math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 6.0s

Alternatives: 19

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: 98.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(\left(im \cdot im\right) \cdot -0.5\right)\\ \mathbf{elif}\;t\_0 \leq -0.0005:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \cos im\\ \mathbf{elif}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 0.995\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\left(re + \mathsf{fma}\left(re \cdot re, 0.5, 1\right)\right) \cdot \cos im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (* (exp re) (* (* im im) -0.5))
     (if (<= t_0 -0.0005)
       (* (fma (fma 0.5 re 1.0) re 1.0) (cos im))
       (if (or (<= t_0 0.0) (not (<= t_0 0.995)))
         (exp re)
         (* (+ re (fma (* re re) 0.5 1.0)) (cos im)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = exp(re) * ((im * im) * -0.5);
	} else if (t_0 <= -0.0005) {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * cos(im);
	} else if ((t_0 <= 0.0) || !(t_0 <= 0.995)) {
		tmp = exp(re);
	} else {
		tmp = (re + fma((re * re), 0.5, 1.0)) * cos(im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(exp(re) * Float64(Float64(im * im) * -0.5));
	elseif (t_0 <= -0.0005)
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * cos(im));
	elseif ((t_0 <= 0.0) || !(t_0 <= 0.995))
		tmp = exp(re);
	else
		tmp = Float64(Float64(re + fma(Float64(re * re), 0.5, 1.0)) * cos(im));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[Exp[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.0005], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 0.995]], $MachinePrecision]], N[Exp[re], $MachinePrecision], N[(N[(re + N[(N[(re * re), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -5.0000000000000001e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -5.0000000000000001e-4 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0 or 0.994999999999999996 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lift-exp.f64 100.0

         \[\leadsto e^{re} \]

   5. Applied rewrites 100.0%

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

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

   1. Initial program 100.0%

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

      1. lift-exp.f64 N/A

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

      2. sinh-+-cosh-rev N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sinh.f64 N/A

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

      6. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 98.8

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

   7. Applied rewrites 98.8%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 98.6%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(\left(im \cdot im\right) \cdot -0.5\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -0.0005:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \cos im\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0 \lor \neg \left(e^{re} \cdot \cos im \leq 0.995\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\left(re + \mathsf{fma}\left(re \cdot re, 0.5, 1\right)\right) \cdot \cos im\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(\left(im \cdot im\right) \cdot -0.5\right)\\ \mathbf{elif}\;t\_0 \leq -0.0005 \lor \neg \left(t\_0 \leq 0 \lor \neg \left(t\_0 \leq 0.995\right)\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (* (exp re) (* (* im im) -0.5))
     (if (or (<= t_0 -0.0005) (not (or (<= t_0 0.0) (not (<= t_0 0.995)))))
       (* (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 = exp(re) * ((im * im) * -0.5);
	} else if ((t_0 <= -0.0005) || !((t_0 <= 0.0) || !(t_0 <= 0.995))) {
		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(exp(re) * Float64(Float64(im * im) * -0.5));
	elseif ((t_0 <= -0.0005) || !((t_0 <= 0.0) || !(t_0 <= 0.995)))
		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[Exp[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, -0.0005], N[Not[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 0.995]], $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 im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -5.0000000000000001e-4 or -0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.994999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 99.2

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

   5. Applied rewrites 99.2%

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

   if -5.0000000000000001e-4 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0 or 0.994999999999999996 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lift-exp.f64 100.0

         \[\leadsto e^{re} \]

   5. Applied rewrites 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(\left(im \cdot im\right) \cdot -0.5\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -0.0005 \lor \neg \left(e^{re} \cdot \cos im \leq 0 \lor \neg \left(e^{re} \cdot \cos im \leq 0.995\right)\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(\left(im \cdot im\right) \cdot -0.5\right)\\ \mathbf{elif}\;t\_0 \leq -0.0005 \lor \neg \left(t\_0 \leq 0 \lor \neg \left(t\_0 \leq 0.995\right)\right):\\ \;\;\;\;\left(re - -1\right) \cdot \cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (* (exp re) (* (* im im) -0.5))
     (if (or (<= t_0 -0.0005) (not (or (<= t_0 0.0) (not (<= t_0 0.995)))))
       (* (- 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 = exp(re) * ((im * im) * -0.5);
	} else if ((t_0 <= -0.0005) || !((t_0 <= 0.0) || !(t_0 <= 0.995))) {
		tmp = (re - -1.0) * cos(im);
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(re) * Math.cos(im);
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = Math.exp(re) * ((im * im) * -0.5);
	} else if ((t_0 <= -0.0005) || !((t_0 <= 0.0) || !(t_0 <= 0.995))) {
		tmp = (re - -1.0) * Math.cos(im);
	} else {
		tmp = Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) * math.cos(im)
	tmp = 0
	if t_0 <= -math.inf:
		tmp = math.exp(re) * ((im * im) * -0.5)
	elif (t_0 <= -0.0005) or not ((t_0 <= 0.0) or not (t_0 <= 0.995)):
		tmp = (re - -1.0) * math.cos(im)
	else:
		tmp = math.exp(re)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(exp(re) * Float64(Float64(im * im) * -0.5));
	elseif ((t_0 <= -0.0005) || !((t_0 <= 0.0) || !(t_0 <= 0.995)))
		tmp = Float64(Float64(re - -1.0) * cos(im));
	else
		tmp = exp(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(re) * cos(im);
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = exp(re) * ((im * im) * -0.5);
	elseif ((t_0 <= -0.0005) || ~(((t_0 <= 0.0) || ~((t_0 <= 0.995)))))
		tmp = (re - -1.0) * cos(im);
	else
		tmp = exp(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[Exp[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, -0.0005], N[Not[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 0.995]], $MachinePrecision]]], $MachinePrecision]], N[(N[(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 im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -5.0000000000000001e-4 or -0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.994999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

         \[\leadsto \left(re - -1 \cdot 1\right) \cdot \cos im \]

      5. metadata-eval N/A

         \[\leadsto \left(re - -1\right) \cdot \cos im \]

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. metadata-eval 98.6

         \[\leadsto \left(re - -1\right) \cdot \cos im \]

   5. Applied rewrites 98.6%

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

   if -5.0000000000000001e-4 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0 or 0.994999999999999996 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lift-exp.f64 100.0

         \[\leadsto e^{re} \]

   5. Applied rewrites 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(\left(im \cdot im\right) \cdot -0.5\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -0.0005 \lor \neg \left(e^{re} \cdot \cos im \leq 0 \lor \neg \left(e^{re} \cdot \cos im \leq 0.995\right)\right):\\ \;\;\;\;\left(re - -1\right) \cdot \cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, 0.16666666666666666, 1\right) \cdot re + \mathsf{fma}\left(re \cdot re, 0.5, 1\right)\right) \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot \mathsf{fma}\left(-0.001388888888888889 \cdot im, im, 0.041666666666666664\right) - 0.5, im \cdot im, 1\right)\\ \mathbf{elif}\;t\_0 \leq -0.0005 \lor \neg \left(t\_0 \leq 0 \lor \neg \left(t\_0 \leq 0.995\right)\right):\\ \;\;\;\;\left(re - -1\right) \cdot \cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (*
      (+
       (* (fma (* re re) 0.16666666666666666 1.0) re)
       (fma (* re re) 0.5 1.0))
      (fma
       (-
        (*
         (* im im)
         (fma (* -0.001388888888888889 im) im 0.041666666666666664))
        0.5)
       (* im im)
       1.0))
     (if (or (<= t_0 -0.0005) (not (or (<= t_0 0.0) (not (<= t_0 0.995)))))
       (* (- 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 = ((fma((re * re), 0.16666666666666666, 1.0) * re) + fma((re * re), 0.5, 1.0)) * fma((((im * im) * fma((-0.001388888888888889 * im), im, 0.041666666666666664)) - 0.5), (im * im), 1.0);
	} else if ((t_0 <= -0.0005) || !((t_0 <= 0.0) || !(t_0 <= 0.995))) {
		tmp = (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(fma(Float64(re * re), 0.16666666666666666, 1.0) * re) + fma(Float64(re * re), 0.5, 1.0)) * fma(Float64(Float64(Float64(im * im) * fma(Float64(-0.001388888888888889 * im), im, 0.041666666666666664)) - 0.5), Float64(im * im), 1.0));
	elseif ((t_0 <= -0.0005) || !((t_0 <= 0.0) || !(t_0 <= 0.995)))
		tmp = Float64(Float64(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[(N[(N[(re * re), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * re), $MachinePrecision] + N[(N[(re * re), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(im * im), $MachinePrecision] * N[(N[(-0.001388888888888889 * im), $MachinePrecision] * im + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, -0.0005], N[Not[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 0.995]], $MachinePrecision]]], $MachinePrecision]], N[(N[(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. Step-by-step derivation

      1. lift-exp.f64 N/A

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

      2. sinh-+-cosh-rev N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sinh.f64 N/A

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

      6. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 52.4

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

   10. Applied rewrites 52.4%

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

   11. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(re \cdot re, \frac{1}{2}, 1\right)\right) \cdot \left(\left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}\right) \cdot {im}^{2} + 1\right) \]

       3. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(re \cdot re, \frac{1}{2}, 1\right)\right) \cdot \left(\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) \cdot {im}^{2} - \frac{1}{2}\right) \cdot {im}^{2} + 1\right) \]

       4. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(re \cdot re, \frac{1}{2}, 1\right)\right) \cdot \left(\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot \left(im \cdot im\right)\right) \cdot {im}^{2} - \frac{1}{2}\right) \cdot {im}^{2} + 1\right) \]

       5. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(re \cdot re, \frac{1}{2}, 1\right)\right) \cdot \left(\left(\left(\frac{-1}{720} \cdot \left(im \cdot im\right) + \frac{1}{24}\right) \cdot {im}^{2} - \frac{1}{2}\right) \cdot {im}^{2} + 1\right) \]

       6. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(re \cdot re, \frac{1}{2}, 1\right)\right) \cdot \left(\left(\left(\frac{-1}{720} \cdot \left(im \cdot im\right) + \frac{1}{24}\right) \cdot \left(im \cdot im\right) - \frac{1}{2}\right) \cdot {im}^{2} + 1\right) \]

       7. lower-fma.f64 N/A

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

   13. Applied rewrites 90.4%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -5.0000000000000001e-4 or -0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.994999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

         \[\leadsto \left(re - -1 \cdot 1\right) \cdot \cos im \]

      5. metadata-eval N/A

         \[\leadsto \left(re - -1\right) \cdot \cos im \]

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. metadata-eval 98.6

         \[\leadsto \left(re - -1\right) \cdot \cos im \]

   5. Applied rewrites 98.6%

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

   if -5.0000000000000001e-4 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0 or 0.994999999999999996 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lift-exp.f64 100.0

         \[\leadsto e^{re} \]

   5. Applied rewrites 100.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, 0.16666666666666666, 1\right) \cdot re + \mathsf{fma}\left(re \cdot re, 0.5, 1\right)\right) \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot \mathsf{fma}\left(-0.001388888888888889 \cdot im, im, 0.041666666666666664\right) - 0.5, im \cdot im, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -0.0005 \lor \neg \left(e^{re} \cdot \cos im \leq 0 \lor \neg \left(e^{re} \cdot \cos im \leq 0.995\right)\right):\\ \;\;\;\;\left(re - -1\right) \cdot \cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, 0.16666666666666666, 1\right) \cdot re + \mathsf{fma}\left(re \cdot re, 0.5, 1\right)\right) \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot \mathsf{fma}\left(-0.001388888888888889 \cdot im, im, 0.041666666666666664\right) - 0.5, im \cdot im, 1\right)\\ \mathbf{elif}\;t\_0 \leq -0.0005 \lor \neg \left(t\_0 \leq 0 \lor \neg \left(t\_0 \leq 0.995\right)\right):\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (*
      (+
       (* (fma (* re re) 0.16666666666666666 1.0) re)
       (fma (* re re) 0.5 1.0))
      (fma
       (-
        (*
         (* im im)
         (fma (* -0.001388888888888889 im) im 0.041666666666666664))
        0.5)
       (* im im)
       1.0))
     (if (or (<= t_0 -0.0005) (not (or (<= t_0 0.0) (not (<= t_0 0.995)))))
       (cos im)
       (exp re)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = ((fma((re * re), 0.16666666666666666, 1.0) * re) + fma((re * re), 0.5, 1.0)) * fma((((im * im) * fma((-0.001388888888888889 * im), im, 0.041666666666666664)) - 0.5), (im * im), 1.0);
	} else if ((t_0 <= -0.0005) || !((t_0 <= 0.0) || !(t_0 <= 0.995))) {
		tmp = cos(im);
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(fma(Float64(re * re), 0.16666666666666666, 1.0) * re) + fma(Float64(re * re), 0.5, 1.0)) * fma(Float64(Float64(Float64(im * im) * fma(Float64(-0.001388888888888889 * im), im, 0.041666666666666664)) - 0.5), Float64(im * im), 1.0));
	elseif ((t_0 <= -0.0005) || !((t_0 <= 0.0) || !(t_0 <= 0.995)))
		tmp = cos(im);
	else
		tmp = exp(re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(re * re), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * re), $MachinePrecision] + N[(N[(re * re), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(im * im), $MachinePrecision] * N[(N[(-0.001388888888888889 * im), $MachinePrecision] * im + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, -0.0005], N[Not[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 0.995]], $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. Step-by-step derivation

      1. lift-exp.f64 N/A

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

      2. sinh-+-cosh-rev N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sinh.f64 N/A

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

      6. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 52.4

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

   10. Applied rewrites 52.4%

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

   11. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(re \cdot re, \frac{1}{2}, 1\right)\right) \cdot \left(\left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}\right) \cdot {im}^{2} + 1\right) \]

       3. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(re \cdot re, \frac{1}{2}, 1\right)\right) \cdot \left(\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) \cdot {im}^{2} - \frac{1}{2}\right) \cdot {im}^{2} + 1\right) \]

       4. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(re \cdot re, \frac{1}{2}, 1\right)\right) \cdot \left(\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot \left(im \cdot im\right)\right) \cdot {im}^{2} - \frac{1}{2}\right) \cdot {im}^{2} + 1\right) \]

       5. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(re \cdot re, \frac{1}{2}, 1\right)\right) \cdot \left(\left(\left(\frac{-1}{720} \cdot \left(im \cdot im\right) + \frac{1}{24}\right) \cdot {im}^{2} - \frac{1}{2}\right) \cdot {im}^{2} + 1\right) \]

       6. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(re \cdot re, \frac{1}{2}, 1\right)\right) \cdot \left(\left(\left(\frac{-1}{720} \cdot \left(im \cdot im\right) + \frac{1}{24}\right) \cdot \left(im \cdot im\right) - \frac{1}{2}\right) \cdot {im}^{2} + 1\right) \]

       7. lower-fma.f64 N/A

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

   13. Applied rewrites 90.4%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -5.0000000000000001e-4 or -0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.994999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lift-cos.f64 97.7

         \[\leadsto \cos im \]

   5. Applied rewrites 97.7%

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

   if -5.0000000000000001e-4 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0 or 0.994999999999999996 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lift-exp.f64 100.0

         \[\leadsto e^{re} \]

   5. Applied rewrites 100.0%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, 0.16666666666666666, 1\right) \cdot re + \mathsf{fma}\left(re \cdot re, 0.5, 1\right)\right) \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot \mathsf{fma}\left(-0.001388888888888889 \cdot im, im, 0.041666666666666664\right) - 0.5, im \cdot im, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -0.0005 \lor \neg \left(e^{re} \cdot \cos im \leq 0 \lor \neg \left(e^{re} \cdot \cos im \leq 0.995\right)\right):\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 53.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, 0.16666666666666666, 1\right) \cdot re + \mathsf{fma}\left(re \cdot re, 0.5, 1\right)\right) \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot \mathsf{fma}\left(-0.001388888888888889 \cdot im, im, 0.041666666666666664\right) - 0.5, im \cdot im, 1\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{1}{1 - \left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664 - 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{re \cdot re - 1}{re - 1} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (*
      (+
       (* (fma (* re re) 0.16666666666666666 1.0) re)
       (fma (* re re) 0.5 1.0))
      (fma
       (-
        (*
         (* im im)
         (fma (* -0.001388888888888889 im) im 0.041666666666666664))
        0.5)
       (* im im)
       1.0))
     (if (<= t_0 0.0)
       (/ 1.0 (- 1.0 (* (* im im) (- (* (* im im) 0.041666666666666664) 0.5))))
       (* (/ (- (* re re) 1.0) (- re 1.0)) 1.0)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = ((fma((re * re), 0.16666666666666666, 1.0) * re) + fma((re * re), 0.5, 1.0)) * fma((((im * im) * fma((-0.001388888888888889 * im), im, 0.041666666666666664)) - 0.5), (im * im), 1.0);
	} else if (t_0 <= 0.0) {
		tmp = 1.0 / (1.0 - ((im * im) * (((im * im) * 0.041666666666666664) - 0.5)));
	} else {
		tmp = (((re * re) - 1.0) / (re - 1.0)) * 1.0;
	}
	return tmp;
}

Julia

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

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(re * re), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * re), $MachinePrecision] + N[(N[(re * re), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(im * im), $MachinePrecision] * N[(N[(-0.001388888888888889 * im), $MachinePrecision] * im + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(1.0 / N[(1.0 - N[(N[(im * im), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(re * re), $MachinePrecision] - 1.0), $MachinePrecision] / N[(re - 1.0), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

      1. lift-exp.f64 N/A

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

      2. sinh-+-cosh-rev N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sinh.f64 N/A

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

      6. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 52.4

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

   10. Applied rewrites 52.4%

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

   11. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(re \cdot re, \frac{1}{2}, 1\right)\right) \cdot \left(\left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}\right) \cdot {im}^{2} + 1\right) \]

       3. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(re \cdot re, \frac{1}{2}, 1\right)\right) \cdot \left(\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) \cdot {im}^{2} - \frac{1}{2}\right) \cdot {im}^{2} + 1\right) \]

       4. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(re \cdot re, \frac{1}{2}, 1\right)\right) \cdot \left(\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot \left(im \cdot im\right)\right) \cdot {im}^{2} - \frac{1}{2}\right) \cdot {im}^{2} + 1\right) \]

       5. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(re \cdot re, \frac{1}{2}, 1\right)\right) \cdot \left(\left(\left(\frac{-1}{720} \cdot \left(im \cdot im\right) + \frac{1}{24}\right) \cdot {im}^{2} - \frac{1}{2}\right) \cdot {im}^{2} + 1\right) \]

       6. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(re \cdot re, \frac{1}{2}, 1\right)\right) \cdot \left(\left(\left(\frac{-1}{720} \cdot \left(im \cdot im\right) + \frac{1}{24}\right) \cdot \left(im \cdot im\right) - \frac{1}{2}\right) \cdot {im}^{2} + 1\right) \]

       7. lower-fma.f64 N/A

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

   13. Applied rewrites 90.4%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lift-cos.f64 35.9

         \[\leadsto \cos im \]

   5. Applied rewrites 35.9%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 1.9

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

   8. Applied rewrites 1.9%

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

      1. lift-fma.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. *-commutative N/A

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

      10. +-commutative N/A

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

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

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

      12. flip-- N/A

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

      13. lower-/.f64 N/A

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

   10. Applied rewrites 1.4%

       \[\leadsto \frac{1 - \left(\left(-im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664 - 0.5\right)\right) \cdot \left(\left(-im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664 - 0.5\right)\right)}{1 + \color{blue}{\left(-im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664 - 0.5\right)}} \]

   11. Taylor expanded in im around 0

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

   13. Applied rewrites 25.2%

       \[\leadsto \frac{1}{1 + \color{blue}{\left(-im \cdot im\right)} \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664 - 0.5\right)} \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

         \[\leadsto \left(re - -1 \cdot 1\right) \cdot \cos im \]

      5. metadata-eval N/A

         \[\leadsto \left(re - -1\right) \cdot \cos im \]

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. metadata-eval 67.7

         \[\leadsto \left(re - -1\right) \cdot \cos im \]

   5. Applied rewrites 67.7%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 43.6%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. pow2 N/A

         \[\leadsto \frac{re \cdot re - 1}{re + -1} \cdot 1 \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{re \cdot re - 1}{re + -1} \cdot 1 \]

      9. lower-+.f64 57.9

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

   10. Applied rewrites 57.9%

       \[\leadsto \frac{re \cdot re - 1}{\color{blue}{re + -1}} \cdot 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 46.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, 0.16666666666666666, 1\right) \cdot re + \mathsf{fma}\left(re \cdot re, 0.5, 1\right)\right) \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot \mathsf{fma}\left(-0.001388888888888889 \cdot im, im, 0.041666666666666664\right) - 0.5, im \cdot im, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\frac{1}{1 - \left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664 - 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{re \cdot re - 1}{re - 1} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(re) * Math.cos(im);
	double t_1 = (im * im) * (((im * im) * 0.041666666666666664) - 0.5);
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = (1.0 - (t_1 * t_1)) / 1.0;
	} else if (t_0 <= 0.0) {
		tmp = 1.0 / (1.0 - t_1);
	} else {
		tmp = (((re * re) - 1.0) / (re - 1.0)) * 1.0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) * math.cos(im)
	t_1 = (im * im) * (((im * im) * 0.041666666666666664) - 0.5)
	tmp = 0
	if t_0 <= -math.inf:
		tmp = (1.0 - (t_1 * t_1)) / 1.0
	elif t_0 <= 0.0:
		tmp = 1.0 / (1.0 - t_1)
	else:
		tmp = (((re * re) - 1.0) / (re - 1.0)) * 1.0
	return tmp

Julia

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

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(re) * cos(im);
	t_1 = (im * im) * (((im * im) * 0.041666666666666664) - 0.5);
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = (1.0 - (t_1 * t_1)) / 1.0;
	elseif (t_0 <= 0.0)
		tmp = 1.0 / (1.0 - t_1);
	else
		tmp = (((re * re) - 1.0) / (re - 1.0)) * 1.0;
	end
	tmp_2 = 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[(im * im), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(1.0 - N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(1.0 / N[(1.0 - t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(re * re), $MachinePrecision] - 1.0), $MachinePrecision] / N[(re - 1.0), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lift-cos.f64 3.1

         \[\leadsto \cos im \]

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 0.1

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

   8. Applied rewrites 0.1%

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

      1. lift-fma.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. *-commutative N/A

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

      10. +-commutative N/A

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

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

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

      12. flip-- N/A

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

      13. lower-/.f64 N/A

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

   10. Applied rewrites 0.1%

       \[\leadsto \frac{1 - \left(\left(-im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664 - 0.5\right)\right) \cdot \left(\left(-im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664 - 0.5\right)\right)}{1 + \color{blue}{\left(-im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664 - 0.5\right)}} \]

   11. Taylor expanded in im around 0

       \[\leadsto \frac{1 - \left(\left(-im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{24} - \frac{1}{2}\right)\right) \cdot \left(\left(-im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{24} - \frac{1}{2}\right)\right)}{1} \]
   12. Step-by-step derivation

   13. Applied rewrites 90.4%

       \[\leadsto \frac{1 - \left(\left(-im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664 - 0.5\right)\right) \cdot \left(\left(-im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664 - 0.5\right)\right)}{1} \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lift-cos.f64 35.9

         \[\leadsto \cos im \]

   5. Applied rewrites 35.9%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 1.9

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

   8. Applied rewrites 1.9%

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

      1. lift-fma.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. *-commutative N/A

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

      10. +-commutative N/A

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

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

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

      12. flip-- N/A

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

      13. lower-/.f64 N/A

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

   10. Applied rewrites 1.4%

       \[\leadsto \frac{1 - \left(\left(-im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664 - 0.5\right)\right) \cdot \left(\left(-im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664 - 0.5\right)\right)}{1 + \color{blue}{\left(-im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664 - 0.5\right)}} \]

   11. Taylor expanded in im around 0

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

   13. Applied rewrites 25.2%

       \[\leadsto \frac{1}{1 + \color{blue}{\left(-im \cdot im\right)} \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664 - 0.5\right)} \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

         \[\leadsto \left(re - -1 \cdot 1\right) \cdot \cos im \]

      5. metadata-eval N/A

         \[\leadsto \left(re - -1\right) \cdot \cos im \]

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. metadata-eval 67.7

         \[\leadsto \left(re - -1\right) \cdot \cos im \]

   5. Applied rewrites 67.7%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 43.6%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. pow2 N/A

         \[\leadsto \frac{re \cdot re - 1}{re + -1} \cdot 1 \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{re \cdot re - 1}{re + -1} \cdot 1 \]

      9. lower-+.f64 57.9

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

   10. Applied rewrites 57.9%

       \[\leadsto \frac{re \cdot re - 1}{\color{blue}{re + -1}} \cdot 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 46.1%

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

Alternative 9: 52.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(re - -1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, im \cdot im, 0.041666666666666664\right) \cdot \left(im \cdot im\right) - 0.5, im \cdot im, 1\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{1}{1 - \left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664 - 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{re \cdot re - 1}{re - 1} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (*
      (- re -1.0)
      (fma
       (-
        (*
         (fma -0.001388888888888889 (* im im) 0.041666666666666664)
         (* im im))
        0.5)
       (* im im)
       1.0))
     (if (<= t_0 0.0)
       (/ 1.0 (- 1.0 (* (* im im) (- (* (* im im) 0.041666666666666664) 0.5))))
       (* (/ (- (* re re) 1.0) (- re 1.0)) 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 - -1.0) * fma(((fma(-0.001388888888888889, (im * im), 0.041666666666666664) * (im * im)) - 0.5), (im * im), 1.0);
	} else if (t_0 <= 0.0) {
		tmp = 1.0 / (1.0 - ((im * im) * (((im * im) * 0.041666666666666664) - 0.5)));
	} else {
		tmp = (((re * re) - 1.0) / (re - 1.0)) * 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(re - -1.0) * fma(Float64(Float64(fma(-0.001388888888888889, Float64(im * im), 0.041666666666666664) * Float64(im * im)) - 0.5), Float64(im * im), 1.0));
	elseif (t_0 <= 0.0)
		tmp = Float64(1.0 / Float64(1.0 - Float64(Float64(im * im) * Float64(Float64(Float64(im * im) * 0.041666666666666664) - 0.5))));
	else
		tmp = Float64(Float64(Float64(Float64(re * re) - 1.0) / Float64(re - 1.0)) * 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(re - -1.0), $MachinePrecision] * N[(N[(N[(N[(-0.001388888888888889 * N[(im * im), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(1.0 / N[(1.0 - N[(N[(im * im), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(re * re), $MachinePrecision] - 1.0), $MachinePrecision] / N[(re - 1.0), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

         \[\leadsto \left(re - -1 \cdot 1\right) \cdot \cos im \]

      5. metadata-eval N/A

         \[\leadsto \left(re - -1\right) \cdot \cos im \]

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. metadata-eval 6.1

         \[\leadsto \left(re - -1\right) \cdot \cos im \]

   5. Applied rewrites 6.1%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

         \[\leadsto \left(re - -1\right) \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}\right) + \color{blue}{1}\right) \]

      2. *-commutative N/A

         \[\leadsto \left(re - -1\right) \cdot \left(\left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}\right) \cdot {im}^{2} + 1\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(re - -1\right) \cdot \mathsf{fma}\left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}, \color{blue}{{im}^{2}}, 1\right) \]

      4. lower--.f64 N/A

         \[\leadsto \left(re - -1\right) \cdot \mathsf{fma}\left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}, {\color{blue}{im}}^{2}, 1\right) \]

      5. *-commutative N/A

         \[\leadsto \left(re - -1\right) \cdot \mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) \cdot {im}^{2} - \frac{1}{2}, {im}^{2}, 1\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(re - -1\right) \cdot \mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) \cdot {im}^{2} - \frac{1}{2}, {im}^{2}, 1\right) \]

      7. +-commutative N/A

         \[\leadsto \left(re - -1\right) \cdot \mathsf{fma}\left(\left(\frac{-1}{720} \cdot {im}^{2} + \frac{1}{24}\right) \cdot {im}^{2} - \frac{1}{2}, {im}^{2}, 1\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(re - -1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, {im}^{2}, \frac{1}{24}\right) \cdot {im}^{2} - \frac{1}{2}, {im}^{2}, 1\right) \]

      9. pow2 N/A

         \[\leadsto \left(re - -1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, im \cdot im, \frac{1}{24}\right) \cdot {im}^{2} - \frac{1}{2}, {im}^{2}, 1\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left(re - -1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, im \cdot im, \frac{1}{24}\right) \cdot {im}^{2} - \frac{1}{2}, {im}^{2}, 1\right) \]

      11. pow2 N/A

          \[\leadsto \left(re - -1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, im \cdot im, \frac{1}{24}\right) \cdot \left(im \cdot im\right) - \frac{1}{2}, {im}^{2}, 1\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left(re - -1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, im \cdot im, \frac{1}{24}\right) \cdot \left(im \cdot im\right) - \frac{1}{2}, {im}^{2}, 1\right) \]

      13. pow2 N/A

          \[\leadsto \left(re - -1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, im \cdot im, \frac{1}{24}\right) \cdot \left(im \cdot im\right) - \frac{1}{2}, im \cdot \color{blue}{im}, 1\right) \]

      14. lift-*.f64 90.4

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

   8. Applied rewrites 90.4%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lift-cos.f64 35.9

         \[\leadsto \cos im \]

   5. Applied rewrites 35.9%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 1.9

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

   8. Applied rewrites 1.9%

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

      1. lift-fma.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. *-commutative N/A

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

      10. +-commutative N/A

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

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

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

      12. flip-- N/A

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

      13. lower-/.f64 N/A

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

   10. Applied rewrites 1.4%

       \[\leadsto \frac{1 - \left(\left(-im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664 - 0.5\right)\right) \cdot \left(\left(-im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664 - 0.5\right)\right)}{1 + \color{blue}{\left(-im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664 - 0.5\right)}} \]

   11. Taylor expanded in im around 0

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

   13. Applied rewrites 25.2%

       \[\leadsto \frac{1}{1 + \color{blue}{\left(-im \cdot im\right)} \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664 - 0.5\right)} \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

         \[\leadsto \left(re - -1 \cdot 1\right) \cdot \cos im \]

      5. metadata-eval N/A

         \[\leadsto \left(re - -1\right) \cdot \cos im \]

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. metadata-eval 67.7

         \[\leadsto \left(re - -1\right) \cdot \cos im \]

   5. Applied rewrites 67.7%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 43.6%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. pow2 N/A

         \[\leadsto \frac{re \cdot re - 1}{re + -1} \cdot 1 \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{re \cdot re - 1}{re + -1} \cdot 1 \]

      9. lower-+.f64 57.9

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

   10. Applied rewrites 57.9%

       \[\leadsto \frac{re \cdot re - 1}{\color{blue}{re + -1}} \cdot 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 46.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;\left(re - -1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, im \cdot im, 0.041666666666666664\right) \cdot \left(im \cdot im\right) - 0.5, im \cdot im, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\frac{1}{1 - \left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664 - 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{re \cdot re - 1}{re - 1} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 52.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(im \cdot im\right) \cdot -0.001388888888888889\right) \cdot \left(im \cdot im\right) - 0.5, im \cdot im, 1\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{1}{1 - \left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664 - 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{re \cdot re - 1}{re - 1} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (fma
      (- (* (* (* im im) -0.001388888888888889) (* im im)) 0.5)
      (* im im)
      1.0)
     (if (<= t_0 0.0)
       (/ 1.0 (- 1.0 (* (* im im) (- (* (* im im) 0.041666666666666664) 0.5))))
       (* (/ (- (* re re) 1.0) (- re 1.0)) 1.0)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(((((im * im) * -0.001388888888888889) * (im * im)) - 0.5), (im * im), 1.0);
	} else if (t_0 <= 0.0) {
		tmp = 1.0 / (1.0 - ((im * im) * (((im * im) * 0.041666666666666664) - 0.5)));
	} else {
		tmp = (((re * re) - 1.0) / (re - 1.0)) * 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 = fma(Float64(Float64(Float64(Float64(im * im) * -0.001388888888888889) * Float64(im * im)) - 0.5), Float64(im * im), 1.0);
	elseif (t_0 <= 0.0)
		tmp = Float64(1.0 / Float64(1.0 - Float64(Float64(im * im) * Float64(Float64(Float64(im * im) * 0.041666666666666664) - 0.5))));
	else
		tmp = Float64(Float64(Float64(Float64(re * re) - 1.0) / Float64(re - 1.0)) * 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(im * im), $MachinePrecision] * -0.001388888888888889), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(1.0 / N[(1.0 - N[(N[(im * im), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(re * re), $MachinePrecision] - 1.0), $MachinePrecision] / N[(re - 1.0), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lift-cos.f64 3.1

         \[\leadsto \cos im \]

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}\right) + 1 \]

      2. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}\right) \cdot {im}^{2} + 1 \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}, {im}^{\color{blue}{2}}, 1\right) \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}, {im}^{2}, 1\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) \cdot {im}^{2} - \frac{1}{2}, {im}^{2}, 1\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) \cdot {im}^{2} - \frac{1}{2}, {im}^{2}, 1\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{720} \cdot {im}^{2} + \frac{1}{24}\right) \cdot {im}^{2} - \frac{1}{2}, {im}^{2}, 1\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, {im}^{2}, \frac{1}{24}\right) \cdot {im}^{2} - \frac{1}{2}, {im}^{2}, 1\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, im \cdot im, \frac{1}{24}\right) \cdot {im}^{2} - \frac{1}{2}, {im}^{2}, 1\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, im \cdot im, \frac{1}{24}\right) \cdot {im}^{2} - \frac{1}{2}, {im}^{2}, 1\right) \]

      11. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, im \cdot im, \frac{1}{24}\right) \cdot \left(im \cdot im\right) - \frac{1}{2}, {im}^{2}, 1\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, im \cdot im, \frac{1}{24}\right) \cdot \left(im \cdot im\right) - \frac{1}{2}, {im}^{2}, 1\right) \]

      13. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, im \cdot im, \frac{1}{24}\right) \cdot \left(im \cdot im\right) - \frac{1}{2}, im \cdot im, 1\right) \]

      14. lift-*.f64 90.4

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

   8. Applied rewrites 90.4%

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

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{720} \cdot {im}^{2}\right) \cdot \left(im \cdot im\right) - \frac{1}{2}, im \cdot im, 1\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left({im}^{2} \cdot \frac{-1}{720}\right) \cdot \left(im \cdot im\right) - \frac{1}{2}, im \cdot im, 1\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left({im}^{2} \cdot \frac{-1}{720}\right) \cdot \left(im \cdot im\right) - \frac{1}{2}, im \cdot im, 1\right) \]

       3. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(im \cdot im\right) \cdot \frac{-1}{720}\right) \cdot \left(im \cdot im\right) - \frac{1}{2}, im \cdot im, 1\right) \]

       4. lift-*.f64 90.4

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

   11. Applied rewrites 90.4%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lift-cos.f64 35.9

         \[\leadsto \cos im \]

   5. Applied rewrites 35.9%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 1.9

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

   8. Applied rewrites 1.9%

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

      1. lift-fma.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. *-commutative N/A

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

      10. +-commutative N/A

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

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

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

      12. flip-- N/A

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

      13. lower-/.f64 N/A

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

   10. Applied rewrites 1.4%

       \[\leadsto \frac{1 - \left(\left(-im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664 - 0.5\right)\right) \cdot \left(\left(-im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664 - 0.5\right)\right)}{1 + \color{blue}{\left(-im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664 - 0.5\right)}} \]

   11. Taylor expanded in im around 0

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

   13. Applied rewrites 25.2%

       \[\leadsto \frac{1}{1 + \color{blue}{\left(-im \cdot im\right)} \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664 - 0.5\right)} \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

         \[\leadsto \left(re - -1 \cdot 1\right) \cdot \cos im \]

      5. metadata-eval N/A

         \[\leadsto \left(re - -1\right) \cdot \cos im \]

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. metadata-eval 67.7

         \[\leadsto \left(re - -1\right) \cdot \cos im \]

   5. Applied rewrites 67.7%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 43.6%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. pow2 N/A

         \[\leadsto \frac{re \cdot re - 1}{re + -1} \cdot 1 \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{re \cdot re - 1}{re + -1} \cdot 1 \]

      9. lower-+.f64 57.9

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

   10. Applied rewrites 57.9%

       \[\leadsto \frac{re \cdot re - 1}{\color{blue}{re + -1}} \cdot 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 46.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(im \cdot im\right) \cdot -0.001388888888888889\right) \cdot \left(im \cdot im\right) - 0.5, im \cdot im, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\frac{1}{1 - \left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664 - 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{re \cdot re - 1}{re - 1} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 36.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -0.0005:\\ \;\;\;\;\mathsf{fma}\left(-0.5, im \cdot im, 1\right)\\ \mathbf{elif}\;t\_0 \leq 0.995:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(im \cdot im\right) \cdot 0.041666666666666664, im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 -0.0005)
     (fma -0.5 (* im im) 1.0)
     (if (<= t_0 0.995)
       1.0
       (fma (* (* im im) 0.041666666666666664) (* im im) 1.0)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -0.0005) {
		tmp = fma(-0.5, (im * im), 1.0);
	} else if (t_0 <= 0.995) {
		tmp = 1.0;
	} else {
		tmp = fma(((im * im) * 0.041666666666666664), (im * im), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= -0.0005)
		tmp = fma(-0.5, Float64(im * im), 1.0);
	elseif (t_0 <= 0.995)
		tmp = 1.0;
	else
		tmp = fma(Float64(Float64(im * im) * 0.041666666666666664), 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, -0.0005], N[(-0.5 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[t$95$0, 0.995], 1.0, N[(N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -5.0000000000000001e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lift-cos.f64 77.4

         \[\leadsto \cos im \]

   5. Applied rewrites 77.4%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}\right) + 1 \]

      2. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}\right) \cdot {im}^{2} + 1 \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}, {im}^{\color{blue}{2}}, 1\right) \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}, {im}^{2}, 1\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) \cdot {im}^{2} - \frac{1}{2}, {im}^{2}, 1\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) \cdot {im}^{2} - \frac{1}{2}, {im}^{2}, 1\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{720} \cdot {im}^{2} + \frac{1}{24}\right) \cdot {im}^{2} - \frac{1}{2}, {im}^{2}, 1\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, {im}^{2}, \frac{1}{24}\right) \cdot {im}^{2} - \frac{1}{2}, {im}^{2}, 1\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, im \cdot im, \frac{1}{24}\right) \cdot {im}^{2} - \frac{1}{2}, {im}^{2}, 1\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, im \cdot im, \frac{1}{24}\right) \cdot {im}^{2} - \frac{1}{2}, {im}^{2}, 1\right) \]

      11. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, im \cdot im, \frac{1}{24}\right) \cdot \left(im \cdot im\right) - \frac{1}{2}, {im}^{2}, 1\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, im \cdot im, \frac{1}{24}\right) \cdot \left(im \cdot im\right) - \frac{1}{2}, {im}^{2}, 1\right) \]

      13. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, im \cdot im, \frac{1}{24}\right) \cdot \left(im \cdot im\right) - \frac{1}{2}, im \cdot im, 1\right) \]

      14. lift-*.f64 22.7

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

   8. Applied rewrites 22.7%

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

   9. Taylor expanded in im around 0

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

   11. Applied rewrites 21.1%

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

   if -5.0000000000000001e-4 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.994999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lift-cos.f64 40.9

         \[\leadsto \cos im \]

   5. Applied rewrites 40.9%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 10.1%

      \[\leadsto 1 \]

   if 0.994999999999999996 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lift-cos.f64 53.1

         \[\leadsto \cos im \]

   5. Applied rewrites 53.1%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 63.9

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

   8. Applied rewrites 63.9%

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

   9. Taylor expanded in im around inf

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

       1. *-commutative N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 63.9

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

   11. Applied rewrites 63.9%

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

Alternative 12: 73.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0:\\ \;\;\;\;\frac{1}{1 - \left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664 - 0.5\right)}\\ \mathbf{elif}\;e^{re} \leq 2:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, 0.16666666666666666, 1\right) \cdot re + \mathsf{fma}\left(re \cdot re, 0.5, 1\right)\right) \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot \mathsf{fma}\left(-0.001388888888888889 \cdot im, im, 0.041666666666666664\right) - 0.5, im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (exp re) 0.0)
   (/ 1.0 (- 1.0 (* (* im im) (- (* (* im im) 0.041666666666666664) 0.5))))
   (if (<= (exp re) 2.0)
     (cos im)
     (*
      (+
       (* (fma (* re re) 0.16666666666666666 1.0) re)
       (fma (* re re) 0.5 1.0))
      (fma
       (-
        (*
         (* im im)
         (fma (* -0.001388888888888889 im) im 0.041666666666666664))
        0.5)
       (* im im)
       1.0)))))

C

double code(double re, double im) {
	double tmp;
	if (exp(re) <= 0.0) {
		tmp = 1.0 / (1.0 - ((im * im) * (((im * im) * 0.041666666666666664) - 0.5)));
	} else if (exp(re) <= 2.0) {
		tmp = cos(im);
	} else {
		tmp = ((fma((re * re), 0.16666666666666666, 1.0) * re) + fma((re * re), 0.5, 1.0)) * fma((((im * im) * fma((-0.001388888888888889 * im), im, 0.041666666666666664)) - 0.5), (im * im), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (exp(re) <= 0.0)
		tmp = Float64(1.0 / Float64(1.0 - Float64(Float64(im * im) * Float64(Float64(Float64(im * im) * 0.041666666666666664) - 0.5))));
	elseif (exp(re) <= 2.0)
		tmp = cos(im);
	else
		tmp = Float64(Float64(Float64(fma(Float64(re * re), 0.16666666666666666, 1.0) * re) + fma(Float64(re * re), 0.5, 1.0)) * fma(Float64(Float64(Float64(im * im) * fma(Float64(-0.001388888888888889 * im), im, 0.041666666666666664)) - 0.5), Float64(im * im), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Exp[re], $MachinePrecision], 0.0], N[(1.0 / N[(1.0 - N[(N[(im * im), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Exp[re], $MachinePrecision], 2.0], N[Cos[im], $MachinePrecision], N[(N[(N[(N[(N[(re * re), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * re), $MachinePrecision] + N[(N[(re * re), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(im * im), $MachinePrecision] * N[(N[(-0.001388888888888889 * im), $MachinePrecision] * im + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 re) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lift-cos.f64 3.1

         \[\leadsto \cos im \]

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 2.5

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

   8. Applied rewrites 2.5%

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

      1. lift-fma.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. *-commutative N/A

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

      10. +-commutative N/A

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

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

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

      12. flip-- N/A

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

      13. lower-/.f64 N/A

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

   10. Applied rewrites 2.0%

       \[\leadsto \frac{1 - \left(\left(-im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664 - 0.5\right)\right) \cdot \left(\left(-im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664 - 0.5\right)\right)}{1 + \color{blue}{\left(-im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664 - 0.5\right)}} \]

   11. Taylor expanded in im around 0

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

   13. Applied rewrites 36.5%

       \[\leadsto \frac{1}{1 + \color{blue}{\left(-im \cdot im\right)} \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664 - 0.5\right)} \]

   if 0.0 < (exp.f64 re) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lift-cos.f64 98.6

         \[\leadsto \cos im \]

   5. Applied rewrites 98.6%

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

   if 2 < (exp.f64 re)

   1. Initial program 100.0%

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

      1. lift-exp.f64 N/A

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

      2. sinh-+-cosh-rev N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sinh.f64 N/A

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

      6. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 98.6

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

   7. Applied rewrites 98.6%

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

   8. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 64.1

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

   10. Applied rewrites 64.1%

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

   11. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(re \cdot re, \frac{1}{2}, 1\right)\right) \cdot \left(\left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}\right) \cdot {im}^{2} + 1\right) \]

       3. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(re \cdot re, \frac{1}{2}, 1\right)\right) \cdot \left(\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) \cdot {im}^{2} - \frac{1}{2}\right) \cdot {im}^{2} + 1\right) \]

       4. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(re \cdot re, \frac{1}{2}, 1\right)\right) \cdot \left(\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot \left(im \cdot im\right)\right) \cdot {im}^{2} - \frac{1}{2}\right) \cdot {im}^{2} + 1\right) \]

       5. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(re \cdot re, \frac{1}{2}, 1\right)\right) \cdot \left(\left(\left(\frac{-1}{720} \cdot \left(im \cdot im\right) + \frac{1}{24}\right) \cdot {im}^{2} - \frac{1}{2}\right) \cdot {im}^{2} + 1\right) \]

       6. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(re \cdot re, \frac{1}{2}, 1\right)\right) \cdot \left(\left(\left(\frac{-1}{720} \cdot \left(im \cdot im\right) + \frac{1}{24}\right) \cdot \left(im \cdot im\right) - \frac{1}{2}\right) \cdot {im}^{2} + 1\right) \]

       7. lower-fma.f64 N/A

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

   13. Applied rewrites 53.1%

       \[\leadsto \left(\mathsf{fma}\left(re \cdot re, 0.16666666666666666, 1\right) \cdot re + \mathsf{fma}\left(re \cdot re, 0.5, 1\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\left(im \cdot im\right) \cdot \mathsf{fma}\left(-0.001388888888888889 \cdot im, im, 0.041666666666666664\right) - 0.5, im \cdot im, 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0:\\ \;\;\;\;\frac{1}{1 - \left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664 - 0.5\right)}\\ \mathbf{elif}\;e^{re} \leq 2:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, 0.16666666666666666, 1\right) \cdot re + \mathsf{fma}\left(re \cdot re, 0.5, 1\right)\right) \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot \mathsf{fma}\left(-0.001388888888888889 \cdot im, im, 0.041666666666666664\right) - 0.5, im \cdot im, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 43.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(im \cdot im\right) \cdot -0.001388888888888889\right) \cdot \left(im \cdot im\right) - 0.5, im \cdot im, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{re \cdot re - 1}{re - 1} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (cos im)) 0.0)
   (fma
    (- (* (* (* im im) -0.001388888888888889) (* im im)) 0.5)
    (* im im)
    1.0)
   (* (/ (- (* re re) 1.0) (- re 1.0)) 1.0)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * cos(im)) <= 0.0) {
		tmp = fma(((((im * im) * -0.001388888888888889) * (im * im)) - 0.5), (im * im), 1.0);
	} else {
		tmp = (((re * re) - 1.0) / (re - 1.0)) * 1.0;
	}
	return tmp;
}

Julia

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

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[(N[(N[(im * im), $MachinePrecision] * -0.001388888888888889), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(N[(re * re), $MachinePrecision] - 1.0), $MachinePrecision] / N[(re - 1.0), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lift-cos.f64 33.0

         \[\leadsto \cos im \]

   5. Applied rewrites 33.0%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}\right) + 1 \]

      2. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}\right) \cdot {im}^{2} + 1 \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}, {im}^{\color{blue}{2}}, 1\right) \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}, {im}^{2}, 1\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) \cdot {im}^{2} - \frac{1}{2}, {im}^{2}, 1\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) \cdot {im}^{2} - \frac{1}{2}, {im}^{2}, 1\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{720} \cdot {im}^{2} + \frac{1}{24}\right) \cdot {im}^{2} - \frac{1}{2}, {im}^{2}, 1\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, {im}^{2}, \frac{1}{24}\right) \cdot {im}^{2} - \frac{1}{2}, {im}^{2}, 1\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, im \cdot im, \frac{1}{24}\right) \cdot {im}^{2} - \frac{1}{2}, {im}^{2}, 1\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, im \cdot im, \frac{1}{24}\right) \cdot {im}^{2} - \frac{1}{2}, {im}^{2}, 1\right) \]

      11. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, im \cdot im, \frac{1}{24}\right) \cdot \left(im \cdot im\right) - \frac{1}{2}, {im}^{2}, 1\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, im \cdot im, \frac{1}{24}\right) \cdot \left(im \cdot im\right) - \frac{1}{2}, {im}^{2}, 1\right) \]

      13. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, im \cdot im, \frac{1}{24}\right) \cdot \left(im \cdot im\right) - \frac{1}{2}, im \cdot im, 1\right) \]

      14. lift-*.f64 10.6

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

   8. Applied rewrites 10.6%

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

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{720} \cdot {im}^{2}\right) \cdot \left(im \cdot im\right) - \frac{1}{2}, im \cdot im, 1\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left({im}^{2} \cdot \frac{-1}{720}\right) \cdot \left(im \cdot im\right) - \frac{1}{2}, im \cdot im, 1\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left({im}^{2} \cdot \frac{-1}{720}\right) \cdot \left(im \cdot im\right) - \frac{1}{2}, im \cdot im, 1\right) \]

       3. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(im \cdot im\right) \cdot \frac{-1}{720}\right) \cdot \left(im \cdot im\right) - \frac{1}{2}, im \cdot im, 1\right) \]

       4. lift-*.f64 10.6

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

   11. Applied rewrites 10.6%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

         \[\leadsto \left(re - -1 \cdot 1\right) \cdot \cos im \]

      5. metadata-eval N/A

         \[\leadsto \left(re - -1\right) \cdot \cos im \]

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. metadata-eval 67.7

         \[\leadsto \left(re - -1\right) \cdot \cos im \]

   5. Applied rewrites 67.7%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 43.6%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. pow2 N/A

         \[\leadsto \frac{re \cdot re - 1}{re + -1} \cdot 1 \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{re \cdot re - 1}{re + -1} \cdot 1 \]

      9. lower-+.f64 57.9

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

   10. Applied rewrites 57.9%

       \[\leadsto \frac{re \cdot re - 1}{\color{blue}{re + -1}} \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(im \cdot im\right) \cdot -0.001388888888888889\right) \cdot \left(im \cdot im\right) - 0.5, im \cdot im, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{re \cdot re - 1}{re - 1} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 41.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lift-cos.f64 33.0

         \[\leadsto \cos im \]

   5. Applied rewrites 33.0%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}\right) + 1 \]

      2. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}\right) \cdot {im}^{2} + 1 \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}, {im}^{\color{blue}{2}}, 1\right) \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}, {im}^{2}, 1\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) \cdot {im}^{2} - \frac{1}{2}, {im}^{2}, 1\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) \cdot {im}^{2} - \frac{1}{2}, {im}^{2}, 1\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{720} \cdot {im}^{2} + \frac{1}{24}\right) \cdot {im}^{2} - \frac{1}{2}, {im}^{2}, 1\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, {im}^{2}, \frac{1}{24}\right) \cdot {im}^{2} - \frac{1}{2}, {im}^{2}, 1\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, im \cdot im, \frac{1}{24}\right) \cdot {im}^{2} - \frac{1}{2}, {im}^{2}, 1\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, im \cdot im, \frac{1}{24}\right) \cdot {im}^{2} - \frac{1}{2}, {im}^{2}, 1\right) \]

      11. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, im \cdot im, \frac{1}{24}\right) \cdot \left(im \cdot im\right) - \frac{1}{2}, {im}^{2}, 1\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, im \cdot im, \frac{1}{24}\right) \cdot \left(im \cdot im\right) - \frac{1}{2}, {im}^{2}, 1\right) \]

      13. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, im \cdot im, \frac{1}{24}\right) \cdot \left(im \cdot im\right) - \frac{1}{2}, im \cdot im, 1\right) \]

      14. lift-*.f64 10.6

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

   8. Applied rewrites 10.6%

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

   9. Taylor expanded in im around 0

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

   11. Applied rewrites 10.0%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

         \[\leadsto \left(re - -1 \cdot 1\right) \cdot \cos im \]

      5. metadata-eval N/A

         \[\leadsto \left(re - -1\right) \cdot \cos im \]

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. metadata-eval 67.7

         \[\leadsto \left(re - -1\right) \cdot \cos im \]

   5. Applied rewrites 67.7%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 43.6%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. pow2 N/A

         \[\leadsto \frac{re \cdot re - 1}{re + -1} \cdot 1 \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{re \cdot re - 1}{re + -1} \cdot 1 \]

      9. lower-+.f64 57.9

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

   10. Applied rewrites 57.9%

       \[\leadsto \frac{re \cdot re - 1}{\color{blue}{re + -1}} \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.0%

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

Alternative 15: 32.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lift-cos.f64 33.0

         \[\leadsto \cos im \]

   5. Applied rewrites 33.0%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}\right) + 1 \]

      2. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}\right) \cdot {im}^{2} + 1 \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}, {im}^{\color{blue}{2}}, 1\right) \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}, {im}^{2}, 1\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) \cdot {im}^{2} - \frac{1}{2}, {im}^{2}, 1\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) \cdot {im}^{2} - \frac{1}{2}, {im}^{2}, 1\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{720} \cdot {im}^{2} + \frac{1}{24}\right) \cdot {im}^{2} - \frac{1}{2}, {im}^{2}, 1\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, {im}^{2}, \frac{1}{24}\right) \cdot {im}^{2} - \frac{1}{2}, {im}^{2}, 1\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, im \cdot im, \frac{1}{24}\right) \cdot {im}^{2} - \frac{1}{2}, {im}^{2}, 1\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, im \cdot im, \frac{1}{24}\right) \cdot {im}^{2} - \frac{1}{2}, {im}^{2}, 1\right) \]

      11. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, im \cdot im, \frac{1}{24}\right) \cdot \left(im \cdot im\right) - \frac{1}{2}, {im}^{2}, 1\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, im \cdot im, \frac{1}{24}\right) \cdot \left(im \cdot im\right) - \frac{1}{2}, {im}^{2}, 1\right) \]

      13. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, im \cdot im, \frac{1}{24}\right) \cdot \left(im \cdot im\right) - \frac{1}{2}, im \cdot im, 1\right) \]

      14. lift-*.f64 10.6

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

   8. Applied rewrites 10.6%

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

   9. Taylor expanded in im around 0

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

   11. Applied rewrites 10.0%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

         \[\leadsto \left(re - -1 \cdot 1\right) \cdot \cos im \]

      5. metadata-eval N/A

         \[\leadsto \left(re - -1\right) \cdot \cos im \]

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. metadata-eval 67.7

         \[\leadsto \left(re - -1\right) \cdot \cos im \]

   5. Applied rewrites 67.7%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 43.6%

      \[\leadsto \left(re - -1\right) \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 97.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.038:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.012:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, 0.16666666666666666, 1\right) \cdot re + \mathsf{fma}\left(re \cdot re, 0.5, 1\right)\right) \cdot \cos im\\ \mathbf{elif}\;re \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \cos im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.038)
   (exp re)
   (if (<= re 0.012)
     (*
      (+
       (* (fma (* re re) 0.16666666666666666 1.0) re)
       (fma (* re re) 0.5 1.0))
      (cos im))
     (if (<= re 1.02e+103)
       (* (exp re) (fma (* im im) -0.5 1.0))
       (*
        (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)
        (cos im))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.038) {
		tmp = exp(re);
	} else if (re <= 0.012) {
		tmp = ((fma((re * re), 0.16666666666666666, 1.0) * re) + fma((re * re), 0.5, 1.0)) * cos(im);
	} else if (re <= 1.02e+103) {
		tmp = exp(re) * fma((im * im), -0.5, 1.0);
	} 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)
	tmp = 0.0
	if (re <= -0.038)
		tmp = exp(re);
	elseif (re <= 0.012)
		tmp = Float64(Float64(Float64(fma(Float64(re * re), 0.16666666666666666, 1.0) * re) + fma(Float64(re * re), 0.5, 1.0)) * cos(im));
	elseif (re <= 1.02e+103)
		tmp = Float64(exp(re) * fma(Float64(im * im), -0.5, 1.0));
	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_] := If[LessEqual[re, -0.038], N[Exp[re], $MachinePrecision], If[LessEqual[re, 0.012], N[(N[(N[(N[(N[(re * re), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * re), $MachinePrecision] + N[(N[(re * re), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.02e+103], N[(N[Exp[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $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 re < -0.0379999999999999991

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lift-exp.f64 100.0

         \[\leadsto e^{re} \]

   5. Applied rewrites 100.0%

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

   if -0.0379999999999999991 < re < 0.012

   1. Initial program 100.0%

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

      1. lift-exp.f64 N/A

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

      2. sinh-+-cosh-rev N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sinh.f64 N/A

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

      6. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 100.0

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

   10. Applied rewrites 100.0%

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

   if 0.012 < re < 1.01999999999999991e103

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 81.8

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

   5. Applied rewrites 81.8%

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

   if 1.01999999999999991e103 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 17: 97.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.03:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;re \leq 0.0086:\\ \;\;\;\;\left(re + \mathsf{fma}\left(re \cdot re, 0.5, 1\right)\right) \cdot \cos im\\ \mathbf{elif}\;re \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \cos im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.03)
   (exp re)
   (if (<= re 0.0086)
     (* (+ re (fma (* re re) 0.5 1.0)) (cos im))
     (if (<= re 1.02e+103)
       (* (exp re) (fma (* im im) -0.5 1.0))
       (*
        (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)
        (cos im))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.03) {
		tmp = exp(re);
	} else if (re <= 0.0086) {
		tmp = (re + fma((re * re), 0.5, 1.0)) * cos(im);
	} else if (re <= 1.02e+103) {
		tmp = exp(re) * fma((im * im), -0.5, 1.0);
	} 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)
	tmp = 0.0
	if (re <= -0.03)
		tmp = exp(re);
	elseif (re <= 0.0086)
		tmp = Float64(Float64(re + fma(Float64(re * re), 0.5, 1.0)) * cos(im));
	elseif (re <= 1.02e+103)
		tmp = Float64(exp(re) * fma(Float64(im * im), -0.5, 1.0));
	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_] := If[LessEqual[re, -0.03], N[Exp[re], $MachinePrecision], If[LessEqual[re, 0.0086], N[(N[(re + N[(N[(re * re), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.02e+103], N[(N[Exp[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $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 re < -0.029999999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lift-exp.f64 100.0

         \[\leadsto e^{re} \]

   5. Applied rewrites 100.0%

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

   if -0.029999999999999999 < re < 0.0086

   1. Initial program 100.0%

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

      1. lift-exp.f64 N/A

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

      2. sinh-+-cosh-rev N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sinh.f64 N/A

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

      6. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 100.0%

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

   if 0.0086 < re < 1.01999999999999991e103

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 81.8

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

   5. Applied rewrites 81.8%

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

   if 1.01999999999999991e103 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 18: 28.7% accurate, 22.9× speedup

Math

\[\begin{array}{l} \\ \left(re - -1\right) \cdot 1 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (- re -1.0) 1.0))

C

double code(double re, double im) {
	return (re - -1.0) * 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 = (re - (-1.0d0)) * 1.0d0
end function

Java

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

Python

def code(re, im):
	return (re - -1.0) * 1.0

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in re around 0

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

   1. +-commutative N/A

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

   2. metadata-eval N/A

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

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

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

   4. metadata-eval N/A

      \[\leadsto \left(re - -1 \cdot 1\right) \cdot \cos im \]

   5. metadata-eval N/A

      \[\leadsto \left(re - -1\right) \cdot \cos im \]

   6. metadata-eval N/A

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

   7. lower--.f64 N/A

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

   8. metadata-eval 52.6

      \[\leadsto \left(re - -1\right) \cdot \cos im \]

5. Applied rewrites 52.6%

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

6. Taylor expanded in im around 0

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

8. Applied rewrites 25.3%

   \[\leadsto \left(re - -1\right) \cdot \color{blue}{1} \]
9. Add Preprocessing

Alternative 19: 28.3% 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. Taylor expanded in re around 0

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

   1. lift-cos.f64 52.0

      \[\leadsto \cos im \]

5. Applied rewrites 52.0%

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

6. Taylor expanded in im around 0

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

8. Applied rewrites 25.2%

   \[\leadsto 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025046 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))