\frac{1}{im \cdot im} - 0.5\right) \cdot im\right) \cdot im \]

   12. Taylor expanded in im around 0

       \[\leadsto \frac{1}{im} \cdot im \]
   13. Step-by-step derivation

   14. Applied rewrites 20.0%

       \[\leadsto \frac{1}{im} \cdot im \]

   if 0.998999999999999999 < (*.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

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

      1. lower-cos.f64 58.0

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

   5. Applied rewrites 58.0%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 68.0%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), \color{blue}{im \cdot im}, 1\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -100:\\ \;\;\;\;\left(1 + re\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\left(-0.5 \cdot im\right) \cdot im\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0.999:\\ \;\;\;\;{im}^{-1} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 39.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -100:\\ \;\;\;\;\left(1 + re\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(-0.5 \cdot im\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;{im}^{-1} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 -100.0)
     (* (+ 1.0 re) (fma (* im im) -0.5 1.0))
     (if (<= t_0 0.0) (* (* -0.5 im) im) (* (pow im -1.0) im)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -100.0) {
		tmp = (1.0 + re) * fma((im * im), -0.5, 1.0);
	} else if (t_0 <= 0.0) {
		tmp = (-0.5 * im) * im;
	} else {
		tmp = pow(im, -1.0) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= -100.0)
		tmp = Float64(Float64(1.0 + re) * fma(Float64(im * im), -0.5, 1.0));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(-0.5 * im) * im);
	else
		tmp = Float64((im ^ -1.0) * im);
	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, -100.0], N[(N[(1.0 + re), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(-0.5 * im), $MachinePrecision] * im), $MachinePrecision], N[(N[Power[im, -1.0], $MachinePrecision] * im), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -100

   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. lower-+.f64 6.8

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

   5. Applied rewrites 6.8%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 68.4

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

   8. Applied rewrites 68.4%

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

   if -100 < (*.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 35.0

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

   5. Applied rewrites 35.0%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 3.2%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 21.3%

       \[\leadsto \left(im \cdot im\right) \cdot -0.5 \]
   12. Step-by-step derivation

   13. Applied rewrites 21.3%

       \[\leadsto \left(-0.5 \cdot im\right) \cdot im \]

   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

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

      1. lower-cos.f64 66.8

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

   5. Applied rewrites 66.8%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 44.4%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 35.5%

       \[\leadsto \left(\left(\frac{1}{im \cdot im} - 0.5\right) \cdot im\right) \cdot im \]

   12. Taylor expanded in im around 0

       \[\leadsto \frac{1}{im} \cdot im \]
   13. Step-by-step derivation

   14. Applied rewrites 48.8%

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

4. Final simplification 39.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -100:\\ \;\;\;\;\left(1 + re\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\left(-0.5 \cdot im\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;{im}^{-1} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 37.7% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((exp(re) * cos(im)) <= 0.0d0) then
        tmp = ((-0.5d0) * im) * im
    else
        tmp = (im ** (-1.0d0)) * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) * Math.cos(im)) <= 0.0) {
		tmp = (-0.5 * im) * im;
	} else {
		tmp = Math.pow(im, -1.0) * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) * math.cos(im)) <= 0.0:
		tmp = (-0.5 * im) * im
	else:
		tmp = math.pow(im, -1.0) * im
	return tmp

Julia

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

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) * cos(im)) <= 0.0)
		tmp = (-0.5 * im) * im;
	else
		tmp = (im ^ -1.0) * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(-0.5 * im), $MachinePrecision] * im), $MachinePrecision], N[(N[Power[im, -1.0], $MachinePrecision] * im), $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 30.1

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

   5. Applied rewrites 30.1%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 9.1%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 24.4%

       \[\leadsto \left(im \cdot im\right) \cdot -0.5 \]
   12. Step-by-step derivation

   13. Applied rewrites 24.4%

       \[\leadsto \left(-0.5 \cdot im\right) \cdot im \]

   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

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

      1. lower-cos.f64 66.8

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

   5. Applied rewrites 66.8%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 44.4%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 35.5%

       \[\leadsto \left(\left(\frac{1}{im \cdot im} - 0.5\right) \cdot im\right) \cdot im \]

   12. Taylor expanded in im around 0

       \[\leadsto \frac{1}{im} \cdot im \]
   13. Step-by-step derivation

   14. Applied rewrites 48.8%

       \[\leadsto \frac{1}{im} \cdot im \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.8%

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

Alternative 13: 35.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * cos(im)) <= 0.0)
		tmp = Float64(Float64(-0.5 * im) * im);
	else
		tmp = fma(Float64(im * im), -0.5, 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[(-0.5 * im), $MachinePrecision] * im), $MachinePrecision], N[(N[(im * im), $MachinePrecision] * -0.5 + 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 30.1

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

   5. Applied rewrites 30.1%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 9.1%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 24.4%

       \[\leadsto \left(im \cdot im\right) \cdot -0.5 \]
   12. Step-by-step derivation

   13. Applied rewrites 24.4%

       \[\leadsto \left(-0.5 \cdot im\right) \cdot im \]

   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

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

      1. lower-cos.f64 66.8

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

   5. Applied rewrites 66.8%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 44.4%

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

Alternative 14: 90.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.0275:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;re \leq 5.2:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \cos im\\ \mathbf{elif}\;re \leq 2.1 \cdot 10^{+94}:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5, im \cdot im, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot re\right) \cdot re\right) \cdot \cos im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.0275)
   (* (exp re) (fma (* im im) -0.5 1.0))
   (if (<= re 5.2)
     (* (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0) (cos im))
     (if (<= re 2.1e+94)
       (*
        (exp re)
        (fma (- (* 0.041666666666666664 (* im im)) 0.5) (* im im) 1.0))
       (* (* (* (fma 0.16666666666666666 re 0.5) re) re) (cos im))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.0275) {
		tmp = exp(re) * fma((im * im), -0.5, 1.0);
	} else if (re <= 5.2) {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * cos(im);
	} else if (re <= 2.1e+94) {
		tmp = exp(re) * fma(((0.041666666666666664 * (im * im)) - 0.5), (im * im), 1.0);
	} else {
		tmp = ((fma(0.16666666666666666, re, 0.5) * re) * re) * cos(im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.0275)
		tmp = Float64(exp(re) * fma(Float64(im * im), -0.5, 1.0));
	elseif (re <= 5.2)
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * cos(im));
	elseif (re <= 2.1e+94)
		tmp = Float64(exp(re) * fma(Float64(Float64(0.041666666666666664 * Float64(im * im)) - 0.5), Float64(im * im), 1.0));
	else
		tmp = Float64(Float64(Float64(fma(0.16666666666666666, re, 0.5) * re) * re) * cos(im));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.0275], N[(N[Exp[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 5.2], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.1e+94], N[(N[Exp[re], $MachinePrecision] * N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -0.0275000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 74.2

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

   5. Applied rewrites 74.2%

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

   if -0.0275000000000000001 < re < 5.20000000000000018

   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. fp-cancel-sign-sub-inv N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. metadata-eval N/A

         \[\leadsto \left(\left(re + re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) + \color{blue}{\left(0 + 1\right)}\right) \cdot \cos im \]

      8. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(re + re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) + 0\right) + 1\right)} \cdot \cos im \]

      9. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + re\right)} + 0\right) + 1\right) \cdot \cos im \]

      10. associate-+r+ N/A

          \[\leadsto \left(\color{blue}{\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(re + 0\right)\right)} + 1\right) \cdot \cos im \]

      11. *-rgt-identity N/A

          \[\leadsto \left(\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(\color{blue}{re \cdot 1} + 0\right)\right) + 1\right) \cdot \cos im \]

      12. mul0-rgt N/A

          \[\leadsto \left(\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(re \cdot 1 + \color{blue}{re \cdot 0}\right)\right) + 1\right) \cdot \cos im \]

      13. distribute-lft-in N/A

          \[\leadsto \left(\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \color{blue}{re \cdot \left(1 + 0\right)}\right) + 1\right) \cdot \cos im \]

      14. metadata-eval N/A

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

      15. distribute-lft-in N/A

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

      16. +-commutative N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 98.9%

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

   if 5.20000000000000018 < re < 2.09999999999999989e94

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-neg-out N/A

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

      5. distribute-lft-neg-out N/A

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

      6. remove-double-neg N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 94.4

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

   5. Applied rewrites 94.4%

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

   if 2.09999999999999989e94 < 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. fp-cancel-sign-sub-inv N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. metadata-eval N/A

         \[\leadsto \left(\left(re + re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) + \color{blue}{\left(0 + 1\right)}\right) \cdot \cos im \]

      8. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(re + re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) + 0\right) + 1\right)} \cdot \cos im \]

      9. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + re\right)} + 0\right) + 1\right) \cdot \cos im \]

      10. associate-+r+ N/A

          \[\leadsto \left(\color{blue}{\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(re + 0\right)\right)} + 1\right) \cdot \cos im \]

      11. *-rgt-identity N/A

          \[\leadsto \left(\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(\color{blue}{re \cdot 1} + 0\right)\right) + 1\right) \cdot \cos im \]

      12. mul0-rgt N/A

          \[\leadsto \left(\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(re \cdot 1 + \color{blue}{re \cdot 0}\right)\right) + 1\right) \cdot \cos im \]

      13. distribute-lft-in N/A

          \[\leadsto \left(\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \color{blue}{re \cdot \left(1 + 0\right)}\right) + 1\right) \cdot \cos im \]

      14. metadata-eval N/A

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

      15. distribute-lft-in N/A

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

      16. +-commutative N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 95.9%

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

   6. Taylor expanded in re around inf

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

   8. Applied rewrites 95.9%

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

Alternative 15: 90.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{if}\;re \leq -0.0275:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 5.2:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \cos im\\ \mathbf{elif}\;re \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot 0.5\right) \cdot \cos im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (fma (* im im) -0.5 1.0))))
   (if (<= re -0.0275)
     t_0
     (if (<= re 5.2)
       (* (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0) (cos im))
       (if (<= re 1.4e+154) t_0 (* (* (* re re) 0.5) (cos im)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * fma((im * im), -0.5, 1.0);
	double tmp;
	if (re <= -0.0275) {
		tmp = t_0;
	} else if (re <= 5.2) {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * cos(im);
	} else if (re <= 1.4e+154) {
		tmp = t_0;
	} else {
		tmp = ((re * re) * 0.5) * cos(im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * fma(Float64(im * im), -0.5, 1.0))
	tmp = 0.0
	if (re <= -0.0275)
		tmp = t_0;
	elseif (re <= 5.2)
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * cos(im));
	elseif (re <= 1.4e+154)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(re * re) * 0.5) * cos(im));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -0.0275], t$95$0, If[LessEqual[re, 5.2], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.4e+154], t$95$0, N[(N[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.0275000000000000001 or 5.20000000000000018 < re < 1.4e154

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 74.7

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

   5. Applied rewrites 74.7%

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

   if -0.0275000000000000001 < re < 5.20000000000000018

   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. fp-cancel-sign-sub-inv N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. metadata-eval N/A

         \[\leadsto \left(\left(re + re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) + \color{blue}{\left(0 + 1\right)}\right) \cdot \cos im \]

      8. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(re + re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) + 0\right) + 1\right)} \cdot \cos im \]

      9. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + re\right)} + 0\right) + 1\right) \cdot \cos im \]

      10. associate-+r+ N/A

          \[\leadsto \left(\color{blue}{\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(re + 0\right)\right)} + 1\right) \cdot \cos im \]

      11. *-rgt-identity N/A

          \[\leadsto \left(\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(\color{blue}{re \cdot 1} + 0\right)\right) + 1\right) \cdot \cos im \]

      12. mul0-rgt N/A

          \[\leadsto \left(\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(re \cdot 1 + \color{blue}{re \cdot 0}\right)\right) + 1\right) \cdot \cos im \]

      13. distribute-lft-in N/A

          \[\leadsto \left(\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \color{blue}{re \cdot \left(1 + 0\right)}\right) + 1\right) \cdot \cos im \]

      14. metadata-eval N/A

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

      15. distribute-lft-in N/A

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

      16. +-commutative N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 98.9%

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

   if 1.4e154 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-neg-out N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. remove-double-neg N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt1-in N/A

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

      10. *-commutative N/A

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

      11. metadata-eval N/A

          \[\leadsto \left(\left(re + re \cdot \left(\frac{1}{2} \cdot re\right)\right) + \color{blue}{\left(0 + 1\right)}\right) \cdot \cos im \]

      12. associate-+r+ N/A

          \[\leadsto \color{blue}{\left(\left(\left(re + re \cdot \left(\frac{1}{2} \cdot re\right)\right) + 0\right) + 1\right)} \cdot \cos im \]

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in re around inf

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

   8. Applied rewrites 100.0%

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

Alternative 16: 90.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{if}\;re \leq -4.2 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 5.2:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \cos im\\ \mathbf{elif}\;re \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot 0.5\right) \cdot \cos im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (fma (* im im) -0.5 1.0))))
   (if (<= re -4.2e-5)
     t_0
     (if (<= re 5.2)
       (* (fma (fma 0.5 re 1.0) re 1.0) (cos im))
       (if (<= re 1.4e+154) t_0 (* (* (* re re) 0.5) (cos im)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * fma((im * im), -0.5, 1.0);
	double tmp;
	if (re <= -4.2e-5) {
		tmp = t_0;
	} else if (re <= 5.2) {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * cos(im);
	} else if (re <= 1.4e+154) {
		tmp = t_0;
	} else {
		tmp = ((re * re) * 0.5) * cos(im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * fma(Float64(im * im), -0.5, 1.0))
	tmp = 0.0
	if (re <= -4.2e-5)
		tmp = t_0;
	elseif (re <= 5.2)
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * cos(im));
	elseif (re <= 1.4e+154)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(re * re) * 0.5) * cos(im));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -4.2e-5], t$95$0, If[LessEqual[re, 5.2], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.4e+154], t$95$0, N[(N[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -4.19999999999999977e-5 or 5.20000000000000018 < re < 1.4e154

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 75.0

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

   5. Applied rewrites 75.0%

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

   if -4.19999999999999977e-5 < re < 5.20000000000000018

   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. fp-cancel-sign-sub-inv N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-neg-out N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. remove-double-neg N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt1-in N/A

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

      10. *-commutative N/A

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

      11. metadata-eval N/A

          \[\leadsto \left(\left(re + re \cdot \left(\frac{1}{2} \cdot re\right)\right) + \color{blue}{\left(0 + 1\right)}\right) \cdot \cos im \]

      12. associate-+r+ N/A

          \[\leadsto \color{blue}{\left(\left(\left(re + re \cdot \left(\frac{1}{2} \cdot re\right)\right) + 0\right) + 1\right)} \cdot \cos im \]

   5. Applied rewrites 98.7%

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

   if 1.4e154 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-neg-out N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. remove-double-neg N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt1-in N/A

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

      10. *-commutative N/A

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

      11. metadata-eval N/A

          \[\leadsto \left(\left(re + re \cdot \left(\frac{1}{2} \cdot re\right)\right) + \color{blue}{\left(0 + 1\right)}\right) \cdot \cos im \]

      12. associate-+r+ N/A

          \[\leadsto \color{blue}{\left(\left(\left(re + re \cdot \left(\frac{1}{2} \cdot re\right)\right) + 0\right) + 1\right)} \cdot \cos im \]

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in re around inf

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

   8. Applied rewrites 100.0%

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

Alternative 17: 90.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{if}\;re \leq -4.2 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 0.00039:\\ \;\;\;\;\left(1 + re\right) \cdot \cos im\\ \mathbf{elif}\;re \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot 0.5\right) \cdot \cos im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (fma (* im im) -0.5 1.0))))
   (if (<= re -4.2e-5)
     t_0
     (if (<= re 0.00039)
       (* (+ 1.0 re) (cos im))
       (if (<= re 1.4e+154) t_0 (* (* (* re re) 0.5) (cos im)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * fma((im * im), -0.5, 1.0);
	double tmp;
	if (re <= -4.2e-5) {
		tmp = t_0;
	} else if (re <= 0.00039) {
		tmp = (1.0 + re) * cos(im);
	} else if (re <= 1.4e+154) {
		tmp = t_0;
	} else {
		tmp = ((re * re) * 0.5) * cos(im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * fma(Float64(im * im), -0.5, 1.0))
	tmp = 0.0
	if (re <= -4.2e-5)
		tmp = t_0;
	elseif (re <= 0.00039)
		tmp = Float64(Float64(1.0 + re) * cos(im));
	elseif (re <= 1.4e+154)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(re * re) * 0.5) * cos(im));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -4.2e-5], t$95$0, If[LessEqual[re, 0.00039], N[(N[(1.0 + re), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.4e+154], t$95$0, N[(N[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -4.19999999999999977e-5 or 3.89999999999999993e-4 < re < 1.4e154

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 73.8

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

   5. Applied rewrites 73.8%

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

   if -4.19999999999999977e-5 < re < 3.89999999999999993e-4

   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. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 1.4e154 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-neg-out N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. remove-double-neg N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt1-in N/A

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

      10. *-commutative N/A

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

      11. metadata-eval N/A

          \[\leadsto \left(\left(re + re \cdot \left(\frac{1}{2} \cdot re\right)\right) + \color{blue}{\left(0 + 1\right)}\right) \cdot \cos im \]

      12. associate-+r+ N/A

          \[\leadsto \color{blue}{\left(\left(\left(re + re \cdot \left(\frac{1}{2} \cdot re\right)\right) + 0\right) + 1\right)} \cdot \cos im \]

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in re around inf

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

   8. Applied rewrites 100.0%

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

Alternative 18: 51.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.041666666666666664 \cdot \left(im \cdot im\right)\\ \mathbf{if}\;re \leq -500:\\ \;\;\;\;\left(1 + re\right) \cdot \left(\left(im \cdot im\right) \cdot t\_0\right)\\ \mathbf{elif}\;re \leq 10^{-9}:\\ \;\;\;\;{im}^{-1} \cdot im\\ \mathbf{elif}\;re \leq 1.15 \cdot 10^{+91}:\\ \;\;\;\;\left(1 + re\right) \cdot \mathsf{fma}\left(t\_0, im \cdot im, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot re\right) \cdot re\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.041666666666666664 (* im im))))
   (if (<= re -500.0)
     (* (+ 1.0 re) (* (* im im) t_0))
     (if (<= re 1e-9)
       (* (pow im -1.0) im)
       (if (<= re 1.15e+91)
         (* (+ 1.0 re) (fma t_0 (* im im) 1.0))
         (*
          (* (* (fma 0.16666666666666666 re 0.5) re) re)
          (fma (* im im) -0.5 1.0)))))))

C

double code(double re, double im) {
	double t_0 = 0.041666666666666664 * (im * im);
	double tmp;
	if (re <= -500.0) {
		tmp = (1.0 + re) * ((im * im) * t_0);
	} else if (re <= 1e-9) {
		tmp = pow(im, -1.0) * im;
	} else if (re <= 1.15e+91) {
		tmp = (1.0 + re) * fma(t_0, (im * im), 1.0);
	} else {
		tmp = ((fma(0.16666666666666666, re, 0.5) * re) * re) * fma((im * im), -0.5, 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.041666666666666664 * Float64(im * im))
	tmp = 0.0
	if (re <= -500.0)
		tmp = Float64(Float64(1.0 + re) * Float64(Float64(im * im) * t_0));
	elseif (re <= 1e-9)
		tmp = Float64((im ^ -1.0) * im);
	elseif (re <= 1.15e+91)
		tmp = Float64(Float64(1.0 + re) * fma(t_0, Float64(im * im), 1.0));
	else
		tmp = Float64(Float64(Float64(fma(0.16666666666666666, re, 0.5) * re) * re) * fma(Float64(im * im), -0.5, 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -500.0], N[(N[(1.0 + re), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1e-9], N[(N[Power[im, -1.0], $MachinePrecision] * im), $MachinePrecision], If[LessEqual[re, 1.15e+91], N[(N[(1.0 + re), $MachinePrecision] * N[(t$95$0 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -500

   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. lower-+.f64 2.3

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

   5. Applied rewrites 2.3%

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

   6. Taylor expanded in im around 0

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. metadata-eval N/A

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

      8. fp-cancel-sub-sign-inv N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 2.0

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

   8. Applied rewrites 2.0%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 45.6%

       \[\leadsto \left(1 + re\right) \cdot \left({im}^{4} \cdot \color{blue}{0.041666666666666664}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 45.6%

       \[\leadsto \left(1 + re\right) \cdot \left(\left(im \cdot im\right) \cdot \left(0.041666666666666664 \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \]

   if -500 < re < 1.00000000000000006e-9

   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 98.3

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 49.5%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 31.2%

       \[\leadsto \left(\left(\frac{1}{im \cdot im} - 0.5\right) \cdot im\right) \cdot im \]

   12. Taylor expanded in im around 0

       \[\leadsto \frac{1}{im} \cdot im \]
   13. Step-by-step derivation

   14. Applied rewrites 53.2%

       \[\leadsto \frac{1}{im} \cdot im \]

   if 1.00000000000000006e-9 < re < 1.14999999999999996e91

   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. lower-+.f64 8.2

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

   5. Applied rewrites 8.2%

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

   6. Taylor expanded in im around 0

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. metadata-eval N/A

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

      8. fp-cancel-sub-sign-inv N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 34.3

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

   8. Applied rewrites 34.3%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 34.3%

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

   if 1.14999999999999996e91 < 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. fp-cancel-sign-sub-inv N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. metadata-eval N/A

         \[\leadsto \left(\left(re + re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) + \color{blue}{\left(0 + 1\right)}\right) \cdot \cos im \]

      8. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(re + re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) + 0\right) + 1\right)} \cdot \cos im \]

      9. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + re\right)} + 0\right) + 1\right) \cdot \cos im \]

      10. associate-+r+ N/A

          \[\leadsto \left(\color{blue}{\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(re + 0\right)\right)} + 1\right) \cdot \cos im \]

      11. *-rgt-identity N/A

          \[\leadsto \left(\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(\color{blue}{re \cdot 1} + 0\right)\right) + 1\right) \cdot \cos im \]

      12. mul0-rgt N/A

          \[\leadsto \left(\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \left(re \cdot 1 + \color{blue}{re \cdot 0}\right)\right) + 1\right) \cdot \cos im \]

      13. distribute-lft-in N/A

          \[\leadsto \left(\left(re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + \color{blue}{re \cdot \left(1 + 0\right)}\right) + 1\right) \cdot \cos im \]

      14. metadata-eval N/A

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

      15. distribute-lft-in N/A

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

      16. +-commutative N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 93.9%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 78.0

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

   8. Applied rewrites 78.0%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 78.0%

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

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -500:\\ \;\;\;\;\left(1 + re\right) \cdot \left(\left(im \cdot im\right) \cdot \left(0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;re \leq 10^{-9}:\\ \;\;\;\;{im}^{-1} \cdot im\\ \mathbf{elif}\;re \leq 1.15 \cdot 10^{+91}:\\ \;\;\;\;\left(1 + re\right) \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right), im \cdot im, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot re\right) \cdot re\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 49.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.041666666666666664 \cdot \left(im \cdot im\right)\\ \mathbf{if}\;re \leq -500:\\ \;\;\;\;\left(1 + re\right) \cdot \left(\left(im \cdot im\right) \cdot t\_0\right)\\ \mathbf{elif}\;re \leq 10^{-9}:\\ \;\;\;\;{im}^{-1} \cdot im\\ \mathbf{elif}\;re \leq 3.6 \cdot 10^{+91}:\\ \;\;\;\;\left(1 + re\right) \cdot \mathsf{fma}\left(t\_0, im \cdot im, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, re, 1\right) \cdot re\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.041666666666666664 (* im im))))
   (if (<= re -500.0)
     (* (+ 1.0 re) (* (* im im) t_0))
     (if (<= re 1e-9)
       (* (pow im -1.0) im)
       (if (<= re 3.6e+91)
         (* (+ 1.0 re) (fma t_0 (* im im) 1.0))
         (* (* (fma 0.5 re 1.0) re) (fma (* im im) -0.5 1.0)))))))

C

double code(double re, double im) {
	double t_0 = 0.041666666666666664 * (im * im);
	double tmp;
	if (re <= -500.0) {
		tmp = (1.0 + re) * ((im * im) * t_0);
	} else if (re <= 1e-9) {
		tmp = pow(im, -1.0) * im;
	} else if (re <= 3.6e+91) {
		tmp = (1.0 + re) * fma(t_0, (im * im), 1.0);
	} else {
		tmp = (fma(0.5, re, 1.0) * re) * fma((im * im), -0.5, 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.041666666666666664 * Float64(im * im))
	tmp = 0.0
	if (re <= -500.0)
		tmp = Float64(Float64(1.0 + re) * Float64(Float64(im * im) * t_0));
	elseif (re <= 1e-9)
		tmp = Float64((im ^ -1.0) * im);
	elseif (re <= 3.6e+91)
		tmp = Float64(Float64(1.0 + re) * fma(t_0, Float64(im * im), 1.0));
	else
		tmp = Float64(Float64(fma(0.5, re, 1.0) * re) * fma(Float64(im * im), -0.5, 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -500.0], N[(N[(1.0 + re), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1e-9], N[(N[Power[im, -1.0], $MachinePrecision] * im), $MachinePrecision], If[LessEqual[re, 3.6e+91], N[(N[(1.0 + re), $MachinePrecision] * N[(t$95$0 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -500

   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. lower-+.f64 2.3

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

   5. Applied rewrites 2.3%

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

   6. Taylor expanded in im around 0

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. metadata-eval N/A

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

      8. fp-cancel-sub-sign-inv N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 2.0

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

   8. Applied rewrites 2.0%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 45.6%

       \[\leadsto \left(1 + re\right) \cdot \left({im}^{4} \cdot \color{blue}{0.041666666666666664}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 45.6%

       \[\leadsto \left(1 + re\right) \cdot \left(\left(im \cdot im\right) \cdot \left(0.041666666666666664 \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \]

   if -500 < re < 1.00000000000000006e-9

   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 98.3

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 49.5%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 31.2%

       \[\leadsto \left(\left(\frac{1}{im \cdot im} - 0.5\right) \cdot im\right) \cdot im \]

   12. Taylor expanded in im around 0

       \[\leadsto \frac{1}{im} \cdot im \]
   13. Step-by-step derivation

   14. Applied rewrites 53.2%

       \[\leadsto \frac{1}{im} \cdot im \]

   if 1.00000000000000006e-9 < re < 3.6e91

   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. lower-+.f64 8.2

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

   5. Applied rewrites 8.2%

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

   6. Taylor expanded in im around 0

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. metadata-eval N/A

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

      8. fp-cancel-sub-sign-inv N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

          \[\leadsto \left(1 + re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{-1}{2}\right), \color{blue}{im \cdot im}, 1\right) \]

      15. lower-*.f64 34.3

          \[\leadsto \left(1 + re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), \color{blue}{im \cdot im}, 1\right) \]

   8. Applied rewrites 34.3%

      \[\leadsto \left(1 + re\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \left(1 + re\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot {im}^{2}, \color{blue}{im} \cdot im, 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 34.3%

       \[\leadsto \left(1 + re\right) \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right), \color{blue}{im} \cdot im, 1\right) \]

   if 3.6e91 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} \cdot \cos im \]
   4. Step-by-step derivation

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(1 - \left(\mathsf{neg}\left(re\right)\right) \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} \cdot \cos im \]

      2. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(re\right)\right)\right)\right) \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} \cdot \cos im \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(re\right)\right)\right)\right) \cdot \left(1 + \frac{1}{2} \cdot re\right) + 1\right)} \cdot \cos im \]

      4. distribute-lft-neg-out N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(re\right)\right) \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)\right)} + 1\right) \cdot \cos im \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{\left(1 + \frac{1}{2} \cdot re\right) \cdot \left(\mathsf{neg}\left(re\right)\right)}\right)\right) + 1\right) \cdot \cos im \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \left(\color{blue}{\left(1 + \frac{1}{2} \cdot re\right) \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(re\right)\right)\right)\right)} + 1\right) \cdot \cos im \]

      7. remove-double-neg N/A

         \[\leadsto \left(\left(1 + \frac{1}{2} \cdot re\right) \cdot \color{blue}{re} + 1\right) \cdot \cos im \]

      8. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot re + 1\right)} \cdot re + 1\right) \cdot \cos im \]

      9. distribute-rgt1-in N/A

         \[\leadsto \left(\color{blue}{\left(re + \left(\frac{1}{2} \cdot re\right) \cdot re\right)} + 1\right) \cdot \cos im \]

      10. *-commutative N/A

          \[\leadsto \left(\left(re + \color{blue}{re \cdot \left(\frac{1}{2} \cdot re\right)}\right) + 1\right) \cdot \cos im \]

      11. metadata-eval N/A

          \[\leadsto \left(\left(re + re \cdot \left(\frac{1}{2} \cdot re\right)\right) + \color{blue}{\left(0 + 1\right)}\right) \cdot \cos im \]

      12. associate-+r+ N/A

          \[\leadsto \color{blue}{\left(\left(\left(re + re \cdot \left(\frac{1}{2} \cdot re\right)\right) + 0\right) + 1\right)} \cdot \cos im \]

   5. Applied rewrites 75.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right)} \cdot \cos im \]

   6. Taylor expanded in re around inf

      \[\leadsto \left({re}^{2} \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{re}\right)}\right) \cdot \cos im \]
   7. Step-by-step derivation

   8. Applied rewrites 75.1%

      \[\leadsto \left(\mathsf{fma}\left(0.5, re, 1\right) \cdot \color{blue}{re}\right) \cdot \cos im \]

   9. Taylor expanded in im around 0

      \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, re, 1\right) \cdot re\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, re, 1\right) \cdot re\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2} + 1\right)} \]

       2. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, re, 1\right) \cdot re\right) \cdot \left(\color{blue}{{im}^{2} \cdot \frac{-1}{2}} + 1\right) \]

       3. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, re, 1\right) \cdot re\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{2}, 1\right)} \]

       4. unpow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, re, 1\right) \cdot re\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{2}, 1\right) \]

       5. lower-*.f64 57.5

          \[\leadsto \left(\mathsf{fma}\left(0.5, re, 1\right) \cdot re\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, -0.5, 1\right) \]

   11. Applied rewrites 57.5%

       \[\leadsto \left(\mathsf{fma}\left(0.5, re, 1\right) \cdot re\right) \cdot \color{blue}{\mathsf{fma}\left(im \cdot im, -0.5, 1\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -500:\\ \;\;\;\;\left(1 + re\right) \cdot \left(\left(im \cdot im\right) \cdot \left(0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;re \leq 10^{-9}:\\ \;\;\;\;{im}^{-1} \cdot im\\ \mathbf{elif}\;re \leq 3.6 \cdot 10^{+91}:\\ \;\;\;\;\left(1 + re\right) \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right), im \cdot im, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, re, 1\right) \cdot re\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 48.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -500:\\ \;\;\;\;\left(1 + re\right) \cdot \left(\left(im \cdot im\right) \cdot \left(0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;re \leq 380:\\ \;\;\;\;{im}^{-1} \cdot im\\ \mathbf{elif}\;re \leq 1.15 \cdot 10^{+91}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, re, 1\right) \cdot re\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -500.0)
   (* (+ 1.0 re) (* (* im im) (* 0.041666666666666664 (* im im))))
   (if (<= re 380.0)
     (* (pow im -1.0) im)
     (if (<= re 1.15e+91)
       (fma (fma 0.041666666666666664 (* im im) -0.5) (* im im) 1.0)
       (* (* (fma 0.5 re 1.0) re) (fma (* im im) -0.5 1.0))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -500.0) {
		tmp = (1.0 + re) * ((im * im) * (0.041666666666666664 * (im * im)));
	} else if (re <= 380.0) {
		tmp = pow(im, -1.0) * im;
	} else if (re <= 1.15e+91) {
		tmp = fma(fma(0.041666666666666664, (im * im), -0.5), (im * im), 1.0);
	} else {
		tmp = (fma(0.5, re, 1.0) * re) * fma((im * im), -0.5, 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -500.0)
		tmp = Float64(Float64(1.0 + re) * Float64(Float64(im * im) * Float64(0.041666666666666664 * Float64(im * im))));
	elseif (re <= 380.0)
		tmp = Float64((im ^ -1.0) * im);
	elseif (re <= 1.15e+91)
		tmp = fma(fma(0.041666666666666664, Float64(im * im), -0.5), Float64(im * im), 1.0);
	else
		tmp = Float64(Float64(fma(0.5, re, 1.0) * re) * fma(Float64(im * im), -0.5, 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -500.0], N[(N[(1.0 + re), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 380.0], N[(N[Power[im, -1.0], $MachinePrecision] * im), $MachinePrecision], If[LessEqual[re, 1.15e+91], N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision] + -0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -500

   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. lower-+.f64 2.3

         \[\leadsto \color{blue}{\left(1 + re\right)} \cdot \cos im \]

   5. Applied rewrites 2.3%

      \[\leadsto \color{blue}{\left(1 + re\right)} \cdot \cos im \]

   6. Taylor expanded in im around 0

      \[\leadsto \left(1 + re\right) \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)} \]
   7. Step-by-step derivation

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \left(1 + re\right) \cdot \color{blue}{\left(1 - \left(\mathsf{neg}\left({im}^{2}\right)\right) \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)} \]

      2. fp-cancel-sub-sign-inv N/A

         \[\leadsto \left(1 + re\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({im}^{2}\right)\right)\right)\right) \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \left(1 + re\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({im}^{2}\right)\right)\right)\right) \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) + 1\right)} \]

      4. remove-double-neg N/A

         \[\leadsto \left(1 + re\right) \cdot \left(\color{blue}{{im}^{2}} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) + 1\right) \]

      5. *-commutative N/A

         \[\leadsto \left(1 + re\right) \cdot \left(\color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot {im}^{2}} + 1\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(1 + re\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}, {im}^{2}, 1\right)} \]

      7. metadata-eval N/A

         \[\leadsto \left(1 + re\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot {im}^{2} - \color{blue}{\frac{1}{2} \cdot 1}, {im}^{2}, 1\right) \]

      8. fp-cancel-sub-sign-inv N/A

         \[\leadsto \left(1 + re\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot 1}, {im}^{2}, 1\right) \]

      9. metadata-eval N/A

         \[\leadsto \left(1 + re\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\frac{-1}{2}} \cdot 1, {im}^{2}, 1\right) \]

      10. metadata-eval N/A

          \[\leadsto \left(1 + re\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot {im}^{2} + \color{blue}{\frac{-1}{2}}, {im}^{2}, 1\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \left(1 + re\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {im}^{2}, \frac{-1}{2}\right)}, {im}^{2}, 1\right) \]

      12. unpow2 N/A

          \[\leadsto \left(1 + re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{-1}{2}\right), {im}^{2}, 1\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(1 + re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{-1}{2}\right), {im}^{2}, 1\right) \]

      14. unpow2 N/A

          \[\leadsto \left(1 + re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{-1}{2}\right), \color{blue}{im \cdot im}, 1\right) \]

      15. lower-*.f64 2.0

          \[\leadsto \left(1 + re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), \color{blue}{im \cdot im}, 1\right) \]

   8. Applied rewrites 2.0%

      \[\leadsto \left(1 + re\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \left(1 + re\right) \cdot \left(\frac{1}{24} \cdot \color{blue}{{im}^{4}}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 45.6%

       \[\leadsto \left(1 + re\right) \cdot \left({im}^{4} \cdot \color{blue}{0.041666666666666664}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 45.6%

       \[\leadsto \left(1 + re\right) \cdot \left(\left(im \cdot im\right) \cdot \left(0.041666666666666664 \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \]

   if -500 < re < 380

   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 96.5

         \[\leadsto \color{blue}{\cos im} \]

   5. Applied rewrites 96.5%

      \[\leadsto \color{blue}{\cos im} \]

   6. Taylor expanded in im around 0

      \[\leadsto 1 + \color{blue}{\frac{-1}{2} \cdot {im}^{2}} \]
   7. Step-by-step derivation

   8. Applied rewrites 48.6%

      \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{-0.5}, 1\right) \]

   9. Taylor expanded in im around inf

      \[\leadsto {im}^{2} \cdot \left(\frac{1}{{im}^{2}} - \color{blue}{\frac{1}{2}}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 30.7%

       \[\leadsto \left(\left(\frac{1}{im \cdot im} - 0.5\right) \cdot im\right) \cdot im \]

   12. Taylor expanded in im around 0

       \[\leadsto \frac{1}{im} \cdot im \]
   13. Step-by-step derivation

   14. Applied rewrites 52.3%

       \[\leadsto \frac{1}{im} \cdot im \]

   if 380 < re < 1.14999999999999996e91

   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. Applied rewrites 3.1%

      \[\leadsto \color{blue}{\cos im} \]

   6. Taylor expanded in im around 0

      \[\leadsto 1 + \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 37.2%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), \color{blue}{im \cdot im}, 1\right) \]

   if 1.14999999999999996e91 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} \cdot \cos im \]
   4. Step-by-step derivation

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(1 - \left(\mathsf{neg}\left(re\right)\right) \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} \cdot \cos im \]

      2. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(re\right)\right)\right)\right) \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} \cdot \cos im \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(re\right)\right)\right)\right) \cdot \left(1 + \frac{1}{2} \cdot re\right) + 1\right)} \cdot \cos im \]

      4. distribute-lft-neg-out N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(re\right)\right) \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)\right)} + 1\right) \cdot \cos im \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{\left(1 + \frac{1}{2} \cdot re\right) \cdot \left(\mathsf{neg}\left(re\right)\right)}\right)\right) + 1\right) \cdot \cos im \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \left(\color{blue}{\left(1 + \frac{1}{2} \cdot re\right) \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(re\right)\right)\right)\right)} + 1\right) \cdot \cos im \]

      7. remove-double-neg N/A

         \[\leadsto \left(\left(1 + \frac{1}{2} \cdot re\right) \cdot \color{blue}{re} + 1\right) \cdot \cos im \]

      8. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot re + 1\right)} \cdot re + 1\right) \cdot \cos im \]

      9. distribute-rgt1-in N/A

         \[\leadsto \left(\color{blue}{\left(re + \left(\frac{1}{2} \cdot re\right) \cdot re\right)} + 1\right) \cdot \cos im \]

      10. *-commutative N/A

          \[\leadsto \left(\left(re + \color{blue}{re \cdot \left(\frac{1}{2} \cdot re\right)}\right) + 1\right) \cdot \cos im \]

      11. metadata-eval N/A

          \[\leadsto \left(\left(re + re \cdot \left(\frac{1}{2} \cdot re\right)\right) + \color{blue}{\left(0 + 1\right)}\right) \cdot \cos im \]

      12. associate-+r+ N/A

          \[\leadsto \color{blue}{\left(\left(\left(re + re \cdot \left(\frac{1}{2} \cdot re\right)\right) + 0\right) + 1\right)} \cdot \cos im \]

   5. Applied rewrites 75.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right)} \cdot \cos im \]

   6. Taylor expanded in re around inf

      \[\leadsto \left({re}^{2} \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{re}\right)}\right) \cdot \cos im \]
   7. Step-by-step derivation

   8. Applied rewrites 75.1%

      \[\leadsto \left(\mathsf{fma}\left(0.5, re, 1\right) \cdot \color{blue}{re}\right) \cdot \cos im \]

   9. Taylor expanded in im around 0

      \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, re, 1\right) \cdot re\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, re, 1\right) \cdot re\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2} + 1\right)} \]

       2. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, re, 1\right) \cdot re\right) \cdot \left(\color{blue}{{im}^{2} \cdot \frac{-1}{2}} + 1\right) \]

       3. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, re, 1\right) \cdot re\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{2}, 1\right)} \]

       4. unpow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, re, 1\right) \cdot re\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{2}, 1\right) \]

       5. lower-*.f64 57.5

          \[\leadsto \left(\mathsf{fma}\left(0.5, re, 1\right) \cdot re\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, -0.5, 1\right) \]

   11. Applied rewrites 57.5%

       \[\leadsto \left(\mathsf{fma}\left(0.5, re, 1\right) \cdot re\right) \cdot \color{blue}{\mathsf{fma}\left(im \cdot im, -0.5, 1\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -500:\\ \;\;\;\;\left(1 + re\right) \cdot \left(\left(im \cdot im\right) \cdot \left(0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;re \leq 380:\\ \;\;\;\;{im}^{-1} \cdot im\\ \mathbf{elif}\;re \leq 1.15 \cdot 10^{+91}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, re, 1\right) \cdot re\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 45.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -510:\\ \;\;\;\;\left(-0.5 \cdot im\right) \cdot im\\ \mathbf{elif}\;re \leq 380:\\ \;\;\;\;{im}^{-1} \cdot im\\ \mathbf{elif}\;re \leq 1.15 \cdot 10^{+91}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, re, 1\right) \cdot re\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -510.0)
   (* (* -0.5 im) im)
   (if (<= re 380.0)
     (* (pow im -1.0) im)
     (if (<= re 1.15e+91)
       (fma (fma 0.041666666666666664 (* im im) -0.5) (* im im) 1.0)
       (* (* (fma 0.5 re 1.0) re) (fma (* im im) -0.5 1.0))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -510.0) {
		tmp = (-0.5 * im) * im;
	} else if (re <= 380.0) {
		tmp = pow(im, -1.0) * im;
	} else if (re <= 1.15e+91) {
		tmp = fma(fma(0.041666666666666664, (im * im), -0.5), (im * im), 1.0);
	} else {
		tmp = (fma(0.5, re, 1.0) * re) * fma((im * im), -0.5, 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -510.0)
		tmp = Float64(Float64(-0.5 * im) * im);
	elseif (re <= 380.0)
		tmp = Float64((im ^ -1.0) * im);
	elseif (re <= 1.15e+91)
		tmp = fma(fma(0.041666666666666664, Float64(im * im), -0.5), Float64(im * im), 1.0);
	else
		tmp = Float64(Float64(fma(0.5, re, 1.0) * re) * fma(Float64(im * im), -0.5, 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -510.0], N[(N[(-0.5 * im), $MachinePrecision] * im), $MachinePrecision], If[LessEqual[re, 380.0], N[(N[Power[im, -1.0], $MachinePrecision] * im), $MachinePrecision], If[LessEqual[re, 1.15e+91], N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision] + -0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -510

   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. Applied rewrites 3.1%

      \[\leadsto \color{blue}{\cos im} \]

   6. Taylor expanded in im around 0

      \[\leadsto 1 + \color{blue}{\frac{-1}{2} \cdot {im}^{2}} \]
   7. Step-by-step derivation

   8. Applied rewrites 2.5%

      \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{-0.5}, 1\right) \]

   9. Taylor expanded in im around inf

      \[\leadsto \frac{-1}{2} \cdot {im}^{\color{blue}{2}} \]
   10. Step-by-step derivation

   11. Applied rewrites 29.8%

       \[\leadsto \left(im \cdot im\right) \cdot -0.5 \]
   12. Step-by-step derivation

   13. Applied rewrites 29.8%

       \[\leadsto \left(-0.5 \cdot im\right) \cdot im \]

   if -510 < re < 380

   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 96.5

         \[\leadsto \color{blue}{\cos im} \]

   5. Applied rewrites 96.5%

      \[\leadsto \color{blue}{\cos im} \]

   6. Taylor expanded in im around 0

      \[\leadsto 1 + \color{blue}{\frac{-1}{2} \cdot {im}^{2}} \]
   7. Step-by-step derivation

   8. Applied rewrites 48.6%

      \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{-0.5}, 1\right) \]

   9. Taylor expanded in im around inf

      \[\leadsto {im}^{2} \cdot \left(\frac{1}{{im}^{2}} - \color{blue}{\frac{1}{2}}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 30.7%

       \[\leadsto \left(\left(\frac{1}{im \cdot im} - 0.5\right) \cdot im\right) \cdot im \]

   12. Taylor expanded in im around 0

       \[\leadsto \frac{1}{im} \cdot im \]
   13. Step-by-step derivation

   14. Applied rewrites 52.3%

       \[\leadsto \frac{1}{im} \cdot im \]

   if 380 < re < 1.14999999999999996e91

   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. Applied rewrites 3.1%

      \[\leadsto \color{blue}{\cos im} \]

   6. Taylor expanded in im around 0

      \[\leadsto 1 + \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 37.2%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), \color{blue}{im \cdot im}, 1\right) \]

   if 1.14999999999999996e91 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} \cdot \cos im \]
   4. Step-by-step derivation

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(1 - \left(\mathsf{neg}\left(re\right)\right) \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} \cdot \cos im \]

      2. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(re\right)\right)\right)\right) \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} \cdot \cos im \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(re\right)\right)\right)\right) \cdot \left(1 + \frac{1}{2} \cdot re\right) + 1\right)} \cdot \cos im \]

      4. distribute-lft-neg-out N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(re\right)\right) \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)\right)} + 1\right) \cdot \cos im \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{\left(1 + \frac{1}{2} \cdot re\right) \cdot \left(\mathsf{neg}\left(re\right)\right)}\right)\right) + 1\right) \cdot \cos im \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \left(\color{blue}{\left(1 + \frac{1}{2} \cdot re\right) \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(re\right)\right)\right)\right)} + 1\right) \cdot \cos im \]

      7. remove-double-neg N/A

         \[\leadsto \left(\left(1 + \frac{1}{2} \cdot re\right) \cdot \color{blue}{re} + 1\right) \cdot \cos im \]

      8. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot re + 1\right)} \cdot re + 1\right) \cdot \cos im \]

      9. distribute-rgt1-in N/A

         \[\leadsto \left(\color{blue}{\left(re + \left(\frac{1}{2} \cdot re\right) \cdot re\right)} + 1\right) \cdot \cos im \]

      10. *-commutative N/A

          \[\leadsto \left(\left(re + \color{blue}{re \cdot \left(\frac{1}{2} \cdot re\right)}\right) + 1\right) \cdot \cos im \]

      11. metadata-eval N/A

          \[\leadsto \left(\left(re + re \cdot \left(\frac{1}{2} \cdot re\right)\right) + \color{blue}{\left(0 + 1\right)}\right) \cdot \cos im \]

      12. associate-+r+ N/A

          \[\leadsto \color{blue}{\left(\left(\left(re + re \cdot \left(\frac{1}{2} \cdot re\right)\right) + 0\right) + 1\right)} \cdot \cos im \]

   5. Applied rewrites 75.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right)} \cdot \cos im \]

   6. Taylor expanded in re around inf

      \[\leadsto \left({re}^{2} \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{re}\right)}\right) \cdot \cos im \]
   7. Step-by-step derivation

   8. Applied rewrites 75.1%

      \[\leadsto \left(\mathsf{fma}\left(0.5, re, 1\right) \cdot \color{blue}{re}\right) \cdot \cos im \]

   9. Taylor expanded in im around 0

      \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, re, 1\right) \cdot re\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, re, 1\right) \cdot re\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2} + 1\right)} \]

       2. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, re, 1\right) \cdot re\right) \cdot \left(\color{blue}{{im}^{2} \cdot \frac{-1}{2}} + 1\right) \]

       3. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, re, 1\right) \cdot re\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{2}, 1\right)} \]

       4. unpow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, re, 1\right) \cdot re\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{2}, 1\right) \]

       5. lower-*.f64 57.5

          \[\leadsto \left(\mathsf{fma}\left(0.5, re, 1\right) \cdot re\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, -0.5, 1\right) \]

   11. Applied rewrites 57.5%

       \[\leadsto \left(\mathsf{fma}\left(0.5, re, 1\right) \cdot re\right) \cdot \color{blue}{\mathsf{fma}\left(im \cdot im, -0.5, 1\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -510:\\ \;\;\;\;\left(-0.5 \cdot im\right) \cdot im\\ \mathbf{elif}\;re \leq 380:\\ \;\;\;\;{im}^{-1} \cdot im\\ \mathbf{elif}\;re \leq 1.15 \cdot 10^{+91}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, re, 1\right) \cdot re\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 11.4% accurate, 18.7× speedup

Math

\[\begin{array}{l} \\ \left(-0.5 \cdot im\right) \cdot im \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (* -0.5 im) im))

C

double code(double re, double im) {
	return (-0.5 * im) * im;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = ((-0.5d0) * im) * im
end function

Java

public static double code(double re, double im) {
	return (-0.5 * im) * im;
}

Python

def code(re, im):
	return (-0.5 * im) * im

Julia

function code(re, im)
	return Float64(Float64(-0.5 * im) * im)
end

MATLAB

function tmp = code(re, im)
	tmp = (-0.5 * im) * im;
end

Wolfram

code[re_, im_] := N[(N[(-0.5 * im), $MachinePrecision] * im), $MachinePrecision]

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 50.2

      \[\leadsto \color{blue}{\cos im} \]

5. Applied rewrites 50.2%

   \[\leadsto \color{blue}{\cos im} \]

6. Taylor expanded in im around 0

   \[\leadsto 1 + \color{blue}{\frac{-1}{2} \cdot {im}^{2}} \]
7. Step-by-step derivation

8. Applied rewrites 28.4%

   \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{-0.5}, 1\right) \]

9. Taylor expanded in im around inf

   \[\leadsto \frac{-1}{2} \cdot {im}^{\color{blue}{2}} \]
10. Step-by-step derivation

11. Applied rewrites 12.0%

    \[\leadsto \left(im \cdot im\right) \cdot -0.5 \]
12. Step-by-step derivation

13. Applied rewrites 12.0%

    \[\leadsto \left(-0.5 \cdot im\right) \cdot im \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024357 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))