math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 3.8s

Alternatives: 17

Speedup: 1.0×

• 24.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 2: 98.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \cos im\\ t_1 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(\left(im \cdot im\right) \cdot -0.5\right)\\ \mathbf{elif}\;t\_1 \leq -0.1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;t\_1 \leq 0.99999998:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (fma (fma 0.5 re 1.0) re 1.0) (cos im)))
        (t_1 (* (exp re) (cos im))))
   (if (<= t_1 (- INFINITY))
     (* (exp re) (* (* im im) -0.5))
     (if (<= t_1 -0.1)
       t_0
       (if (<= t_1 0.0) (exp re) (if (<= t_1 0.99999998) t_0 (exp re)))))))

C

double code(double re, double im) {
	double t_0 = fma(fma(0.5, re, 1.0), re, 1.0) * cos(im);
	double t_1 = exp(re) * cos(im);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = exp(re) * ((im * im) * -0.5);
	} else if (t_1 <= -0.1) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = exp(re);
	} else if (t_1 <= 0.99999998) {
		tmp = t_0;
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * cos(im))
	t_1 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(exp(re) * Float64(Float64(im * im) * -0.5));
	elseif (t_1 <= -0.1)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = exp(re);
	elseif (t_1 <= 0.99999998)
		tmp = t_0;
	else
		tmp = exp(re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[Exp[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.1], t$95$0, If[LessEqual[t$95$1, 0.0], N[Exp[re], $MachinePrecision], If[LessEqual[t$95$1, 0.99999998], t$95$0, 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)) < -0.10000000000000001 or 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.999999980000000011

   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 97.9

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

   5. Applied rewrites 97.9%

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

   if -0.10000000000000001 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0 or 0.999999980000000011 < (*.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 98.5

         \[\leadsto e^{re} \]

   5. Applied rewrites 98.5%

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

Alternative 3: 98.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(re - -1\right) \cdot \cos im\\ t_1 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(\left(im \cdot im\right) \cdot -0.5\right)\\ \mathbf{elif}\;t\_1 \leq -0.1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;t\_1 \leq 0.99999998:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (- re -1.0) (cos im))) (t_1 (* (exp re) (cos im))))
   (if (<= t_1 (- INFINITY))
     (* (exp re) (* (* im im) -0.5))
     (if (<= t_1 -0.1)
       t_0
       (if (<= t_1 0.0) (exp re) (if (<= t_1 0.99999998) t_0 (exp re)))))))

C

double code(double re, double im) {
	double t_0 = (re - -1.0) * cos(im);
	double t_1 = exp(re) * cos(im);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = exp(re) * ((im * im) * -0.5);
	} else if (t_1 <= -0.1) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = exp(re);
	} else if (t_1 <= 0.99999998) {
		tmp = t_0;
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = (re - -1.0) * Math.cos(im);
	double t_1 = Math.exp(re) * Math.cos(im);
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = Math.exp(re) * ((im * im) * -0.5);
	} else if (t_1 <= -0.1) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = Math.exp(re);
	} else if (t_1 <= 0.99999998) {
		tmp = t_0;
	} else {
		tmp = Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (re - -1.0) * math.cos(im)
	t_1 = math.exp(re) * math.cos(im)
	tmp = 0
	if t_1 <= -math.inf:
		tmp = math.exp(re) * ((im * im) * -0.5)
	elif t_1 <= -0.1:
		tmp = t_0
	elif t_1 <= 0.0:
		tmp = math.exp(re)
	elif t_1 <= 0.99999998:
		tmp = t_0
	else:
		tmp = math.exp(re)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(re - -1.0) * cos(im))
	t_1 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(exp(re) * Float64(Float64(im * im) * -0.5));
	elseif (t_1 <= -0.1)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = exp(re);
	elseif (t_1 <= 0.99999998)
		tmp = t_0;
	else
		tmp = exp(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (re - -1.0) * cos(im);
	t_1 = exp(re) * cos(im);
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = exp(re) * ((im * im) * -0.5);
	elseif (t_1 <= -0.1)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = exp(re);
	elseif (t_1 <= 0.99999998)
		tmp = t_0;
	else
		tmp = exp(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(re - -1.0), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[Exp[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.1], t$95$0, If[LessEqual[t$95$1, 0.0], N[Exp[re], $MachinePrecision], If[LessEqual[t$95$1, 0.99999998], t$95$0, 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)) < -0.10000000000000001 or 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.999999980000000011

   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 97.3

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

   5. Applied rewrites 97.3%

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

   if -0.10000000000000001 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0 or 0.999999980000000011 < (*.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 98.5

         \[\leadsto e^{re} \]

   5. Applied rewrites 98.5%

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

Alternative 4: 98.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ t_1 := \left(re - -1\right) \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot 0.5\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(-0.001388888888888889, im \cdot im, 0.041666666666666664\right) \cdot im\right) \cdot im - 0.5, im \cdot im, 1\right)\\ \mathbf{elif}\;t\_0 \leq -0.1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;t\_0 \leq 0.99999998:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))) (t_1 (* (- re -1.0) (cos im))))
   (if (<= t_0 (- INFINITY))
     (*
      (* (* re re) 0.5)
      (fma
       (-
        (*
         (* (fma -0.001388888888888889 (* im im) 0.041666666666666664) im)
         im)
        0.5)
       (* im im)
       1.0))
     (if (<= t_0 -0.1)
       t_1
       (if (<= t_0 0.0) (exp re) (if (<= t_0 0.99999998) t_1 (exp re)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double t_1 = (re - -1.0) * cos(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = ((re * re) * 0.5) * fma((((fma(-0.001388888888888889, (im * im), 0.041666666666666664) * im) * im) - 0.5), (im * im), 1.0);
	} else if (t_0 <= -0.1) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = exp(re);
	} else if (t_0 <= 0.99999998) {
		tmp = t_1;
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	t_1 = Float64(Float64(re - -1.0) * cos(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(re * re) * 0.5) * fma(Float64(Float64(Float64(fma(-0.001388888888888889, Float64(im * im), 0.041666666666666664) * im) * im) - 0.5), Float64(im * im), 1.0));
	elseif (t_0 <= -0.1)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = exp(re);
	elseif (t_0 <= 0.99999998)
		tmp = t_1;
	else
		tmp = exp(re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(re - -1.0), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[(N[(N[(N[(-0.001388888888888889 * N[(im * im), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.1], t$95$1, If[LessEqual[t$95$0, 0.0], N[Exp[re], $MachinePrecision], If[LessEqual[t$95$0, 0.99999998], t$95$1, N[Exp[re], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} \cdot \cos im \]
   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 53.6

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

   5. Applied rewrites 53.6%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 96.4

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

   8. Applied rewrites 96.4%

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

   9. Taylor expanded in re around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 96.4

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

   11. Applied rewrites 96.4%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.10000000000000001 or 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.999999980000000011

   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 97.3

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

   5. Applied rewrites 97.3%

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

   if -0.10000000000000001 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0 or 0.999999980000000011 < (*.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 98.5

         \[\leadsto e^{re} \]

   5. Applied rewrites 98.5%

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

Alternative 5: 97.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot 0.5\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(-0.001388888888888889, im \cdot im, 0.041666666666666664\right) \cdot im\right) \cdot im - 0.5, im \cdot im, 1\right)\\ \mathbf{elif}\;t\_0 \leq -0.1:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;t\_0 \leq 0.9999:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (*
      (* (* re re) 0.5)
      (fma
       (-
        (*
         (* (fma -0.001388888888888889 (* im im) 0.041666666666666664) im)
         im)
        0.5)
       (* im im)
       1.0))
     (if (<= t_0 -0.1)
       (cos im)
       (if (<= t_0 0.0) (exp re) (if (<= t_0 0.9999) (cos im) (exp re)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = ((re * re) * 0.5) * fma((((fma(-0.001388888888888889, (im * im), 0.041666666666666664) * im) * im) - 0.5), (im * im), 1.0);
	} else if (t_0 <= -0.1) {
		tmp = cos(im);
	} else if (t_0 <= 0.0) {
		tmp = exp(re);
	} else if (t_0 <= 0.9999) {
		tmp = cos(im);
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(re * re) * 0.5) * fma(Float64(Float64(Float64(fma(-0.001388888888888889, Float64(im * im), 0.041666666666666664) * im) * im) - 0.5), Float64(im * im), 1.0));
	elseif (t_0 <= -0.1)
		tmp = cos(im);
	elseif (t_0 <= 0.0)
		tmp = exp(re);
	elseif (t_0 <= 0.9999)
		tmp = cos(im);
	else
		tmp = exp(re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[(N[(N[(N[(-0.001388888888888889 * N[(im * im), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.1], N[Cos[im], $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[Exp[re], $MachinePrecision], If[LessEqual[t$95$0, 0.9999], N[Cos[im], $MachinePrecision], N[Exp[re], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-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 53.6

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

   5. Applied rewrites 53.6%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 96.4

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

   8. Applied rewrites 96.4%

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

   9. Taylor expanded in re around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 96.4

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

   11. Applied rewrites 96.4%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.10000000000000001 or 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.99990000000000001

   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 96.3

         \[\leadsto \cos im \]

   5. Applied rewrites 96.3%

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

   if -0.10000000000000001 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0 or 0.99990000000000001 < (*.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 98.3

         \[\leadsto e^{re} \]

   5. Applied rewrites 98.3%

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

Alternative 6: 73.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot 0.5\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(-0.001388888888888889, im \cdot im, 0.041666666666666664\right) \cdot im\right) \cdot im - 0.5, im \cdot im, 1\right)\\ \mathbf{elif}\;t\_0 \leq -0.1:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;1 \cdot \left(\left(im \cdot im\right) \cdot -0.5\right)\\ \mathbf{elif}\;t\_0 \leq 0.9999:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (*
      (* (* re re) 0.5)
      (fma
       (-
        (*
         (* (fma -0.001388888888888889 (* im im) 0.041666666666666664) im)
         im)
        0.5)
       (* im im)
       1.0))
     (if (<= t_0 -0.1)
       (cos im)
       (if (<= t_0 0.0)
         (* 1.0 (* (* im im) -0.5))
         (if (<= t_0 0.9999)
           (cos im)
           (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = ((re * re) * 0.5) * fma((((fma(-0.001388888888888889, (im * im), 0.041666666666666664) * im) * im) - 0.5), (im * im), 1.0);
	} else if (t_0 <= -0.1) {
		tmp = cos(im);
	} else if (t_0 <= 0.0) {
		tmp = 1.0 * ((im * im) * -0.5);
	} else if (t_0 <= 0.9999) {
		tmp = cos(im);
	} else {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(re * re) * 0.5) * fma(Float64(Float64(Float64(fma(-0.001388888888888889, Float64(im * im), 0.041666666666666664) * im) * im) - 0.5), Float64(im * im), 1.0));
	elseif (t_0 <= -0.1)
		tmp = cos(im);
	elseif (t_0 <= 0.0)
		tmp = Float64(1.0 * Float64(Float64(im * im) * -0.5));
	elseif (t_0 <= 0.9999)
		tmp = cos(im);
	else
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[(N[(N[(N[(-0.001388888888888889 * N[(im * im), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.1], N[Cos[im], $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(1.0 * N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.9999], N[Cos[im], $MachinePrecision], N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-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 53.6

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

   5. Applied rewrites 53.6%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 96.4

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

   8. Applied rewrites 96.4%

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

   9. Taylor expanded in re around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 96.4

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

   11. Applied rewrites 96.4%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.10000000000000001 or 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.99990000000000001

   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 96.3

         \[\leadsto \cos im \]

   5. Applied rewrites 96.3%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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 71.8

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

   5. Applied rewrites 71.8%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 1.8

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

   8. Applied rewrites 1.8%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 2.6%

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

   12. Taylor expanded in im around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 25.5

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

   14. Applied rewrites 25.5%

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

   if 0.99990000000000001 < (*.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 99.4

         \[\leadsto e^{re} \]

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 85.9

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

   8. Applied rewrites 85.9%

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

Alternative 7: 53.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-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 53.6

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

   5. Applied rewrites 53.6%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 96.4

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

   8. Applied rewrites 96.4%

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

   9. Taylor expanded in re around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 96.4

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

   11. Applied rewrites 96.4%

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

   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 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 50.3

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

   5. Applied rewrites 50.3%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 2.6

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

   8. Applied rewrites 2.6%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 3.1%

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

   12. Taylor expanded in im around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 18.8

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

   14. Applied rewrites 18.8%

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

   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 im around 0

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

      1. lift-exp.f64 81.3

         \[\leadsto e^{re} \]

   5. Applied rewrites 81.3%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 70.6

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

   8. Applied rewrites 70.6%

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

Alternative 8: 52.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -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 5.4

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

   5. Applied rewrites 5.4%

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

   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. pow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 92.0

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

   8. Applied rewrites 92.0%

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

   9. Taylor expanded in re around inf

      \[\leadsto re \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{720}, im \cdot im, \frac{1}{24}\right) \cdot im\right) \cdot im - \frac{1}{2}, im \cdot im, 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 92.0%

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

   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 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 50.3

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

   5. Applied rewrites 50.3%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 2.6

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

   8. Applied rewrites 2.6%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 3.1%

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

   12. Taylor expanded in im around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 18.8

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

   14. Applied rewrites 18.8%

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

   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 im around 0

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

      1. lift-exp.f64 81.3

         \[\leadsto e^{re} \]

   5. Applied rewrites 81.3%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 70.6

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

   8. Applied rewrites 70.6%

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

Alternative 9: 52.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -0.9:\\ \;\;\;\;\mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.16666666666666666, re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;1 \cdot \left(\left(im \cdot im\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 -0.9)
     (*
      (fma (* (* re re) 0.16666666666666666) re 1.0)
      (fma (* im im) -0.5 1.0))
     (if (<= t_0 0.0)
       (* 1.0 (* (* im im) -0.5))
       (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -0.9) {
		tmp = fma(((re * re) * 0.16666666666666666), re, 1.0) * fma((im * im), -0.5, 1.0);
	} else if (t_0 <= 0.0) {
		tmp = 1.0 * ((im * im) * -0.5);
	} else {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= -0.9)
		tmp = Float64(fma(Float64(Float64(re * re) * 0.16666666666666666), re, 1.0) * fma(Float64(im * im), -0.5, 1.0));
	elseif (t_0 <= 0.0)
		tmp = Float64(1.0 * Float64(Float64(im * im) * -0.5));
	else
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.9], N[(N[(N[(N[(re * re), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(1.0 * N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.900000000000000022

   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 65.6

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

   5. Applied rewrites 65.6%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 60.4

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

   8. Applied rewrites 60.4%

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

   9. Taylor expanded in re around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 60.4

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

   11. Applied rewrites 60.4%

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

   if -0.900000000000000022 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      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 55.3

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

   5. Applied rewrites 55.3%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 2.4

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

   8. Applied rewrites 2.4%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 3.0%

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

   12. Taylor expanded in im around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 20.3

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

   14. Applied rewrites 20.3%

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

   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 im around 0

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

      1. lift-exp.f64 81.3

         \[\leadsto e^{re} \]

   5. Applied rewrites 81.3%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 70.6

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

   8. Applied rewrites 70.6%

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

Alternative 10: 52.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -0.85:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;1 \cdot \left(\left(im \cdot im\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 -0.85)
     (* (fma (fma 0.5 re 1.0) re 1.0) (fma (* im im) -0.5 1.0))
     (if (<= t_0 0.0)
       (* 1.0 (* (* im im) -0.5))
       (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -0.85) {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * fma((im * im), -0.5, 1.0);
	} else if (t_0 <= 0.0) {
		tmp = 1.0 * ((im * im) * -0.5);
	} else {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= -0.85)
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * fma(Float64(im * im), -0.5, 1.0));
	elseif (t_0 <= 0.0)
		tmp = Float64(1.0 * Float64(Float64(im * im) * -0.5));
	else
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.85], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(1.0 * N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.849999999999999978

   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 60.7

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

   5. Applied rewrites 60.7%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 55.9

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

   8. Applied rewrites 55.9%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 53.3%

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

   if -0.849999999999999978 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      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 56.6

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

   5. Applied rewrites 56.6%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 2.3

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

   8. Applied rewrites 2.3%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 2.9%

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

   12. Taylor expanded in im around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 20.7

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

   14. Applied rewrites 20.7%

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

   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 im around 0

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

      1. lift-exp.f64 81.3

         \[\leadsto e^{re} \]

   5. Applied rewrites 81.3%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 70.6

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

   8. Applied rewrites 70.6%

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

Alternative 11: 52.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(1.0 * t$95$0), $MachinePrecision], N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 92.0

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

   8. Applied rewrites 92.0%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 87.6%

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

   12. Taylor expanded in im around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 87.6

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

   14. Applied rewrites 87.6%

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

   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 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 50.3

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

   5. Applied rewrites 50.3%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 2.6

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

   8. Applied rewrites 2.6%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 3.1%

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

   12. Taylor expanded in im around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 18.8

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

   14. Applied rewrites 18.8%

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

   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 im around 0

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

      1. lift-exp.f64 81.3

         \[\leadsto e^{re} \]

   5. Applied rewrites 81.3%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 70.6

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

   8. Applied rewrites 70.6%

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

Alternative 12: 51.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -0.9:\\ \;\;\;\;\left(1 + re\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;1 \cdot \left(\left(im \cdot im\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 -0.9)
     (* (+ 1.0 re) (fma (* im im) -0.5 1.0))
     (if (<= t_0 0.0)
       (* 1.0 (* (* im im) -0.5))
       (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -0.9) {
		tmp = (1.0 + re) * fma((im * im), -0.5, 1.0);
	} else if (t_0 <= 0.0) {
		tmp = 1.0 * ((im * im) * -0.5);
	} else {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= -0.9)
		tmp = Float64(Float64(1.0 + re) * fma(Float64(im * im), -0.5, 1.0));
	elseif (t_0 <= 0.0)
		tmp = Float64(1.0 * Float64(Float64(im * im) * -0.5));
	else
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.9], N[(N[(1.0 + re), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(1.0 * N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.900000000000000022

   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 65.6

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

   5. Applied rewrites 65.6%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 60.4

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

   8. Applied rewrites 60.4%

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

   9. Taylor expanded in re around 0

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

       1. lower-+.f64 49.8

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

   11. Applied rewrites 49.8%

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

   if -0.900000000000000022 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      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 55.3

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

   5. Applied rewrites 55.3%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 2.4

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

   8. Applied rewrites 2.4%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 3.0%

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

   12. Taylor expanded in im around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 20.3

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

   14. Applied rewrites 20.3%

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

   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 im around 0

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

      1. lift-exp.f64 81.3

         \[\leadsto e^{re} \]

   5. Applied rewrites 81.3%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 70.6

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

   8. Applied rewrites 70.6%

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

Alternative 13: 50.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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 57.6

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

   5. Applied rewrites 57.6%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 15.7

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

   8. Applied rewrites 15.7%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 10.4%

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

   12. Taylor expanded in im around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 23.8

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

   14. Applied rewrites 23.8%

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

   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 im around 0

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

      1. lift-exp.f64 81.3

         \[\leadsto e^{re} \]

   5. Applied rewrites 81.3%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 70.6

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

   8. Applied rewrites 70.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), \color{blue}{re}, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 47.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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 57.6

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

   5. Applied rewrites 57.6%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 15.7

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

   8. Applied rewrites 15.7%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 10.4%

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

   12. Taylor expanded in im around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 23.8

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

   14. Applied rewrites 23.8%

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

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-cos.f64 84.4

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

   5. Applied rewrites 84.4%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-fma.f64 66.0

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

   8. Applied rewrites 66.0%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), \color{blue}{re}, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 38.0% accurate, 15.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in re around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. associate-*r* N/A

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

   5. distribute-rgt1-in N/A

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

   6. +-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. lift-cos.f64 64.1

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

5. Applied rewrites 64.1%

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

6. Taylor expanded in im around 0

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

   1. *-commutative N/A

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

   2. +-commutative N/A

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

   3. +-commutative N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-fma.f64 38.0

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

8. Applied rewrites 38.0%

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

Alternative 16: 29.0% accurate, 51.5× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 1.0d0 + re
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0

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

   1. lift-exp.f64 70.6

      \[\leadsto e^{re} \]

5. Applied rewrites 70.6%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 28.6%

   \[\leadsto 1 \]

9. Taylor expanded in re around 0

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

    1. lower-+.f64 29.0

       \[\leadsto 1 + re \]

11. Applied rewrites 29.0%

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

Alternative 17: 28.6% accurate, 206.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 1.0)

C

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

Fortran

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 im around 0

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

   1. lift-exp.f64 70.6

      \[\leadsto e^{re} \]

5. Applied rewrites 70.6%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 28.6%

   \[\leadsto 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025089 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))