math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 7.3s

Alternatives: 20

Speedup: 1.0×

• 24.7% 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: 99.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(\left(im \cdot im\right) \cdot -0.5\right)\\ \mathbf{elif}\;t\_0 \leq -5 \cdot 10^{-50}:\\ \;\;\;\;\frac{\cos im}{\mathsf{fma}\left(\left(-0.16666666666666666 \cdot re\right) \cdot re - 1, re, 1\right)}\\ \mathbf{elif}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 0.9998\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos im}{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, 0.5\right) \cdot re - 1, re, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (* (exp re) (* (* im im) -0.5))
     (if (<= t_0 -5e-50)
       (/ (cos im) (fma (- (* (* -0.16666666666666666 re) re) 1.0) re 1.0))
       (if (or (<= t_0 0.0) (not (<= t_0 0.9998)))
         (exp re)
         (/
          (cos im)
          (fma (- (* (fma -0.16666666666666666 re 0.5) re) 1.0) re 1.0)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = exp(re) * ((im * im) * -0.5);
	} else if (t_0 <= -5e-50) {
		tmp = cos(im) / fma((((-0.16666666666666666 * re) * re) - 1.0), re, 1.0);
	} else if ((t_0 <= 0.0) || !(t_0 <= 0.9998)) {
		tmp = exp(re);
	} else {
		tmp = cos(im) / fma(((fma(-0.16666666666666666, re, 0.5) * re) - 1.0), re, 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(exp(re) * Float64(Float64(im * im) * -0.5));
	elseif (t_0 <= -5e-50)
		tmp = Float64(cos(im) / fma(Float64(Float64(Float64(-0.16666666666666666 * re) * re) - 1.0), re, 1.0));
	elseif ((t_0 <= 0.0) || !(t_0 <= 0.9998))
		tmp = exp(re);
	else
		tmp = Float64(cos(im) / fma(Float64(Float64(fma(-0.16666666666666666, re, 0.5) * re) - 1.0), re, 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[Exp[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -5e-50], N[(N[Cos[im], $MachinePrecision] / N[(N[(N[(N[(-0.16666666666666666 * re), $MachinePrecision] * re), $MachinePrecision] - 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 0.9998]], $MachinePrecision]], N[Exp[re], $MachinePrecision], N[(N[Cos[im], $MachinePrecision] / N[(N[(N[(N[(-0.16666666666666666 * re + 0.5), $MachinePrecision] * re), $MachinePrecision] - 1.0), $MachinePrecision] * re + 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 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 100.0

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

   5. Applied rewrites 100.0%

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

   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 100.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -4.99999999999999968e-50

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. remove-double-neg N/A

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

      4. rec-exp N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. sin-PI/2 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. sin-PI/2 N/A

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

      12. *-commutative N/A

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

      13. *-lft-identity N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 99.9

          \[\leadsto \frac{\cos im}{e^{\color{blue}{-re}}} \]

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{\cos im}{e^{-re}}} \]

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 96.4

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

   7. Applied rewrites 96.4%

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

   8. Taylor expanded in re around inf

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

   10. Applied rewrites 96.4%

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

   if -4.99999999999999968e-50 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0 or 0.99980000000000002 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. remove-double-neg N/A

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

      4. rec-exp N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. sin-PI/2 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. sin-PI/2 N/A

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

      12. *-commutative N/A

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

      13. *-lft-identity N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 100.0

          \[\leadsto \frac{\cos im}{e^{\color{blue}{-re}}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\cos im}{e^{-re}}} \]

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{neg}\left(re\right)}}} \]
   6. Step-by-step derivation

      1. exp-neg N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{re}}}} \]

      2. remove-double-div N/A

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

      3. lower-exp.f64 100.0

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

   7. Applied rewrites 100.0%

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

   if 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.99980000000000002

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. remove-double-neg N/A

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

      4. rec-exp N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. sin-PI/2 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. sin-PI/2 N/A

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

      12. *-commutative N/A

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

      13. *-lft-identity N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 100.0

          \[\leadsto \frac{\cos im}{e^{\color{blue}{-re}}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\cos im}{e^{-re}}} \]

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 97.4

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

   7. Applied rewrites 97.4%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(\left(im \cdot im\right) \cdot -0.5\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -5 \cdot 10^{-50}:\\ \;\;\;\;\frac{\cos im}{\mathsf{fma}\left(\left(-0.16666666666666666 \cdot re\right) \cdot re - 1, re, 1\right)}\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0 \lor \neg \left(e^{re} \cdot \cos im \leq 0.9998\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos im}{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, 0.5\right) \cdot re - 1, re, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(\left(im \cdot im\right) \cdot -0.5\right)\\ \mathbf{elif}\;t\_0 \leq -5 \cdot 10^{-50}:\\ \;\;\;\;\frac{\cos im}{\mathsf{fma}\left(\left(-0.16666666666666666 \cdot re\right) \cdot re - 1, re, 1\right)}\\ \mathbf{elif}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 0.9998\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \cos im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (* (exp re) (* (* im im) -0.5))
     (if (<= t_0 -5e-50)
       (/ (cos im) (fma (- (* (* -0.16666666666666666 re) re) 1.0) re 1.0))
       (if (or (<= t_0 0.0) (not (<= t_0 0.9998)))
         (exp re)
         (*
          (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)
          (cos im)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = exp(re) * ((im * im) * -0.5);
	} else if (t_0 <= -5e-50) {
		tmp = cos(im) / fma((((-0.16666666666666666 * re) * re) - 1.0), re, 1.0);
	} else if ((t_0 <= 0.0) || !(t_0 <= 0.9998)) {
		tmp = exp(re);
	} else {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * cos(im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(exp(re) * Float64(Float64(im * im) * -0.5));
	elseif (t_0 <= -5e-50)
		tmp = Float64(cos(im) / fma(Float64(Float64(Float64(-0.16666666666666666 * re) * re) - 1.0), re, 1.0));
	elseif ((t_0 <= 0.0) || !(t_0 <= 0.9998))
		tmp = exp(re);
	else
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * cos(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, (-Infinity)], N[(N[Exp[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -5e-50], N[(N[Cos[im], $MachinePrecision] / N[(N[(N[(N[(-0.16666666666666666 * re), $MachinePrecision] * re), $MachinePrecision] - 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 0.9998]], $MachinePrecision]], N[Exp[re], $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Cos[im], $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 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 100.0

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

   5. Applied rewrites 100.0%

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

   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 100.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -4.99999999999999968e-50

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. remove-double-neg N/A

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

      4. rec-exp N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. sin-PI/2 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. sin-PI/2 N/A

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

      12. *-commutative N/A

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

      13. *-lft-identity N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 99.9

          \[\leadsto \frac{\cos im}{e^{\color{blue}{-re}}} \]

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{\cos im}{e^{-re}}} \]

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 96.4

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

   7. Applied rewrites 96.4%

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

   8. Taylor expanded in re around inf

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

   10. Applied rewrites 96.4%

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

   if -4.99999999999999968e-50 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0 or 0.99980000000000002 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. remove-double-neg N/A

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

      4. rec-exp N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. sin-PI/2 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. sin-PI/2 N/A

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

      12. *-commutative N/A

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

      13. *-lft-identity N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 100.0

          \[\leadsto \frac{\cos im}{e^{\color{blue}{-re}}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\cos im}{e^{-re}}} \]

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{neg}\left(re\right)}}} \]
   6. Step-by-step derivation

      1. exp-neg N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{re}}}} \]

      2. remove-double-div N/A

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

      3. lower-exp.f64 100.0

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

   7. Applied rewrites 100.0%

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

   if 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.99980000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

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

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

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

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

      6. +-commutative N/A

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

      7. remove-double-neg N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 97.4

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

   5. Applied rewrites 97.4%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(\left(im \cdot im\right) \cdot -0.5\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -5 \cdot 10^{-50}:\\ \;\;\;\;\frac{\cos im}{\mathsf{fma}\left(\left(-0.16666666666666666 \cdot re\right) \cdot re - 1, re, 1\right)}\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0 \lor \neg \left(e^{re} \cdot \cos im \leq 0.9998\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \cos im\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(\left(im \cdot im\right) \cdot -0.5\right)\\ \mathbf{elif}\;t\_0 \leq -5 \cdot 10^{-50}:\\ \;\;\;\;\frac{\cos im}{1 - re}\\ \mathbf{elif}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 0.9998\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \cos im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (* (exp re) (* (* im im) -0.5))
     (if (<= t_0 -5e-50)
       (/ (cos im) (- 1.0 re))
       (if (or (<= t_0 0.0) (not (<= t_0 0.9998)))
         (exp re)
         (*
          (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)
          (cos im)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = exp(re) * ((im * im) * -0.5);
	} else if (t_0 <= -5e-50) {
		tmp = cos(im) / (1.0 - re);
	} else if ((t_0 <= 0.0) || !(t_0 <= 0.9998)) {
		tmp = exp(re);
	} else {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * cos(im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(exp(re) * Float64(Float64(im * im) * -0.5));
	elseif (t_0 <= -5e-50)
		tmp = Float64(cos(im) / Float64(1.0 - re));
	elseif ((t_0 <= 0.0) || !(t_0 <= 0.9998))
		tmp = exp(re);
	else
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * cos(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, (-Infinity)], N[(N[Exp[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -5e-50], N[(N[Cos[im], $MachinePrecision] / N[(1.0 - re), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 0.9998]], $MachinePrecision]], N[Exp[re], $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Cos[im], $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 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 100.0

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

   5. Applied rewrites 100.0%

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

   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 100.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -4.99999999999999968e-50

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. remove-double-neg N/A

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

      4. rec-exp N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. sin-PI/2 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. sin-PI/2 N/A

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

      12. *-commutative N/A

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

      13. *-lft-identity N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 99.9

          \[\leadsto \frac{\cos im}{e^{\color{blue}{-re}}} \]

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{\cos im}{e^{-re}}} \]

   5. Taylor expanded in re around 0

      \[\leadsto \frac{\cos im}{\color{blue}{1 + -1 \cdot re}} \]
   6. Step-by-step derivation

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

         \[\leadsto \frac{\cos im}{1 - \color{blue}{re}} \]

      4. lower--.f64 96.4

         \[\leadsto \frac{\cos im}{\color{blue}{1 - re}} \]

   7. Applied rewrites 96.4%

      \[\leadsto \frac{\cos im}{\color{blue}{1 - re}} \]

   if -4.99999999999999968e-50 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0 or 0.99980000000000002 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. remove-double-neg N/A

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

      4. rec-exp N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. sin-PI/2 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. sin-PI/2 N/A

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

      12. *-commutative N/A

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

      13. *-lft-identity N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 100.0

          \[\leadsto \frac{\cos im}{e^{\color{blue}{-re}}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\cos im}{e^{-re}}} \]

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{neg}\left(re\right)}}} \]
   6. Step-by-step derivation

      1. exp-neg N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{re}}}} \]

      2. remove-double-div N/A

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

      3. lower-exp.f64 100.0

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

   7. Applied rewrites 100.0%

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

   if 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.99980000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

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

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

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

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

      6. +-commutative N/A

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

      7. remove-double-neg N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 97.4

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

   5. Applied rewrites 97.4%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(\left(im \cdot im\right) \cdot -0.5\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -5 \cdot 10^{-50}:\\ \;\;\;\;\frac{\cos im}{1 - re}\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0 \lor \neg \left(e^{re} \cdot \cos im \leq 0.9998\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \cos im\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(\left(im \cdot im\right) \cdot -0.5\right)\\ \mathbf{elif}\;t\_0 \leq -5 \cdot 10^{-50}:\\ \;\;\;\;\frac{\cos im}{1 - re}\\ \mathbf{elif}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 0.9998\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \cos im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (* (exp re) (* (* im im) -0.5))
     (if (<= t_0 -5e-50)
       (/ (cos im) (- 1.0 re))
       (if (or (<= t_0 0.0) (not (<= t_0 0.9998)))
         (exp re)
         (* (fma (fma 0.5 re 1.0) re 1.0) (cos im)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = exp(re) * ((im * im) * -0.5);
	} else if (t_0 <= -5e-50) {
		tmp = cos(im) / (1.0 - re);
	} else if ((t_0 <= 0.0) || !(t_0 <= 0.9998)) {
		tmp = exp(re);
	} else {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * cos(im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(exp(re) * Float64(Float64(im * im) * -0.5));
	elseif (t_0 <= -5e-50)
		tmp = Float64(cos(im) / Float64(1.0 - re));
	elseif ((t_0 <= 0.0) || !(t_0 <= 0.9998))
		tmp = exp(re);
	else
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * cos(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, (-Infinity)], N[(N[Exp[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -5e-50], N[(N[Cos[im], $MachinePrecision] / N[(1.0 - re), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 0.9998]], $MachinePrecision]], N[Exp[re], $MachinePrecision], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Cos[im], $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 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 100.0

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

   5. Applied rewrites 100.0%

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

   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 100.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -4.99999999999999968e-50

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. remove-double-neg N/A

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

      4. rec-exp N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. sin-PI/2 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. sin-PI/2 N/A

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

      12. *-commutative N/A

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

      13. *-lft-identity N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 99.9

          \[\leadsto \frac{\cos im}{e^{\color{blue}{-re}}} \]

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{\cos im}{e^{-re}}} \]

   5. Taylor expanded in re around 0

      \[\leadsto \frac{\cos im}{\color{blue}{1 + -1 \cdot re}} \]
   6. Step-by-step derivation

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

         \[\leadsto \frac{\cos im}{1 - \color{blue}{re}} \]

      4. lower--.f64 96.4

         \[\leadsto \frac{\cos im}{\color{blue}{1 - re}} \]

   7. Applied rewrites 96.4%

      \[\leadsto \frac{\cos im}{\color{blue}{1 - re}} \]

   if -4.99999999999999968e-50 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0 or 0.99980000000000002 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. remove-double-neg N/A

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

      4. rec-exp N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. sin-PI/2 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. sin-PI/2 N/A

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

      12. *-commutative N/A

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

      13. *-lft-identity N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 100.0

          \[\leadsto \frac{\cos im}{e^{\color{blue}{-re}}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\cos im}{e^{-re}}} \]

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{neg}\left(re\right)}}} \]
   6. Step-by-step derivation

      1. exp-neg N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{re}}}} \]

      2. remove-double-div N/A

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

      3. lower-exp.f64 100.0

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

   7. Applied rewrites 100.0%

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

   if 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.99980000000000002

   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. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 96.5

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

   5. Applied rewrites 96.5%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(\left(im \cdot im\right) \cdot -0.5\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -5 \cdot 10^{-50}:\\ \;\;\;\;\frac{\cos im}{1 - re}\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0 \lor \neg \left(e^{re} \cdot \cos im \leq 0.9998\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \cos im\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(\left(im \cdot im\right) \cdot -0.5\right)\\ \mathbf{elif}\;t\_0 \leq -5 \cdot 10^{-50} \lor \neg \left(t\_0 \leq 0 \lor \neg \left(t\_0 \leq 0.9998\right)\right):\\ \;\;\;\;\frac{\cos im}{1 - re}\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (* (exp re) (* (* im im) -0.5))
     (if (or (<= t_0 -5e-50) (not (or (<= t_0 0.0) (not (<= t_0 0.9998)))))
       (/ (cos im) (- 1.0 re))
       (exp re)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = exp(re) * ((im * im) * -0.5);
	} else if ((t_0 <= -5e-50) || !((t_0 <= 0.0) || !(t_0 <= 0.9998))) {
		tmp = cos(im) / (1.0 - re);
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(re) * Math.cos(im);
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = Math.exp(re) * ((im * im) * -0.5);
	} else if ((t_0 <= -5e-50) || !((t_0 <= 0.0) || !(t_0 <= 0.9998))) {
		tmp = Math.cos(im) / (1.0 - re);
	} else {
		tmp = Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) * math.cos(im)
	tmp = 0
	if t_0 <= -math.inf:
		tmp = math.exp(re) * ((im * im) * -0.5)
	elif (t_0 <= -5e-50) or not ((t_0 <= 0.0) or not (t_0 <= 0.9998)):
		tmp = math.cos(im) / (1.0 - re)
	else:
		tmp = math.exp(re)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(exp(re) * Float64(Float64(im * im) * -0.5));
	elseif ((t_0 <= -5e-50) || !((t_0 <= 0.0) || !(t_0 <= 0.9998)))
		tmp = Float64(cos(im) / Float64(1.0 - re));
	else
		tmp = exp(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(re) * cos(im);
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = exp(re) * ((im * im) * -0.5);
	elseif ((t_0 <= -5e-50) || ~(((t_0 <= 0.0) || ~((t_0 <= 0.9998)))))
		tmp = cos(im) / (1.0 - re);
	else
		tmp = exp(re);
	end
	tmp_2 = 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[Exp[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, -5e-50], N[Not[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 0.9998]], $MachinePrecision]]], $MachinePrecision]], N[(N[Cos[im], $MachinePrecision] / N[(1.0 - re), $MachinePrecision]), $MachinePrecision], N[Exp[re], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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 100.0

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

   5. Applied rewrites 100.0%

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

   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 100.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -4.99999999999999968e-50 or 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.99980000000000002

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. remove-double-neg N/A

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

      4. rec-exp N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. sin-PI/2 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. sin-PI/2 N/A

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

      12. *-commutative N/A

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

      13. *-lft-identity N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 100.0

          \[\leadsto \frac{\cos im}{e^{\color{blue}{-re}}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\cos im}{e^{-re}}} \]

   5. Taylor expanded in re around 0

      \[\leadsto \frac{\cos im}{\color{blue}{1 + -1 \cdot re}} \]
   6. Step-by-step derivation

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

         \[\leadsto \frac{\cos im}{1 - \color{blue}{re}} \]

      4. lower--.f64 95.9

         \[\leadsto \frac{\cos im}{\color{blue}{1 - re}} \]

   7. Applied rewrites 95.9%

      \[\leadsto \frac{\cos im}{\color{blue}{1 - re}} \]

   if -4.99999999999999968e-50 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0 or 0.99980000000000002 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. remove-double-neg N/A

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

      4. rec-exp N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. sin-PI/2 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. sin-PI/2 N/A

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

      12. *-commutative N/A

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

      13. *-lft-identity N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 100.0

          \[\leadsto \frac{\cos im}{e^{\color{blue}{-re}}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\cos im}{e^{-re}}} \]

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{neg}\left(re\right)}}} \]
   6. Step-by-step derivation

      1. exp-neg N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{re}}}} \]

      2. remove-double-div N/A

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

      3. lower-exp.f64 100.0

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

   7. Applied rewrites 100.0%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(\left(im \cdot im\right) \cdot -0.5\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -5 \cdot 10^{-50} \lor \neg \left(e^{re} \cdot \cos im \leq 0 \lor \neg \left(e^{re} \cdot \cos im \leq 0.9998\right)\right):\\ \;\;\;\;\frac{\cos im}{1 - re}\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq -5 \cdot 10^{-50} \lor \neg \left(t\_0 \leq 0 \lor \neg \left(t\_0 \leq 0.9998\right)\right):\\ \;\;\;\;\frac{\cos im}{1 - re}\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)
      (fma (* im im) -0.5 1.0))
     (if (or (<= t_0 -5e-50) (not (or (<= t_0 0.0) (not (<= t_0 0.9998)))))
       (/ (cos im) (- 1.0 re))
       (exp re)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * fma((im * im), -0.5, 1.0);
	} else if ((t_0 <= -5e-50) || !((t_0 <= 0.0) || !(t_0 <= 0.9998))) {
		tmp = cos(im) / (1.0 - re);
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * fma(Float64(im * im), -0.5, 1.0));
	elseif ((t_0 <= -5e-50) || !((t_0 <= 0.0) || !(t_0 <= 0.9998)))
		tmp = Float64(cos(im) / Float64(1.0 - re));
	else
		tmp = exp(re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, -5e-50], N[Not[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 0.9998]], $MachinePrecision]]], $MachinePrecision]], N[(N[Cos[im], $MachinePrecision] / N[(1.0 - re), $MachinePrecision]), $MachinePrecision], N[Exp[re], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} \cdot \cos im \]
   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. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 47.9

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

   5. Applied rewrites 47.9%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. /-rgt-identity N/A

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

      3. /-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 86.0

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

   8. Applied rewrites 86.0%

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

   9. Taylor expanded in re around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. +-commutative N/A

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

       4. distribute-rgt1-in N/A

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

       5. distribute-rgt1-in N/A

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

       6. +-commutative N/A

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

       7. lower-fma.f64 N/A

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

       8. +-commutative N/A

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

       9. *-commutative N/A

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

       10. lower-fma.f64 N/A

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

       11. +-commutative N/A

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

       12. lower-fma.f64 90.7

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

   11. Applied rewrites 90.7%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -4.99999999999999968e-50 or 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.99980000000000002

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. remove-double-neg N/A

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

      4. rec-exp N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. sin-PI/2 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. sin-PI/2 N/A

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

      12. *-commutative N/A

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

      13. *-lft-identity N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 100.0

          \[\leadsto \frac{\cos im}{e^{\color{blue}{-re}}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\cos im}{e^{-re}}} \]

   5. Taylor expanded in re around 0

      \[\leadsto \frac{\cos im}{\color{blue}{1 + -1 \cdot re}} \]
   6. Step-by-step derivation

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

         \[\leadsto \frac{\cos im}{1 - \color{blue}{re}} \]

      4. lower--.f64 95.9

         \[\leadsto \frac{\cos im}{\color{blue}{1 - re}} \]

   7. Applied rewrites 95.9%

      \[\leadsto \frac{\cos im}{\color{blue}{1 - re}} \]

   if -4.99999999999999968e-50 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0 or 0.99980000000000002 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. remove-double-neg N/A

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

      4. rec-exp N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. sin-PI/2 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. sin-PI/2 N/A

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

      12. *-commutative N/A

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

      13. *-lft-identity N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 100.0

          \[\leadsto \frac{\cos im}{e^{\color{blue}{-re}}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\cos im}{e^{-re}}} \]

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{neg}\left(re\right)}}} \]
   6. Step-by-step derivation

      1. exp-neg N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{re}}}} \]

      2. remove-double-div N/A

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

      3. lower-exp.f64 100.0

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

   7. Applied rewrites 100.0%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -5 \cdot 10^{-50} \lor \neg \left(e^{re} \cdot \cos im \leq 0 \lor \neg \left(e^{re} \cdot \cos im \leq 0.9998\right)\right):\\ \;\;\;\;\frac{\cos im}{1 - re}\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq -0.05 \lor \neg \left(t\_0 \leq 0 \lor \neg \left(t\_0 \leq 0.9998\right)\right):\\ \;\;\;\;\left(1 + re\right) \cdot \cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)
      (fma (* im im) -0.5 1.0))
     (if (or (<= t_0 -0.05) (not (or (<= t_0 0.0) (not (<= t_0 0.9998)))))
       (* (+ 1.0 re) (cos im))
       (exp re)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * fma((im * im), -0.5, 1.0);
	} else if ((t_0 <= -0.05) || !((t_0 <= 0.0) || !(t_0 <= 0.9998))) {
		tmp = (1.0 + re) * cos(im);
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * fma(Float64(im * im), -0.5, 1.0));
	elseif ((t_0 <= -0.05) || !((t_0 <= 0.0) || !(t_0 <= 0.9998)))
		tmp = Float64(Float64(1.0 + re) * cos(im));
	else
		tmp = exp(re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, -0.05], N[Not[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 0.9998]], $MachinePrecision]]], $MachinePrecision]], N[(N[(1.0 + re), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], N[Exp[re], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} \cdot \cos im \]
   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. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 47.9

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

   5. Applied rewrites 47.9%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. /-rgt-identity N/A

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

      3. /-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 86.0

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

   8. Applied rewrites 86.0%

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

   9. Taylor expanded in re around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. +-commutative N/A

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

       4. distribute-rgt1-in N/A

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

       5. distribute-rgt1-in N/A

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

       6. +-commutative N/A

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

       7. lower-fma.f64 N/A

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

       8. +-commutative N/A

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

       9. *-commutative N/A

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

       10. lower-fma.f64 N/A

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

       11. +-commutative N/A

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

       12. lower-fma.f64 90.7

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

   11. Applied rewrites 90.7%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.050000000000000003 or 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.99980000000000002

   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 97.2

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

   5. Applied rewrites 97.2%

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

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0 or 0.99980000000000002 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. remove-double-neg N/A

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

      4. rec-exp N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. sin-PI/2 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. sin-PI/2 N/A

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

      12. *-commutative N/A

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

      13. *-lft-identity N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 100.0

          \[\leadsto \frac{\cos im}{e^{\color{blue}{-re}}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\cos im}{e^{-re}}} \]

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{neg}\left(re\right)}}} \]
   6. Step-by-step derivation

      1. exp-neg N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{re}}}} \]

      2. remove-double-div N/A

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

      3. lower-exp.f64 99.4

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

   7. Applied rewrites 99.4%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -0.05 \lor \neg \left(e^{re} \cdot \cos im \leq 0 \lor \neg \left(e^{re} \cdot \cos im \leq 0.9998\right)\right):\\ \;\;\;\;\left(1 + re\right) \cdot \cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq -5 \cdot 10^{-50} \lor \neg \left(t\_0 \leq 0 \lor \neg \left(t\_0 \leq 0.9998\right)\right):\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)
      (fma (* im im) -0.5 1.0))
     (if (or (<= t_0 -5e-50) (not (or (<= t_0 0.0) (not (<= t_0 0.9998)))))
       (cos im)
       (exp re)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * fma((im * im), -0.5, 1.0);
	} else if ((t_0 <= -5e-50) || !((t_0 <= 0.0) || !(t_0 <= 0.9998))) {
		tmp = cos(im);
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * fma(Float64(im * im), -0.5, 1.0));
	elseif ((t_0 <= -5e-50) || !((t_0 <= 0.0) || !(t_0 <= 0.9998)))
		tmp = cos(im);
	else
		tmp = exp(re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, -5e-50], N[Not[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 0.9998]], $MachinePrecision]]], $MachinePrecision]], N[Cos[im], $MachinePrecision], N[Exp[re], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} \cdot \cos im \]
   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. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 47.9

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

   5. Applied rewrites 47.9%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. /-rgt-identity N/A

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

      3. /-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 86.0

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

   8. Applied rewrites 86.0%

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

   9. Taylor expanded in re around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. +-commutative N/A

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

       4. distribute-rgt1-in N/A

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

       5. distribute-rgt1-in N/A

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

       6. +-commutative N/A

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

       7. lower-fma.f64 N/A

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

       8. +-commutative N/A

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

       9. *-commutative N/A

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

       10. lower-fma.f64 N/A

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

       11. +-commutative N/A

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

       12. lower-fma.f64 90.7

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

   11. Applied rewrites 90.7%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -4.99999999999999968e-50 or 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.99980000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-cos.f64 94.8

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

   5. Applied rewrites 94.8%

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

   if -4.99999999999999968e-50 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0 or 0.99980000000000002 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. remove-double-neg N/A

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

      4. rec-exp N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. sin-PI/2 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. sin-PI/2 N/A

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

      12. *-commutative N/A

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

      13. *-lft-identity N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 100.0

          \[\leadsto \frac{\cos im}{e^{\color{blue}{-re}}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\cos im}{e^{-re}}} \]

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{neg}\left(re\right)}}} \]
   6. Step-by-step derivation

      1. exp-neg N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{re}}}} \]

      2. remove-double-div N/A

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

      3. lower-exp.f64 100.0

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

   7. Applied rewrites 100.0%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -5 \cdot 10^{-50} \lor \neg \left(e^{re} \cdot \cos im \leq 0 \lor \neg \left(e^{re} \cdot \cos im \leq 0.9998\right)\right):\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 82.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ t_1 := e^{re} \cdot \cos im\\ t_2 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_2 \cdot t\_0\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-50}:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, 0.5\right) \cdot re - 1, re, 1\right)}\\ \mathbf{elif}\;t\_1 \leq 0.9998:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5, im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (fma (* im im) -0.5 1.0))
        (t_1 (* (exp re) (cos im)))
        (t_2 (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)))
   (if (<= t_1 (- INFINITY))
     (* t_2 t_0)
     (if (<= t_1 -5e-50)
       (cos im)
       (if (<= t_1 0.0)
         (/ t_0 (fma (- (* (fma -0.16666666666666666 re 0.5) re) 1.0) re 1.0))
         (if (<= t_1 0.9998)
           (cos im)
           (*
            t_2
            (fma
             (- (* 0.041666666666666664 (* im im)) 0.5)
             (* im im)
             1.0))))))))

C

double code(double re, double im) {
	double t_0 = fma((im * im), -0.5, 1.0);
	double t_1 = exp(re) * cos(im);
	double t_2 = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2 * t_0;
	} else if (t_1 <= -5e-50) {
		tmp = cos(im);
	} else if (t_1 <= 0.0) {
		tmp = t_0 / fma(((fma(-0.16666666666666666, re, 0.5) * re) - 1.0), re, 1.0);
	} else if (t_1 <= 0.9998) {
		tmp = cos(im);
	} else {
		tmp = t_2 * fma(((0.041666666666666664 * (im * im)) - 0.5), (im * im), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(Float64(im * im), -0.5, 1.0)
	t_1 = Float64(exp(re) * cos(im))
	t_2 = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(t_2 * t_0);
	elseif (t_1 <= -5e-50)
		tmp = cos(im);
	elseif (t_1 <= 0.0)
		tmp = Float64(t_0 / fma(Float64(Float64(fma(-0.16666666666666666, re, 0.5) * re) - 1.0), re, 1.0));
	elseif (t_1 <= 0.9998)
		tmp = cos(im);
	else
		tmp = Float64(t_2 * fma(Float64(Float64(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[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(t$95$2 * t$95$0), $MachinePrecision], If[LessEqual[t$95$1, -5e-50], N[Cos[im], $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(t$95$0 / N[(N[(N[(N[(-0.16666666666666666 * re + 0.5), $MachinePrecision] * re), $MachinePrecision] - 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9998], N[Cos[im], $MachinePrecision], N[(t$95$2 * N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $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

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} \cdot \cos im \]
   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. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 47.9

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

   5. Applied rewrites 47.9%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. /-rgt-identity N/A

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

      3. /-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 86.0

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

   8. Applied rewrites 86.0%

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

   9. Taylor expanded in re around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. +-commutative N/A

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

       4. distribute-rgt1-in N/A

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

       5. distribute-rgt1-in N/A

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

       6. +-commutative N/A

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

       7. lower-fma.f64 N/A

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

       8. +-commutative N/A

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

       9. *-commutative N/A

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

       10. lower-fma.f64 N/A

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

       11. +-commutative N/A

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

       12. lower-fma.f64 90.7

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

   11. Applied rewrites 90.7%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -4.99999999999999968e-50 or 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.99980000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-cos.f64 94.8

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

   5. Applied rewrites 94.8%

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

   if -4.99999999999999968e-50 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. remove-double-neg N/A

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

      4. rec-exp N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. sin-PI/2 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. sin-PI/2 N/A

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

      12. *-commutative N/A

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

      13. *-lft-identity N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 100.0

          \[\leadsto \frac{\cos im}{e^{\color{blue}{-re}}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\cos im}{e^{-re}}} \]

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 54.6

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

   7. Applied rewrites 54.6%

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

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 37.1

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

   10. Applied rewrites 37.1%

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

   if 0.99980000000000002 < (*.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. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 78.5

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

   5. Applied rewrites 78.5%

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

   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. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 84.4

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

   8. Applied rewrites 84.4%

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

   9. Taylor expanded in re around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. +-commutative N/A

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

       4. distribute-rgt1-in N/A

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

       5. distribute-rgt1-in N/A

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

       6. +-commutative N/A

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

       7. lower-fma.f64 N/A

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

       8. +-commutative N/A

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

       9. *-commutative N/A

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

       10. lower-fma.f64 N/A

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

       11. +-commutative N/A

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

       12. lower-fma.f64 93.4

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

   11. Applied rewrites 93.4%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -5 \cdot 10^{-50}:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(im \cdot im, -0.5, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, 0.5\right) \cdot re - 1, re, 1\right)}\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0.9998:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5, im \cdot im, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 58.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ t_1 := e^{re} \cdot \cos im\\ t_2 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_2 \cdot t\_0\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, 0.5\right) \cdot re - 1, re, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5, im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (fma (* im im) -0.5 1.0))
        (t_1 (* (exp re) (cos im)))
        (t_2 (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)))
   (if (<= t_1 (- INFINITY))
     (* t_2 t_0)
     (if (<= t_1 0.0)
       (/ t_0 (fma (- (* (fma -0.16666666666666666 re 0.5) re) 1.0) re 1.0))
       (*
        t_2
        (fma (- (* 0.041666666666666664 (* im im)) 0.5) (* im im) 1.0))))))

C

double code(double re, double im) {
	double t_0 = fma((im * im), -0.5, 1.0);
	double t_1 = exp(re) * cos(im);
	double t_2 = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2 * t_0;
	} else if (t_1 <= 0.0) {
		tmp = t_0 / fma(((fma(-0.16666666666666666, re, 0.5) * re) - 1.0), re, 1.0);
	} else {
		tmp = t_2 * fma(((0.041666666666666664 * (im * im)) - 0.5), (im * im), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(Float64(im * im), -0.5, 1.0)
	t_1 = Float64(exp(re) * cos(im))
	t_2 = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(t_2 * t_0);
	elseif (t_1 <= 0.0)
		tmp = Float64(t_0 / fma(Float64(Float64(fma(-0.16666666666666666, re, 0.5) * re) - 1.0), re, 1.0));
	else
		tmp = Float64(t_2 * fma(Float64(Float64(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[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(t$95$2 * t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(t$95$0 / N[(N[(N[(N[(-0.16666666666666666 * re + 0.5), $MachinePrecision] * re), $MachinePrecision] - 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} \cdot \cos im \]
   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. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 47.9

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

   5. Applied rewrites 47.9%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. /-rgt-identity N/A

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

      3. /-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 86.0

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

   8. Applied rewrites 86.0%

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

   9. Taylor expanded in re around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. +-commutative N/A

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

       4. distribute-rgt1-in N/A

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

       5. distribute-rgt1-in N/A

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

       6. +-commutative N/A

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

       7. lower-fma.f64 N/A

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

       8. +-commutative N/A

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

       9. *-commutative N/A

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

       10. lower-fma.f64 N/A

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

       11. +-commutative N/A

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

       12. lower-fma.f64 90.7

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

   11. Applied rewrites 90.7%

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

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

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. remove-double-neg N/A

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

      4. rec-exp N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. sin-PI/2 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. sin-PI/2 N/A

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

      12. *-commutative N/A

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

      13. *-lft-identity N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 100.0

          \[\leadsto \frac{\cos im}{e^{\color{blue}{-re}}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\cos im}{e^{-re}}} \]

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 68.2

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

   7. Applied rewrites 68.2%

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

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 26.3

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

   10. Applied rewrites 26.3%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} \cdot \cos im \]
   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. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 83.4

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

   5. Applied rewrites 83.4%

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

   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. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 64.1

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

   8. Applied rewrites 64.1%

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

   9. Taylor expanded in re around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. +-commutative N/A

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

       4. distribute-rgt1-in N/A

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

       5. distribute-rgt1-in N/A

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

       6. +-commutative N/A

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

       7. lower-fma.f64 N/A

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

       8. +-commutative N/A

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

       9. *-commutative N/A

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

       10. lower-fma.f64 N/A

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

       11. +-commutative N/A

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

       12. lower-fma.f64 70.8

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

   11. Applied rewrites 70.8%

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

Alternative 12: 56.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} \cdot \cos im \]
   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. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 47.9

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

   5. Applied rewrites 47.9%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. /-rgt-identity N/A

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

      3. /-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 86.0

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

   8. Applied rewrites 86.0%

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

   9. Taylor expanded in re around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. +-commutative N/A

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

       4. distribute-rgt1-in N/A

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

       5. distribute-rgt1-in N/A

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

       6. +-commutative N/A

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

       7. lower-fma.f64 N/A

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

       8. +-commutative N/A

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

       9. *-commutative N/A

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

       10. lower-fma.f64 N/A

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

       11. +-commutative N/A

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

       12. lower-fma.f64 90.7

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

   11. Applied rewrites 90.7%

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

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

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. remove-double-neg N/A

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

      4. rec-exp N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. sin-PI/2 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. sin-PI/2 N/A

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

      12. *-commutative N/A

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

      13. *-lft-identity N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 100.0

          \[\leadsto \frac{\cos im}{e^{\color{blue}{-re}}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\cos im}{e^{-re}}} \]

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 68.2

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

   7. Applied rewrites 68.2%

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

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 26.3

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

   10. Applied rewrites 26.3%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} \cdot \cos im \]
   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. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 83.4

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

   5. Applied rewrites 83.4%

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

   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. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 64.1

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

   8. Applied rewrites 64.1%

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

Alternative 13: 50.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -0.9995:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(im \cdot im\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5, im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 -0.9995)
     (*
      (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)
      (fma (* im im) -0.5 1.0))
     (if (<= t_0 0.0)
       (* (* im im) -0.5)
       (*
        (fma (fma 0.5 re 1.0) re 1.0)
        (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 <= -0.9995) {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * fma((im * im), -0.5, 1.0);
	} else if (t_0 <= 0.0) {
		tmp = (im * im) * -0.5;
	} else {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * 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 <= -0.9995)
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * fma(Float64(im * im), -0.5, 1.0));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(im * im) * -0.5);
	else
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * fma(Float64(Float64(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, -0.9995], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.99950000000000006

   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. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 50.4

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

   5. Applied rewrites 50.4%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. /-rgt-identity N/A

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

      3. /-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 82.2

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

   8. Applied rewrites 82.2%

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

   9. Taylor expanded in re around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. +-commutative N/A

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

       4. distribute-rgt1-in N/A

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

       5. distribute-rgt1-in N/A

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

       6. +-commutative N/A

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

       7. lower-fma.f64 N/A

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

       8. +-commutative N/A

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

       9. *-commutative N/A

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

       10. lower-fma.f64 N/A

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

       11. +-commutative N/A

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

       12. lower-fma.f64 86.7

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

   11. Applied rewrites 86.7%

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

   if -0.99950000000000006 < (*.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 32.1

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

   5. Applied rewrites 32.1%

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

   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 2.9%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 19.2%

       \[\leadsto \left(im \cdot im\right) \cdot -0.5 \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} \cdot \cos im \]
   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. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 83.4

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

   5. Applied rewrites 83.4%

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

   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. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 64.1

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

   8. Applied rewrites 64.1%

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

4. Final simplification 52.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -0.9995:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\left(im \cdot im\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5, im \cdot im, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 41.8% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (* (+ 1.0 re) (fma (* im im) -0.5 1.0))
     (if (<= t_0 0.0)
       (* (* im im) -0.5)
       (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 = (1.0 + re) * fma((im * im), -0.5, 1.0);
	} else if (t_0 <= 0.0) {
		tmp = (im * im) * -0.5;
	} else {
		tmp = 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(1.0 + re) * fma(Float64(im * im), -0.5, 1.0));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(im * im) * -0.5);
	else
		tmp = fma(Float64(Float64(Float64(0.041666666666666664 * Float64(im * im)) - 0.5) * 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[(1.0 + re), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision], N[(N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * im), $MachinePrecision] * im + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-+.f64 5.4

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

   5. Applied rewrites 5.4%

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

   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. /-rgt-identity N/A

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

      3. /-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 67.4

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

   8. Applied rewrites 67.4%

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

   if -inf.0 < (*.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 32.9

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

   5. Applied rewrites 32.9%

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

   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 2.9%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 19.1%

       \[\leadsto \left(im \cdot im\right) \cdot -0.5 \]

   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 62.2

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

   5. Applied rewrites 62.2%

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

   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 44.5%

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

   10. Applied rewrites 44.5%

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

4. Final simplification 38.3%

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

Alternative 15: 35.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * cos(im)) <= 0.0) {
		tmp = (im * im) * -0.5;
	} 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(im * im) * -0.5);
	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[(im * im), $MachinePrecision] * -0.5), $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 27.0

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

   5. Applied rewrites 27.0%

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

   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 9.2%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 22.1%

       \[\leadsto \left(im \cdot im\right) \cdot -0.5 \]

   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 62.2

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

   5. Applied rewrites 62.2%

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

   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 37.6%

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

4. Final simplification 31.6%

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

Alternative 16: 44.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 2 \cdot 10^{-49}:\\ \;\;\;\;\left(im \cdot im\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (exp re) 2e-49)
   (* (* im im) -0.5)
   (* (fma (fma 0.5 re 1.0) re 1.0) (fma (* im im) -0.5 1.0))))

C

double code(double re, double im) {
	double tmp;
	if (exp(re) <= 2e-49) {
		tmp = (im * im) * -0.5;
	} else {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * fma((im * im), -0.5, 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (exp(re) <= 2e-49)
		tmp = Float64(Float64(im * im) * -0.5);
	else
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * fma(Float64(im * im), -0.5, 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Exp[re], $MachinePrecision], 2e-49], N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 1.99999999999999987e-49

   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 3.2

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

   5. Applied rewrites 3.2%

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

   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 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 26.0%

       \[\leadsto \left(im \cdot im\right) \cdot -0.5 \]

   if 1.99999999999999987e-49 < (exp.f64 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. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 81.9

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

   5. Applied rewrites 81.9%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, re, 1\right), re, 1\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2} + 1\right)} \]

      2. /-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, re, 1\right), re, 1\right) \cdot \left(\color{blue}{\frac{\frac{-1}{2} \cdot {im}^{2}}{1}} + 1\right) \]

      3. /-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, re, 1\right), re, 1\right) \cdot \left(\color{blue}{\frac{-1}{2} \cdot {im}^{2}} + 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, re, 1\right), re, 1\right) \cdot \left(\color{blue}{{im}^{2} \cdot \frac{-1}{2}} + 1\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, re, 1\right), re, 1\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{2}, 1\right)} \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{2}, 1\right) \]

      7. lower-*.f64 51.5

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, -0.5, 1\right) \]

   8. Applied rewrites 51.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \color{blue}{\mathsf{fma}\left(im \cdot im, -0.5, 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 2 \cdot 10^{-49}:\\ \;\;\;\;\left(im \cdot im\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 46.8% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.6:\\ \;\;\;\;\left(im \cdot im\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.6)
   (* (* im im) -0.5)
   (*
    (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)
    (fma (* im im) -0.5 1.0))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.6) {
		tmp = (im * im) * -0.5;
	} else {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * fma((im * im), -0.5, 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.6)
		tmp = Float64(Float64(im * im) * -0.5);
	else
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * fma(Float64(im * im), -0.5, 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.6], N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.6000000000000001

   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 3.2

         \[\leadsto \color{blue}{\cos im} \]

   5. Applied rewrites 3.2%

      \[\leadsto \color{blue}{\cos im} \]

   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 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 26.0%

       \[\leadsto \left(im \cdot im\right) \cdot -0.5 \]

   if -1.6000000000000001 < 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. remove-double-neg N/A

         \[\leadsto \left(\color{blue}{re} \cdot \left(1 + \frac{1}{2} \cdot re\right) + 1\right) \cdot \cos im \]

      5. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(1 + \frac{1}{2} \cdot re\right) \cdot re} + 1\right) \cdot \cos im \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{1}{2} \cdot re, re, 1\right)} \cdot \cos im \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{2} \cdot re + 1}, re, 1\right) \cdot \cos im \]

      8. lower-fma.f64 81.9

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.5, re, 1\right)}, re, 1\right) \cdot \cos im \]

   5. Applied rewrites 81.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right)} \cdot \cos im \]

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, re, 1\right), re, 1\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, re, 1\right), re, 1\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2} + 1\right)} \]

      2. /-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, re, 1\right), re, 1\right) \cdot \left(\color{blue}{\frac{\frac{-1}{2} \cdot {im}^{2}}{1}} + 1\right) \]

      3. /-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, re, 1\right), re, 1\right) \cdot \left(\color{blue}{\frac{-1}{2} \cdot {im}^{2}} + 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, re, 1\right), re, 1\right) \cdot \left(\color{blue}{{im}^{2} \cdot \frac{-1}{2}} + 1\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, re, 1\right), re, 1\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{2}, 1\right)} \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{2}, 1\right) \]

      7. lower-*.f64 51.5

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, -0.5, 1\right) \]

   8. Applied rewrites 51.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \color{blue}{\mathsf{fma}\left(im \cdot im, -0.5, 1\right)} \]

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)} \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{2}, 1\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + 1\right)} \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{2}, 1\right) \]

       2. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) \cdot re} + 1\right) \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{2}, 1\right) \]

       3. +-commutative N/A

          \[\leadsto \left(\color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right) + 1\right)} \cdot re + 1\right) \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{2}, 1\right) \]

       4. distribute-rgt1-in N/A

          \[\leadsto \left(\color{blue}{\left(re + \left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) \cdot re\right)} + 1\right) \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{2}, 1\right) \]

       5. distribute-rgt1-in N/A

          \[\leadsto \left(\color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right) + 1\right) \cdot re} + 1\right) \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{2}, 1\right) \]

       6. +-commutative N/A

          \[\leadsto \left(\color{blue}{\left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)} \cdot re + 1\right) \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{2}, 1\right) \]

       7. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right), re, 1\right)} \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{2}, 1\right) \]

       8. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right) + 1}, re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{2}, 1\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot re\right) \cdot re} + 1, re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{2}, 1\right) \]

       10. lower-fma.f64 N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{6} \cdot re, re, 1\right)}, re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{2}, 1\right) \]

       11. +-commutative N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{6} \cdot re + \frac{1}{2}}, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{2}, 1\right) \]

       12. lower-fma.f64 56.3

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.16666666666666666, re, 0.5\right)}, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right) \]

   11. Applied rewrites 56.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right)} \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.6:\\ \;\;\;\;\left(im \cdot im\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 44.3% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -110:\\ \;\;\;\;\left(im \cdot im\right) \cdot -0.5\\ \mathbf{elif}\;re \leq 60000000:\\ \;\;\;\;\mathsf{fma}\left(\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5\right) \cdot im, im, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot 0.5\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -110.0)
   (* (* im im) -0.5)
   (if (<= re 60000000.0)
     (fma (* (- (* 0.041666666666666664 (* im im)) 0.5) im) im 1.0)
     (* (* (* re re) 0.5) (fma (* im im) -0.5 1.0)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -110.0) {
		tmp = (im * im) * -0.5;
	} else if (re <= 60000000.0) {
		tmp = fma((((0.041666666666666664 * (im * im)) - 0.5) * im), im, 1.0);
	} else {
		tmp = ((re * re) * 0.5) * fma((im * im), -0.5, 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -110.0)
		tmp = Float64(Float64(im * im) * -0.5);
	elseif (re <= 60000000.0)
		tmp = fma(Float64(Float64(Float64(0.041666666666666664 * Float64(im * im)) - 0.5) * im), im, 1.0);
	else
		tmp = Float64(Float64(Float64(re * re) * 0.5) * fma(Float64(im * im), -0.5, 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -110.0], N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[re, 60000000.0], N[(N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * im), $MachinePrecision] * im + 1.0), $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -110

   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 3.2

         \[\leadsto \color{blue}{\cos im} \]

   5. Applied rewrites 3.2%

      \[\leadsto \color{blue}{\cos im} \]

   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 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 26.0%

       \[\leadsto \left(im \cdot im\right) \cdot -0.5 \]

   if -110 < re < 6e7

   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 95.9

         \[\leadsto \color{blue}{\cos im} \]

   5. Applied rewrites 95.9%

      \[\leadsto \color{blue}{\cos im} \]

   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 48.0%

      \[\leadsto \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5, \color{blue}{im \cdot im}, 1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 48.0%

       \[\leadsto \mathsf{fma}\left(\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5\right) \cdot im, im, 1\right) \]

   if 6e7 < 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. remove-double-neg N/A

         \[\leadsto \left(\color{blue}{re} \cdot \left(1 + \frac{1}{2} \cdot re\right) + 1\right) \cdot \cos im \]

      5. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(1 + \frac{1}{2} \cdot re\right) \cdot re} + 1\right) \cdot \cos im \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{1}{2} \cdot re, re, 1\right)} \cdot \cos im \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{2} \cdot re + 1}, re, 1\right) \cdot \cos im \]

      8. lower-fma.f64 55.4

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.5, re, 1\right)}, re, 1\right) \cdot \cos im \]

   5. Applied rewrites 55.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right)} \cdot \cos im \]

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, re, 1\right), re, 1\right) \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {im}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, re, 1\right), re, 1\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot {im}^{2} + 1\right)} \]

      2. /-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, re, 1\right), re, 1\right) \cdot \left(\color{blue}{\frac{\frac{-1}{2} \cdot {im}^{2}}{1}} + 1\right) \]

      3. /-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, re, 1\right), re, 1\right) \cdot \left(\color{blue}{\frac{-1}{2} \cdot {im}^{2}} + 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, re, 1\right), re, 1\right) \cdot \left(\color{blue}{{im}^{2} \cdot \frac{-1}{2}} + 1\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, re, 1\right), re, 1\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{2}, 1\right)} \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{2}, 1\right) \]

      7. lower-*.f64 56.9

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, -0.5, 1\right) \]

   8. Applied rewrites 56.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \color{blue}{\mathsf{fma}\left(im \cdot im, -0.5, 1\right)} \]

   9. Taylor expanded in re around inf

      \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{{re}^{2}}\right) \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{2}, 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 56.9%

       \[\leadsto \left(\left(re \cdot re\right) \cdot \color{blue}{0.5}\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -110:\\ \;\;\;\;\left(im \cdot im\right) \cdot -0.5\\ \mathbf{elif}\;re \leq 60000000:\\ \;\;\;\;\mathsf{fma}\left(\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5\right) \cdot im, im, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot 0.5\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 37.6% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.4:\\ \;\;\;\;\left(im \cdot im\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.4) (* (* im im) -0.5) (* (+ 1.0 re) (fma (* im im) -0.5 1.0))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.4) {
		tmp = (im * im) * -0.5;
	} else {
		tmp = (1.0 + re) * fma((im * im), -0.5, 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.4)
		tmp = Float64(Float64(im * im) * -0.5);
	else
		tmp = Float64(Float64(1.0 + re) * fma(Float64(im * im), -0.5, 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.4], N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision], N[(N[(1.0 + re), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.3999999999999999

   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 3.2

         \[\leadsto \color{blue}{\cos im} \]

   5. Applied rewrites 3.2%

      \[\leadsto \color{blue}{\cos im} \]

   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 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 26.0%

       \[\leadsto \left(im \cdot im\right) \cdot -0.5 \]

   if -1.3999999999999999 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re\right)} \cdot \cos im \]
   4. Step-by-step derivation

      1. lower-+.f64 62.7

         \[\leadsto \color{blue}{\left(1 + re\right)} \cdot \cos im \]

   5. Applied rewrites 62.7%

      \[\leadsto \color{blue}{\left(1 + re\right)} \cdot \cos im \]

   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. /-rgt-identity N/A

         \[\leadsto \left(1 + re\right) \cdot \left(\color{blue}{\frac{\frac{-1}{2} \cdot {im}^{2}}{1}} + 1\right) \]

      3. /-rgt-identity N/A

         \[\leadsto \left(1 + re\right) \cdot \left(\color{blue}{\frac{-1}{2} \cdot {im}^{2}} + 1\right) \]

      4. *-commutative N/A

         \[\leadsto \left(1 + re\right) \cdot \left(\color{blue}{{im}^{2} \cdot \frac{-1}{2}} + 1\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(1 + re\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{2}, 1\right)} \]

      6. unpow2 N/A

         \[\leadsto \left(1 + re\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{2}, 1\right) \]

      7. lower-*.f64 37.6

         \[\leadsto \left(1 + re\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, -0.5, 1\right) \]

   8. Applied rewrites 37.6%

      \[\leadsto \left(1 + re\right) \cdot \color{blue}{\mathsf{fma}\left(im \cdot im, -0.5, 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.4:\\ \;\;\;\;\left(im \cdot im\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 11.5% accurate, 18.7× speedup

Math

\[\begin{array}{l} \\ \left(im \cdot im\right) \cdot -0.5 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (* im im) -0.5))

C

double code(double re, double im) {
	return (im * im) * -0.5;
}

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 = (im * im) * (-0.5d0)
end function

Java

public static double code(double re, double im) {
	return (im * im) * -0.5;
}

Python

def code(re, im):
	return (im * im) * -0.5

Julia

function code(re, im)
	return Float64(Float64(im * im) * -0.5)
end

MATLAB

function tmp = code(re, im)
	tmp = (im * im) * -0.5;
end

Wolfram

code[re_, im_] := N[(N[(im * im), $MachinePrecision] * -0.5), $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 48.5

      \[\leadsto \color{blue}{\cos im} \]

5. Applied rewrites 48.5%

   \[\leadsto \color{blue}{\cos im} \]

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 26.5%

   \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{-0.5}, 1\right) \]

9. Taylor expanded in im around inf

   \[\leadsto \frac{-1}{2} \cdot {im}^{\color{blue}{2}} \]
10. Step-by-step derivation

11. Applied rewrites 9.6%

    \[\leadsto \left(im \cdot im\right) \cdot -0.5 \]

12. Final simplification 9.6%

    \[\leadsto \left(im \cdot im\right) \cdot -0.5 \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024359 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))