math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 6.8s

Alternatives: 20

Speedup: 1.0×

• 25.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 2: 98.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \mathbf{elif}\;t\_0 \leq -0.1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \cos im\\ \mathbf{elif}\;t\_0 \leq 10^{-76} \lor \neg \left(t\_0 \leq 0.9999999956798714\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos im}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, 0.5\right), re, -1\right), re, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (*
      (* (* re re) 0.5)
      (fma
       (fma
        (fma -0.001388888888888889 (* im im) 0.041666666666666664)
        (* im im)
        -0.5)
       (* im im)
       1.0))
     (if (<= t_0 -0.1)
       (* (fma (fma 0.5 re 1.0) re 1.0) (cos im))
       (if (or (<= t_0 1e-76) (not (<= t_0 0.9999999956798714)))
         (exp re)
         (/
          (cos im)
          (fma (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 = ((re * re) * 0.5) * fma(fma(fma(-0.001388888888888889, (im * im), 0.041666666666666664), (im * im), -0.5), (im * im), 1.0);
	} else if (t_0 <= -0.1) {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * cos(im);
	} else if ((t_0 <= 1e-76) || !(t_0 <= 0.9999999956798714)) {
		tmp = exp(re);
	} else {
		tmp = cos(im) / fma(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(Float64(Float64(re * re) * 0.5) * fma(fma(fma(-0.001388888888888889, Float64(im * im), 0.041666666666666664), Float64(im * im), -0.5), Float64(im * im), 1.0));
	elseif (t_0 <= -0.1)
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * cos(im));
	elseif ((t_0 <= 1e-76) || !(t_0 <= 0.9999999956798714))
		tmp = exp(re);
	else
		tmp = Float64(cos(im) / fma(fma(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[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[(N[(-0.001388888888888889 * N[(im * im), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + -0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.1], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, 1e-76], N[Not[LessEqual[t$95$0, 0.9999999956798714]], $MachinePrecision]], N[Exp[re], $MachinePrecision], N[(N[Cos[im], $MachinePrecision] / N[(N[(N[(-0.16666666666666666 * re + 0.5), $MachinePrecision] * re + -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 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

   5. Applied rewrites 30.8%

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

   6. Taylor expanded in re around inf

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

   8. Applied rewrites 30.8%

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

   9. Taylor expanded in im around 0

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

   11. Applied rewrites 100.0%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 100.0%

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

   if -0.10000000000000001 < (*.f64 (exp.f64 re) (cos.f64 im)) < 9.99999999999999927e-77 or 0.999999995679871412 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 99.5%

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

   if 9.99999999999999927e-77 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.999999995679871412

   1. Initial program 99.9%

      \[e^{re} \cdot \cos im \]
   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. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

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

      5. sinh-cosh N/A

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

      6. sinh---cosh-rev N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. lift-cos.f64 N/A

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

      10. sin-PI/2 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. sin-PI/2 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-exp.f64 N/A

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

      17. lower-neg.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 98.3%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 98.3

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

   9. Applied rewrites 98.3%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -0.1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \cos im\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 10^{-76} \lor \neg \left(e^{re} \cdot \cos im \leq 0.9999999956798714\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos im}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, 0.5\right), re, -1\right), re, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \mathbf{elif}\;t\_0 \leq -0.1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \cos im\\ \mathbf{elif}\;t\_0 \leq 10^{-76} \lor \neg \left(t\_0 \leq 0.9999999956798714\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \cos im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (*
      (* (* re re) 0.5)
      (fma
       (fma
        (fma -0.001388888888888889 (* im im) 0.041666666666666664)
        (* im im)
        -0.5)
       (* im im)
       1.0))
     (if (<= t_0 -0.1)
       (* (fma (fma 0.5 re 1.0) re 1.0) (cos im))
       (if (or (<= t_0 1e-76) (not (<= t_0 0.9999999956798714)))
         (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 = ((re * re) * 0.5) * fma(fma(fma(-0.001388888888888889, (im * im), 0.041666666666666664), (im * im), -0.5), (im * im), 1.0);
	} else if (t_0 <= -0.1) {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * cos(im);
	} else if ((t_0 <= 1e-76) || !(t_0 <= 0.9999999956798714)) {
		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(Float64(Float64(re * re) * 0.5) * fma(fma(fma(-0.001388888888888889, Float64(im * im), 0.041666666666666664), Float64(im * im), -0.5), Float64(im * im), 1.0));
	elseif (t_0 <= -0.1)
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * cos(im));
	elseif ((t_0 <= 1e-76) || !(t_0 <= 0.9999999956798714))
		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[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[(N[(-0.001388888888888889 * N[(im * im), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + -0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.1], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, 1e-76], N[Not[LessEqual[t$95$0, 0.9999999956798714]], $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 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

   5. Applied rewrites 30.8%

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

   6. Taylor expanded in re around inf

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

   8. Applied rewrites 30.8%

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

   9. Taylor expanded in im around 0

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

   11. Applied rewrites 100.0%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 100.0%

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

   if -0.10000000000000001 < (*.f64 (exp.f64 re) (cos.f64 im)) < 9.99999999999999927e-77 or 0.999999995679871412 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 99.5%

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

   if 9.99999999999999927e-77 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.999999995679871412

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 98.3%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -0.1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \cos im\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 10^{-76} \lor \neg \left(e^{re} \cdot \cos im \leq 0.9999999956798714\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \cos im\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \mathbf{elif}\;t\_0 \leq -0.1 \lor \neg \left(t\_0 \leq 10^{-76} \lor \neg \left(t\_0 \leq 0.9999999956798714\right)\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (*
      (* (* re re) 0.5)
      (fma
       (fma
        (fma -0.001388888888888889 (* im im) 0.041666666666666664)
        (* im im)
        -0.5)
       (* im im)
       1.0))
     (if (or (<= t_0 -0.1)
             (not (or (<= t_0 1e-76) (not (<= t_0 0.9999999956798714)))))
       (* (fma (fma 0.5 re 1.0) re 1.0) (cos im))
       (exp re)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = ((re * re) * 0.5) * fma(fma(fma(-0.001388888888888889, (im * im), 0.041666666666666664), (im * im), -0.5), (im * im), 1.0);
	} else if ((t_0 <= -0.1) || !((t_0 <= 1e-76) || !(t_0 <= 0.9999999956798714))) {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * cos(im);
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(re * re) * 0.5) * fma(fma(fma(-0.001388888888888889, Float64(im * im), 0.041666666666666664), Float64(im * im), -0.5), Float64(im * im), 1.0));
	elseif ((t_0 <= -0.1) || !((t_0 <= 1e-76) || !(t_0 <= 0.9999999956798714)))
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * 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[(re * re), $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[(N[(-0.001388888888888889 * N[(im * im), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + -0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, -0.1], N[Not[Or[LessEqual[t$95$0, 1e-76], N[Not[LessEqual[t$95$0, 0.9999999956798714]], $MachinePrecision]]], $MachinePrecision]], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $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

   5. Applied rewrites 30.8%

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

   6. Taylor expanded in re around inf

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

   8. Applied rewrites 30.8%

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

   9. Taylor expanded in im around 0

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

   11. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.10000000000000001 or 9.99999999999999927e-77 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.999999995679871412

   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

   5. Applied rewrites 98.6%

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

   if -0.10000000000000001 < (*.f64 (exp.f64 re) (cos.f64 im)) < 9.99999999999999927e-77 or 0.999999995679871412 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 99.5%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -0.1 \lor \neg \left(e^{re} \cdot \cos im \leq 10^{-76} \lor \neg \left(e^{re} \cdot \cos im \leq 0.9999999956798714\right)\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \mathbf{elif}\;t\_0 \leq -0.1 \lor \neg \left(t\_0 \leq 10^{-76} \lor \neg \left(t\_0 \leq 0.9999999956798714\right)\right):\\ \;\;\;\;\left(1 + re\right) \cdot \cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (*
      (* (* re re) 0.5)
      (fma
       (fma
        (fma -0.001388888888888889 (* im im) 0.041666666666666664)
        (* im im)
        -0.5)
       (* im im)
       1.0))
     (if (or (<= t_0 -0.1)
             (not (or (<= t_0 1e-76) (not (<= t_0 0.9999999956798714)))))
       (* (+ 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 = ((re * re) * 0.5) * fma(fma(fma(-0.001388888888888889, (im * im), 0.041666666666666664), (im * im), -0.5), (im * im), 1.0);
	} else if ((t_0 <= -0.1) || !((t_0 <= 1e-76) || !(t_0 <= 0.9999999956798714))) {
		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(Float64(Float64(re * re) * 0.5) * fma(fma(fma(-0.001388888888888889, Float64(im * im), 0.041666666666666664), Float64(im * im), -0.5), Float64(im * im), 1.0));
	elseif ((t_0 <= -0.1) || !((t_0 <= 1e-76) || !(t_0 <= 0.9999999956798714)))
		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[(re * re), $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[(N[(-0.001388888888888889 * N[(im * im), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + -0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, -0.1], N[Not[Or[LessEqual[t$95$0, 1e-76], N[Not[LessEqual[t$95$0, 0.9999999956798714]], $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

   5. Applied rewrites 30.8%

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

   6. Taylor expanded in re around inf

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

   8. Applied rewrites 30.8%

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

   9. Taylor expanded in im around 0

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

   11. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.10000000000000001 or 9.99999999999999927e-77 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.999999995679871412

   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

   5. Applied rewrites 97.6%

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

   if -0.10000000000000001 < (*.f64 (exp.f64 re) (cos.f64 im)) < 9.99999999999999927e-77 or 0.999999995679871412 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 99.5%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -0.1 \lor \neg \left(e^{re} \cdot \cos im \leq 10^{-76} \lor \neg \left(e^{re} \cdot \cos im \leq 0.9999999956798714\right)\right):\\ \;\;\;\;\left(1 + re\right) \cdot \cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \mathbf{elif}\;t\_0 \leq -0.1 \lor \neg \left(t\_0 \leq 10^{-76} \lor \neg \left(t\_0 \leq 0.9999999956798714\right)\right):\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (*
      (* (* re re) 0.5)
      (fma
       (fma
        (fma -0.001388888888888889 (* im im) 0.041666666666666664)
        (* im im)
        -0.5)
       (* im im)
       1.0))
     (if (or (<= t_0 -0.1)
             (not (or (<= t_0 1e-76) (not (<= t_0 0.9999999956798714)))))
       (cos im)
       (exp re)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = ((re * re) * 0.5) * fma(fma(fma(-0.001388888888888889, (im * im), 0.041666666666666664), (im * im), -0.5), (im * im), 1.0);
	} else if ((t_0 <= -0.1) || !((t_0 <= 1e-76) || !(t_0 <= 0.9999999956798714))) {
		tmp = cos(im);
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(re * re) * 0.5) * fma(fma(fma(-0.001388888888888889, Float64(im * im), 0.041666666666666664), Float64(im * im), -0.5), Float64(im * im), 1.0));
	elseif ((t_0 <= -0.1) || !((t_0 <= 1e-76) || !(t_0 <= 0.9999999956798714)))
		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[(re * re), $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[(N[(-0.001388888888888889 * N[(im * im), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + -0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, -0.1], N[Not[Or[LessEqual[t$95$0, 1e-76], N[Not[LessEqual[t$95$0, 0.9999999956798714]], $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

   5. Applied rewrites 30.8%

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

   6. Taylor expanded in re around inf

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

   8. Applied rewrites 30.8%

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

   9. Taylor expanded in im around 0

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

   11. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.10000000000000001 or 9.99999999999999927e-77 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.999999995679871412

   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

   5. Applied rewrites 94.3%

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

   if -0.10000000000000001 < (*.f64 (exp.f64 re) (cos.f64 im)) < 9.99999999999999927e-77 or 0.999999995679871412 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 99.5%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -0.1 \lor \neg \left(e^{re} \cdot \cos im \leq 10^{-76} \lor \neg \left(e^{re} \cdot \cos im \leq 0.9999999956798714\right)\right):\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \mathbf{elif}\;t\_0 \leq -0.1:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;t\_0 \leq 10^{-76}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, 0.5\right), re, -1\right), re, 1\right)}\\ \mathbf{elif}\;t\_0 \leq 0.99999:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (*
      (* (* re re) 0.5)
      (fma
       (fma
        (fma -0.001388888888888889 (* im im) 0.041666666666666664)
        (* im im)
        -0.5)
       (* im im)
       1.0))
     (if (<= t_0 -0.1)
       (cos im)
       (if (<= t_0 1e-76)
         (/ 1.0 (fma (fma (fma -0.16666666666666666 re 0.5) re -1.0) re 1.0))
         (if (<= t_0 0.99999)
           (cos im)
           (*
            (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)
            (fma
             (fma 0.041666666666666664 (* im im) -0.5)
             (* im im)
             1.0))))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = ((re * re) * 0.5) * fma(fma(fma(-0.001388888888888889, (im * im), 0.041666666666666664), (im * im), -0.5), (im * im), 1.0);
	} else if (t_0 <= -0.1) {
		tmp = cos(im);
	} else if (t_0 <= 1e-76) {
		tmp = 1.0 / fma(fma(fma(-0.16666666666666666, re, 0.5), re, -1.0), re, 1.0);
	} else if (t_0 <= 0.99999) {
		tmp = cos(im);
	} else {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * fma(fma(0.041666666666666664, (im * im), -0.5), (im * im), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(re * re) * 0.5) * fma(fma(fma(-0.001388888888888889, Float64(im * im), 0.041666666666666664), Float64(im * im), -0.5), Float64(im * im), 1.0));
	elseif (t_0 <= -0.1)
		tmp = cos(im);
	elseif (t_0 <= 1e-76)
		tmp = Float64(1.0 / fma(fma(fma(-0.16666666666666666, re, 0.5), re, -1.0), re, 1.0));
	elseif (t_0 <= 0.99999)
		tmp = cos(im);
	else
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * fma(fma(0.041666666666666664, Float64(im * im), -0.5), Float64(im * im), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[(N[(-0.001388888888888889 * N[(im * im), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + -0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.1], N[Cos[im], $MachinePrecision], If[LessEqual[t$95$0, 1e-76], N[(1.0 / N[(N[(N[(-0.16666666666666666 * re + 0.5), $MachinePrecision] * re + -1.0), $MachinePrecision] * re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.99999], N[Cos[im], $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision] + -0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

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

   5. Applied rewrites 30.8%

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

   6. Taylor expanded in re around inf

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

   8. Applied rewrites 30.8%

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

   9. Taylor expanded in im around 0

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

   11. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.10000000000000001 or 9.99999999999999927e-77 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.999990000000000046

   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

   5. Applied rewrites 94.2%

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

   if -0.10000000000000001 < (*.f64 (exp.f64 re) (cos.f64 im)) < 9.99999999999999927e-77

   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. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

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

      5. sinh-cosh N/A

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

      6. sinh---cosh-rev N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. lift-cos.f64 N/A

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

      10. sin-PI/2 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. sin-PI/2 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-exp.f64 N/A

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

      17. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 63.9%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 63.9

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

   9. Applied rewrites 63.9%

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

   10. Taylor expanded in im around 0

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

   12. Applied rewrites 63.9%

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

   if 0.999990000000000046 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 77.0%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 84.7%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -0.1:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 10^{-76}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, 0.5\right), re, -1\right), re, 1\right)}\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0.99999:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right)\\ \mathbf{if}\;t\_0 \leq -0.1:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \mathbf{elif}\;t\_0 \leq 0.99999:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, 0.5\right), re, -1\right), re, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im)))
        (t_1 (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)))
   (if (<= t_0 -0.1)
     (*
      t_1
      (fma
       (fma
        (fma -0.001388888888888889 (* im im) 0.041666666666666664)
        (* im im)
        -0.5)
       (* im im)
       1.0))
     (if (<= t_0 0.99999)
       (/ 1.0 (fma (fma (fma -0.16666666666666666 re 0.5) re -1.0) re 1.0))
       (*
        t_1
        (fma (fma 0.041666666666666664 (* im im) -0.5) (* im im) 1.0))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double t_1 = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0);
	double tmp;
	if (t_0 <= -0.1) {
		tmp = t_1 * fma(fma(fma(-0.001388888888888889, (im * im), 0.041666666666666664), (im * im), -0.5), (im * im), 1.0);
	} else if (t_0 <= 0.99999) {
		tmp = 1.0 / fma(fma(fma(-0.16666666666666666, re, 0.5), re, -1.0), re, 1.0);
	} else {
		tmp = t_1 * fma(fma(0.041666666666666664, (im * im), -0.5), (im * im), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	t_1 = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0)
	tmp = 0.0
	if (t_0 <= -0.1)
		tmp = Float64(t_1 * fma(fma(fma(-0.001388888888888889, Float64(im * im), 0.041666666666666664), Float64(im * im), -0.5), Float64(im * im), 1.0));
	elseif (t_0 <= 0.99999)
		tmp = Float64(1.0 / fma(fma(fma(-0.16666666666666666, re, 0.5), re, -1.0), re, 1.0));
	else
		tmp = Float64(t_1 * fma(fma(0.041666666666666664, Float64(im * im), -0.5), Float64(im * im), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, -0.1], N[(t$95$1 * N[(N[(N[(-0.001388888888888889 * N[(im * im), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + -0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.99999], N[(1.0 / N[(N[(N[(-0.16666666666666666 * re + 0.5), $MachinePrecision] * re + -1.0), $MachinePrecision] * re + 1.0), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[(0.041666666666666664 * N[(im * im), $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)) < -0.10000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 87.2%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 36.7%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, -0.5\right), im \cdot im, 1\right)} \]

   if -0.10000000000000001 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.999990000000000046

   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. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

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

      5. sinh-cosh N/A

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

      6. sinh---cosh-rev N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. lift-cos.f64 N/A

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

      10. sin-PI/2 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. sin-PI/2 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-exp.f64 N/A

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

      17. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 76.8%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 76.8

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

   9. Applied rewrites 76.8%

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

   10. Taylor expanded in im around 0

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

   12. Applied rewrites 48.1%

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

   if 0.999990000000000046 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 77.0%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 84.7%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -0.1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0.99999:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, 0.5\right), re, -1\right), re, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 66.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -0.68:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \mathbf{elif}\;t\_0 \leq 0.99999:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, 0.5\right), re, -1\right), re, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 -0.68)
     (*
      (* (* re re) 0.5)
      (fma
       (fma
        (fma -0.001388888888888889 (* im im) 0.041666666666666664)
        (* im im)
        -0.5)
       (* im im)
       1.0))
     (if (<= t_0 0.99999)
       (/ 1.0 (fma (fma (fma -0.16666666666666666 re 0.5) re -1.0) re 1.0))
       (*
        (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)
        (fma (fma 0.041666666666666664 (* im im) -0.5) (* im im) 1.0))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -0.68) {
		tmp = ((re * re) * 0.5) * fma(fma(fma(-0.001388888888888889, (im * im), 0.041666666666666664), (im * im), -0.5), (im * im), 1.0);
	} else if (t_0 <= 0.99999) {
		tmp = 1.0 / fma(fma(fma(-0.16666666666666666, re, 0.5), re, -1.0), re, 1.0);
	} else {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * fma(fma(0.041666666666666664, (im * im), -0.5), (im * im), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= -0.68)
		tmp = Float64(Float64(Float64(re * re) * 0.5) * fma(fma(fma(-0.001388888888888889, Float64(im * im), 0.041666666666666664), Float64(im * im), -0.5), Float64(im * im), 1.0));
	elseif (t_0 <= 0.99999)
		tmp = Float64(1.0 / fma(fma(fma(-0.16666666666666666, re, 0.5), re, -1.0), re, 1.0));
	else
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * fma(fma(0.041666666666666664, Float64(im * im), -0.5), Float64(im * im), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.68], N[(N[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[(N[(-0.001388888888888889 * N[(im * im), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + -0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.99999], N[(1.0 / N[(N[(N[(-0.16666666666666666 * re + 0.5), $MachinePrecision] * re + -1.0), $MachinePrecision] * re + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision] + -0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.680000000000000049

   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

   5. Applied rewrites 67.6%

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

   6. Taylor expanded in re around inf

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

   8. Applied rewrites 17.3%

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

   9. Taylor expanded in im around 0

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

   11. Applied rewrites 48.4%

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

   if -0.680000000000000049 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.999990000000000046

   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. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

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

      5. sinh-cosh N/A

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

      6. sinh---cosh-rev N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. lift-cos.f64 N/A

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

      10. sin-PI/2 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. sin-PI/2 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-exp.f64 N/A

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

      17. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 79.3%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 79.3

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

   9. Applied rewrites 79.3%

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

   10. Taylor expanded in im around 0

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

   12. Applied rewrites 43.2%

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

   if 0.999990000000000046 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 77.0%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 84.7%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -0.68:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0.99999:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, 0.5\right), re, -1\right), re, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 66.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right)\\ \mathbf{if}\;t\_0 \leq -0.1:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq 0.99999:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, 0.5\right), re, -1\right), re, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im)))
        (t_1 (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)))
   (if (<= t_0 -0.1)
     (* t_1 (fma (* im im) -0.5 1.0))
     (if (<= t_0 0.99999)
       (/ 1.0 (fma (fma (fma -0.16666666666666666 re 0.5) re -1.0) re 1.0))
       (*
        t_1
        (fma (fma 0.041666666666666664 (* im im) -0.5) (* im im) 1.0))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double t_1 = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0);
	double tmp;
	if (t_0 <= -0.1) {
		tmp = t_1 * fma((im * im), -0.5, 1.0);
	} else if (t_0 <= 0.99999) {
		tmp = 1.0 / fma(fma(fma(-0.16666666666666666, re, 0.5), re, -1.0), re, 1.0);
	} else {
		tmp = t_1 * fma(fma(0.041666666666666664, (im * im), -0.5), (im * im), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	t_1 = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0)
	tmp = 0.0
	if (t_0 <= -0.1)
		tmp = Float64(t_1 * fma(Float64(im * im), -0.5, 1.0));
	elseif (t_0 <= 0.99999)
		tmp = Float64(1.0 / fma(fma(fma(-0.16666666666666666, re, 0.5), re, -1.0), re, 1.0));
	else
		tmp = Float64(t_1 * fma(fma(0.041666666666666664, Float64(im * im), -0.5), Float64(im * im), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, -0.1], N[(t$95$1 * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.99999], N[(1.0 / N[(N[(N[(-0.16666666666666666 * re + 0.5), $MachinePrecision] * re + -1.0), $MachinePrecision] * re + 1.0), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[(0.041666666666666664 * N[(im * im), $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)) < -0.10000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 87.2%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 34.7%

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

   if -0.10000000000000001 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.999990000000000046

   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. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

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

      5. sinh-cosh N/A

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

      6. sinh---cosh-rev N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. lift-cos.f64 N/A

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

      10. sin-PI/2 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. sin-PI/2 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-exp.f64 N/A

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

      17. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 76.8%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 76.8

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

   9. Applied rewrites 76.8%

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

   10. Taylor expanded in im around 0

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

   12. Applied rewrites 48.1%

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

   if 0.999990000000000046 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 77.0%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 84.7%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -0.1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0.99999:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, 0.5\right), re, -1\right), re, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 63.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -0.1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq 0.99999:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, 0.5\right), re, -1\right), re, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 -0.1)
     (*
      (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)
      (fma (* im im) -0.5 1.0))
     (if (<= t_0 0.99999)
       (/ 1.0 (fma (fma (fma -0.16666666666666666 re 0.5) re -1.0) re 1.0))
       (*
        (fma (fma 0.5 re 1.0) re 1.0)
        (fma (fma 0.041666666666666664 (* im im) -0.5) (* im im) 1.0))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -0.1) {
		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 <= 0.99999) {
		tmp = 1.0 / fma(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(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 <= -0.1)
		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 <= 0.99999)
		tmp = Float64(1.0 / fma(fma(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(fma(0.041666666666666664, Float64(im * im), -0.5), Float64(im * im), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.1], 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, 0.99999], N[(1.0 / N[(N[(N[(-0.16666666666666666 * re + 0.5), $MachinePrecision] * re + -1.0), $MachinePrecision] * re + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision] + -0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 87.2%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 34.7%

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

   if -0.10000000000000001 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.999990000000000046

   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. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

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

      5. sinh-cosh N/A

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

      6. sinh---cosh-rev N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. lift-cos.f64 N/A

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

      10. sin-PI/2 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. sin-PI/2 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-exp.f64 N/A

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

      17. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 76.8%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 76.8

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

   9. Applied rewrites 76.8%

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

   10. Taylor expanded in im around 0

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

   12. Applied rewrites 48.1%

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

   if 0.999990000000000046 < (*.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

   5. Applied rewrites 71.8%

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

   6. Taylor expanded in re around inf

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

   8. Applied rewrites 19.7%

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

   9. Taylor expanded in im around 0

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

   11. Applied rewrites 27.5%

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

   12. Taylor expanded in re around 0

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

   14. Applied rewrites 79.5%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -0.1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0.99999:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, 0.5\right), re, -1\right), re, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, -0.5\right), im \cdot im, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 63.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -0.1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, 0.5\right), re, -1\right), re, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot 0.5\right) \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right), im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 -0.1)
     (*
      (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)
      (fma (* im im) -0.5 1.0))
     (if (<= t_0 2.0)
       (/ 1.0 (fma (fma (fma -0.16666666666666666 re 0.5) re -1.0) re 1.0))
       (*
        (* (* re re) 0.5)
        (fma (* 0.041666666666666664 (* im im)) (* im im) 1.0))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -0.1) {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * fma((im * im), -0.5, 1.0);
	} else if (t_0 <= 2.0) {
		tmp = 1.0 / fma(fma(fma(-0.16666666666666666, re, 0.5), re, -1.0), re, 1.0);
	} else {
		tmp = ((re * re) * 0.5) * fma((0.041666666666666664 * (im * im)), (im * im), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= -0.1)
		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 <= 2.0)
		tmp = Float64(1.0 / fma(fma(fma(-0.16666666666666666, re, 0.5), re, -1.0), re, 1.0));
	else
		tmp = Float64(Float64(Float64(re * re) * 0.5) * fma(Float64(0.041666666666666664 * Float64(im * im)), 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, -0.1], 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, 2.0], N[(1.0 / N[(N[(N[(-0.16666666666666666 * re + 0.5), $MachinePrecision] * re + -1.0), $MachinePrecision] * re + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $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)) < -0.10000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 87.2%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 34.7%

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

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

   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. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

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

      5. sinh-cosh N/A

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

      6. sinh---cosh-rev N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. lift-cos.f64 N/A

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

      10. sin-PI/2 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. sin-PI/2 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-exp.f64 N/A

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

      17. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 85.4%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 85.4

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

   9. Applied rewrites 85.4%

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

   10. Taylor expanded in im around 0

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

   12. Applied rewrites 67.2%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 38.5%

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

   6. Taylor expanded in re around inf

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

   8. Applied rewrites 38.5%

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

   9. Taylor expanded in im around 0

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

   11. Applied rewrites 55.5%

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

   12. Taylor expanded in im around inf

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

   14. Applied rewrites 55.5%

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

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -0.1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 2:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, 0.5\right), re, -1\right), re, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot 0.5\right) \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right), im \cdot im, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 64.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -0.1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq 0.03:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, 0.5\right), re, -1\right), re, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot re, re, 1\right), re, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 -0.1)
     (*
      (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)
      (fma (* im im) -0.5 1.0))
     (if (<= t_0 0.03)
       (/ 1.0 (fma (fma (fma -0.16666666666666666 re 0.5) re -1.0) re 1.0))
       (fma (fma (* 0.16666666666666666 re) 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 <= -0.1) {
		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 <= 0.03) {
		tmp = 1.0 / fma(fma(fma(-0.16666666666666666, re, 0.5), re, -1.0), re, 1.0);
	} else {
		tmp = fma(fma((0.16666666666666666 * re), 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 <= -0.1)
		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 <= 0.03)
		tmp = Float64(1.0 / fma(fma(fma(-0.16666666666666666, re, 0.5), re, -1.0), re, 1.0));
	else
		tmp = fma(fma(Float64(0.16666666666666666 * re), 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, -0.1], 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, 0.03], N[(1.0 / N[(N[(N[(-0.16666666666666666 * re + 0.5), $MachinePrecision] * re + -1.0), $MachinePrecision] * re + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.16666666666666666 * re), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 87.2%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 34.7%

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

   if -0.10000000000000001 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.029999999999999999

   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. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

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

      5. sinh-cosh N/A

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

      6. sinh---cosh-rev N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. lift-cos.f64 N/A

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

      10. sin-PI/2 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. sin-PI/2 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-exp.f64 N/A

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

      17. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 64.4%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 64.4

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

   9. Applied rewrites 64.4%

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

   10. Taylor expanded in im around 0

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

   12. Applied rewrites 63.1%

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

   if 0.029999999999999999 < (*.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. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

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

      5. sinh-cosh N/A

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

      6. sinh---cosh-rev N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. lift-cos.f64 N/A

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

      10. sin-PI/2 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. sin-PI/2 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-exp.f64 N/A

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

      17. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 80.3%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 63.0%

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

   11. Taylor expanded in re around inf

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

   13. Applied rewrites 63.0%

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -0.1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0.03:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, 0.5\right), re, -1\right), re, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot re, re, 1\right), re, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 63.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -0.4:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot 0.5\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq 0.03:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, 0.5\right), re, -1\right), re, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot re, re, 1\right), re, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 -0.4)
     (* (* (* re re) 0.5) (fma (* im im) -0.5 1.0))
     (if (<= t_0 0.03)
       (/ 1.0 (fma (fma (fma -0.16666666666666666 re 0.5) re -1.0) re 1.0))
       (fma (fma (* 0.16666666666666666 re) 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 <= -0.4) {
		tmp = ((re * re) * 0.5) * fma((im * im), -0.5, 1.0);
	} else if (t_0 <= 0.03) {
		tmp = 1.0 / fma(fma(fma(-0.16666666666666666, re, 0.5), re, -1.0), re, 1.0);
	} else {
		tmp = fma(fma((0.16666666666666666 * re), 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 <= -0.4)
		tmp = Float64(Float64(Float64(re * re) * 0.5) * fma(Float64(im * im), -0.5, 1.0));
	elseif (t_0 <= 0.03)
		tmp = Float64(1.0 / fma(fma(fma(-0.16666666666666666, re, 0.5), re, -1.0), re, 1.0));
	else
		tmp = fma(fma(Float64(0.16666666666666666 * re), 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, -0.4], N[(N[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.03], N[(1.0 / N[(N[(N[(-0.16666666666666666 * re + 0.5), $MachinePrecision] * re + -1.0), $MachinePrecision] * re + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.16666666666666666 * re), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.40000000000000002

   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

   5. Applied rewrites 75.3%

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

   6. Taylor expanded in re around inf

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

   8. Applied rewrites 14.1%

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

   9. Taylor expanded in im around 0

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

   11. Applied rewrites 33.4%

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

   if -0.40000000000000002 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.029999999999999999

   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. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

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

      5. sinh-cosh N/A

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

      6. sinh---cosh-rev N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. lift-cos.f64 N/A

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

      10. sin-PI/2 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. sin-PI/2 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-exp.f64 N/A

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

      17. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 65.5%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 65.5

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

   9. Applied rewrites 65.5%

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

   10. Taylor expanded in im around 0

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

   12. Applied rewrites 61.2%

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

   if 0.029999999999999999 < (*.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. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

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

      5. sinh-cosh N/A

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

      6. sinh---cosh-rev N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. lift-cos.f64 N/A

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

      10. sin-PI/2 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. sin-PI/2 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-exp.f64 N/A

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

      17. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 80.3%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 63.0%

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

   11. Taylor expanded in re around inf

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

   13. Applied rewrites 63.0%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -0.4:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot 0.5\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0.03:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, 0.5\right), re, -1\right), re, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot re, re, 1\right), re, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 47.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * cos(im)) <= 0.0)
		tmp = Float64(Float64(Float64(re * re) * 0.5) * fma(Float64(im * im), -0.5, 1.0));
	else
		tmp = fma(fma(Float64(0.16666666666666666 * re), re, 1.0), re, 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[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.16666666666666666 * re), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 32.9%

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

   6. Taylor expanded in re around inf

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

   8. Applied rewrites 6.8%

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

   9. Taylor expanded in im around 0

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

   11. Applied rewrites 14.3%

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

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

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing
   3. 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. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

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

      5. sinh-cosh N/A

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

      6. sinh---cosh-rev N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. lift-cos.f64 N/A

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

      10. sin-PI/2 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. sin-PI/2 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-exp.f64 N/A

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

      17. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 79.4%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 62.3%

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

   11. Taylor expanded in re around inf

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

   13. Applied rewrites 62.3%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot re, re, 1\right), re, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 41.6% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot re, re, 1\right), re, 1\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (fma (fma (* 0.16666666666666666 re) re 1.0) re 1.0))

C

double code(double re, double im) {
	return fma(fma((0.16666666666666666 * re), re, 1.0), re, 1.0);
}

Julia

function code(re, im)
	return fma(fma(Float64(0.16666666666666666 * re), re, 1.0), re, 1.0)
end

Wolfram

code[re_, im_] := N[(N[(N[(0.16666666666666666 * re), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision]

Derivation

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. sinh-+-cosh-rev N/A

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

   4. flip-+ N/A

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

   5. sinh-cosh N/A

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

   6. sinh---cosh-rev N/A

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

   7. associate-*l/ N/A

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

   8. *-commutative N/A

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

   9. lift-cos.f64 N/A

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

   10. sin-PI/2 N/A

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

   11. lower-/.f64 N/A

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

   12. lift-cos.f64 N/A

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

   13. sin-PI/2 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lower-exp.f64 N/A

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

   17. lower-neg.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in im around 0

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

7. Applied rewrites 70.9%

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

8. Taylor expanded in re around 0

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

10. Applied rewrites 37.1%

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

11. Taylor expanded in re around inf

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

13. Applied rewrites 37.1%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot re, re, 1\right), re, 1\right) \]
14. Add Preprocessing

Alternative 17: 41.3% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.16666666666666666, re, 1\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (fma (* (* re re) 0.16666666666666666) re 1.0))

C

double code(double re, double im) {
	return fma(((re * re) * 0.16666666666666666), re, 1.0);
}

Julia

function code(re, im)
	return fma(Float64(Float64(re * re) * 0.16666666666666666), re, 1.0)
end

Wolfram

code[re_, im_] := N[(N[(N[(re * re), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * re + 1.0), $MachinePrecision]

Derivation

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. sinh-+-cosh-rev N/A

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

   4. flip-+ N/A

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

   5. sinh-cosh N/A

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

   6. sinh---cosh-rev N/A

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

   7. associate-*l/ N/A

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

   8. *-commutative N/A

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

   9. lift-cos.f64 N/A

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

   10. sin-PI/2 N/A

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

   11. lower-/.f64 N/A

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

   12. lift-cos.f64 N/A

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

   13. sin-PI/2 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lower-exp.f64 N/A

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

   17. lower-neg.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in im around 0

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

7. Applied rewrites 70.9%

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

8. Taylor expanded in re around 0

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

10. Applied rewrites 37.1%

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

11. Taylor expanded in re around inf

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

13. Applied rewrites 37.1%

    \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.16666666666666666, re, 1\right) \]
14. Add Preprocessing

Alternative 18: 38.4% accurate, 15.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (fma (fma 0.5 re 1.0) re 1.0))

C

double code(double re, double im) {
	return fma(fma(0.5, re, 1.0), re, 1.0);
}

Julia

function code(re, im)
	return fma(fma(0.5, re, 1.0), re, 1.0)
end

Wolfram

code[re_, im_] := N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision]

Derivation

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. sinh-+-cosh-rev N/A

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

   4. flip-+ N/A

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

   5. sinh-cosh N/A

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

   6. sinh---cosh-rev N/A

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

   7. associate-*l/ N/A

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

   8. *-commutative N/A

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

   9. lift-cos.f64 N/A

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

   10. sin-PI/2 N/A

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

   11. lower-/.f64 N/A

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

   12. lift-cos.f64 N/A

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

   13. sin-PI/2 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lower-exp.f64 N/A

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

   17. lower-neg.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in im around 0

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

7. Applied rewrites 70.9%

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

8. Taylor expanded in re around 0

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

10. Applied rewrites 34.9%

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

Alternative 19: 29.5% accurate, 51.5× speedup

Math

\[\begin{array}{l} \\ 1 + re \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (+ 1.0 re))

C

double code(double re, double im) {
	return 1.0 + re;
}

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 = 1.0d0 + re
end function

Java

public static double code(double re, double im) {
	return 1.0 + re;
}

Python

def code(re, im):
	return 1.0 + re

Julia

function code(re, im)
	return Float64(1.0 + re)
end

MATLAB

function tmp = code(re, im)
	tmp = 1.0 + re;
end

Wolfram

code[re_, im_] := N[(1.0 + re), $MachinePrecision]

Derivation

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. sinh-+-cosh-rev N/A

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

   4. flip-+ N/A

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

   5. sinh-cosh N/A

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

   6. sinh---cosh-rev N/A

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

   7. associate-*l/ N/A

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

   8. *-commutative N/A

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

   9. lift-cos.f64 N/A

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

   10. sin-PI/2 N/A

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

   11. lower-/.f64 N/A

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

   12. lift-cos.f64 N/A

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

   13. sin-PI/2 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lower-exp.f64 N/A

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

   17. lower-neg.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in im around 0

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

7. Applied rewrites 70.9%

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

8. Taylor expanded in re around 0

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

10. Applied rewrites 28.2%

    \[\leadsto 1 + \color{blue}{re} \]
11. Add Preprocessing

Alternative 20: 29.1% accurate, 206.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 1.0)

C

double code(double re, double im) {
	return 1.0;
}

Fortran

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 = 1.0d0
end function

Java

public static double code(double re, double im) {
	return 1.0;
}

Python

def code(re, im):
	return 1.0

Julia

function code(re, im)
	return 1.0
end

MATLAB

function tmp = code(re, im)
	tmp = 1.0;
end

Wolfram

code[re_, im_] := 1.0

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \cos im \]
2. Add Preprocessing
3. 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. sinh-+-cosh-rev N/A

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

   4. flip-+ N/A

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

   5. sinh-cosh N/A

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

   6. sinh---cosh-rev N/A

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

   7. associate-*l/ N/A

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

   8. *-commutative N/A

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

   9. lift-cos.f64 N/A

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

   10. sin-PI/2 N/A

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

   11. lower-/.f64 N/A

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

   12. lift-cos.f64 N/A

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

   13. sin-PI/2 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lower-exp.f64 N/A

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

   17. lower-neg.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in im around 0

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

7. Applied rewrites 70.9%

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

8. Taylor expanded in re around 0

   \[\leadsto 1 \]
9. Step-by-step derivation

10. Applied rewrites 28.2%

    \[\leadsto 1 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025026 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))