math.cos on complex, imaginary part

Percentage Accurate: 66.3% → 99.9%

Time: 5.2s

Alternatives: 17

Speedup: 1.4×

• 50.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp(-im) - exp(im));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (exp(-im) - exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (Math.exp(-im) - Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.sin(re)) * (math.exp(-im) - math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (exp(-im) - exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 66.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp(-im) - exp(im));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (exp(-im) - exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (Math.exp(-im) - Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.sin(re)) * (math.exp(-im) - math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (exp(-im) - exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(\left(2 \cdot \sinh \left(-im\right)\right) \cdot \sin re\right) \cdot 0.5 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (* (* 2.0 (sinh (- im))) (sin re)) 0.5))

C

double code(double re, double im) {
	return ((2.0 * sinh(-im)) * sin(re)) * 0.5;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = ((2.0d0 * sinh(-im)) * sin(re)) * 0.5d0
end function

Java

public static double code(double re, double im) {
	return ((2.0 * Math.sinh(-im)) * Math.sin(re)) * 0.5;
}

Python

def code(re, im):
	return ((2.0 * math.sinh(-im)) * math.sin(re)) * 0.5

Julia

function code(re, im)
	return Float64(Float64(Float64(2.0 * sinh(Float64(-im))) * sin(re)) * 0.5)
end

MATLAB

function tmp = code(re, im)
	tmp = ((2.0 * sinh(-im)) * sin(re)) * 0.5;
end

Wolfram

code[re_, im_] := N[(N[(N[(2.0 * N[Sinh[(-im)], $MachinePrecision]), $MachinePrecision] * N[Sin[re], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 66.3%

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

3. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\left(\left(2 \cdot \sinh \left(-im\right)\right) \cdot \sin re\right) \cdot 0.5} \]
4. Add Preprocessing

Alternative 2: 93.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -0.0215:\\ \;\;\;\;\left(re \cdot \left(0.5 + -0.08333333333333333 \cdot {re}^{2}\right)\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;im \leq 1.25:\\ \;\;\;\;-1 \cdot \left(im \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(1 - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -0.0215)
   (*
    (* re (+ 0.5 (* -0.08333333333333333 (pow re 2.0))))
    (- (exp (- im)) (exp im)))
   (if (<= im 1.25)
     (* -1.0 (* im (sin re)))
     (* (* 0.5 (sin re)) (- 1.0 (exp im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -0.0215) {
		tmp = (re * (0.5 + (-0.08333333333333333 * pow(re, 2.0)))) * (exp(-im) - exp(im));
	} else if (im <= 1.25) {
		tmp = -1.0 * (im * sin(re));
	} else {
		tmp = (0.5 * sin(re)) * (1.0 - exp(im));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= (-0.0215d0)) then
        tmp = (re * (0.5d0 + ((-0.08333333333333333d0) * (re ** 2.0d0)))) * (exp(-im) - exp(im))
    else if (im <= 1.25d0) then
        tmp = (-1.0d0) * (im * sin(re))
    else
        tmp = (0.5d0 * sin(re)) * (1.0d0 - exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -0.0215) {
		tmp = (re * (0.5 + (-0.08333333333333333 * Math.pow(re, 2.0)))) * (Math.exp(-im) - Math.exp(im));
	} else if (im <= 1.25) {
		tmp = -1.0 * (im * Math.sin(re));
	} else {
		tmp = (0.5 * Math.sin(re)) * (1.0 - Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -0.0215:
		tmp = (re * (0.5 + (-0.08333333333333333 * math.pow(re, 2.0)))) * (math.exp(-im) - math.exp(im))
	elif im <= 1.25:
		tmp = -1.0 * (im * math.sin(re))
	else:
		tmp = (0.5 * math.sin(re)) * (1.0 - math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -0.0215)
		tmp = Float64(Float64(re * Float64(0.5 + Float64(-0.08333333333333333 * (re ^ 2.0)))) * Float64(exp(Float64(-im)) - exp(im)));
	elseif (im <= 1.25)
		tmp = Float64(-1.0 * Float64(im * sin(re)));
	else
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(1.0 - exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -0.0215)
		tmp = (re * (0.5 + (-0.08333333333333333 * (re ^ 2.0)))) * (exp(-im) - exp(im));
	elseif (im <= 1.25)
		tmp = -1.0 * (im * sin(re));
	else
		tmp = (0.5 * sin(re)) * (1.0 - exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -0.0215], N[(N[(re * N[(0.5 + N[(-0.08333333333333333 * N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.25], N[(-1.0 * N[(im * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < -0.021499999999999998

   1. Initial program 66.3%

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

   2. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(e^{-im} - e^{im}\right) \]

   4. Applied rewrites 52.2%

      \[\leadsto \color{blue}{\left(re \cdot \left(0.5 + -0.08333333333333333 \cdot {re}^{2}\right)\right)} \cdot \left(e^{-im} - e^{im}\right) \]

   if -0.021499999999999998 < im < 1.25

   1. Initial program 66.3%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 51.0%

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

   if 1.25 < im

   1. Initial program 66.3%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 40.4%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{1} - e^{im}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 92.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -6.3:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{-im} - 2\right)\\ \mathbf{elif}\;im \leq 1.25:\\ \;\;\;\;-1 \cdot \left(im \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(1 - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -6.3)
   (* (* 0.5 re) (- (exp (- im)) 2.0))
   (if (<= im 1.25)
     (* -1.0 (* im (sin re)))
     (* (* 0.5 (sin re)) (- 1.0 (exp im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -6.3) {
		tmp = (0.5 * re) * (exp(-im) - 2.0);
	} else if (im <= 1.25) {
		tmp = -1.0 * (im * sin(re));
	} else {
		tmp = (0.5 * sin(re)) * (1.0 - exp(im));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= (-6.3d0)) then
        tmp = (0.5d0 * re) * (exp(-im) - 2.0d0)
    else if (im <= 1.25d0) then
        tmp = (-1.0d0) * (im * sin(re))
    else
        tmp = (0.5d0 * sin(re)) * (1.0d0 - exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -6.3) {
		tmp = (0.5 * re) * (Math.exp(-im) - 2.0);
	} else if (im <= 1.25) {
		tmp = -1.0 * (im * Math.sin(re));
	} else {
		tmp = (0.5 * Math.sin(re)) * (1.0 - Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -6.3:
		tmp = (0.5 * re) * (math.exp(-im) - 2.0)
	elif im <= 1.25:
		tmp = -1.0 * (im * math.sin(re))
	else:
		tmp = (0.5 * math.sin(re)) * (1.0 - math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -6.3)
		tmp = Float64(Float64(0.5 * re) * Float64(exp(Float64(-im)) - 2.0));
	elseif (im <= 1.25)
		tmp = Float64(-1.0 * Float64(im * sin(re)));
	else
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(1.0 - exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -6.3)
		tmp = (0.5 * re) * (exp(-im) - 2.0);
	elseif (im <= 1.25)
		tmp = -1.0 * (im * sin(re));
	else
		tmp = (0.5 * sin(re)) * (1.0 - exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -6.3], N[(N[(0.5 * re), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.25], N[(-1.0 * N[(im * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < -6.29999999999999982

   1. Initial program 66.3%

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

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 52.7%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 37.3%

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

   9. Applied rewrites 21.8%

      \[\leadsto \left(0.5 \cdot re\right) \cdot \left(e^{-im} - \color{blue}{2}\right) \]

   if -6.29999999999999982 < im < 1.25

   1. Initial program 66.3%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 51.0%

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

   if 1.25 < im

   1. Initial program 66.3%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 40.4%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{1} - e^{im}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 86.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -6.3:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{-im} - 2\right)\\ \mathbf{elif}\;im \leq 8.5:\\ \;\;\;\;-1 \cdot \left(im \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(1 + -1 \cdot im\right) - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -6.3)
   (* (* 0.5 re) (- (exp (- im)) 2.0))
   (if (<= im 8.5)
     (* -1.0 (* im (sin re)))
     (*
      (* (fma re -0.08333333333333333 0.5) re)
      (- (+ 1.0 (* -1.0 im)) (exp im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -6.3) {
		tmp = (0.5 * re) * (exp(-im) - 2.0);
	} else if (im <= 8.5) {
		tmp = -1.0 * (im * sin(re));
	} else {
		tmp = (fma(re, -0.08333333333333333, 0.5) * re) * ((1.0 + (-1.0 * im)) - exp(im));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -6.3)
		tmp = Float64(Float64(0.5 * re) * Float64(exp(Float64(-im)) - 2.0));
	elseif (im <= 8.5)
		tmp = Float64(-1.0 * Float64(im * sin(re)));
	else
		tmp = Float64(Float64(fma(re, -0.08333333333333333, 0.5) * re) * Float64(Float64(1.0 + Float64(-1.0 * im)) - exp(im)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, -6.3], N[(N[(0.5 * re), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 8.5], N[(-1.0 * N[(im * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(re * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(1.0 + N[(-1.0 * im), $MachinePrecision]), $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < -6.29999999999999982

   1. Initial program 66.3%

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

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 52.7%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 37.3%

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

   9. Applied rewrites 21.8%

      \[\leadsto \left(0.5 \cdot re\right) \cdot \left(e^{-im} - \color{blue}{2}\right) \]

   if -6.29999999999999982 < im < 8.5

   1. Initial program 66.3%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 51.0%

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

   if 8.5 < im

   1. Initial program 66.3%

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

   2. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(e^{-im} - e^{im}\right) \]

   4. Applied rewrites 52.2%

      \[\leadsto \color{blue}{\left(re \cdot \left(0.5 + -0.08333333333333333 \cdot {re}^{2}\right)\right)} \cdot \left(e^{-im} - e^{im}\right) \]

   6. Applied rewrites 52.5%

      \[\leadsto \left(\mathsf{fma}\left(re, -0.08333333333333333, 0.5\right) \cdot \color{blue}{re}\right) \cdot \left(e^{-im} - e^{im}\right) \]

   7. Taylor expanded in im around 0

      \[\leadsto \left(\mathsf{fma}\left(re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\color{blue}{\left(1 + -1 \cdot im\right)} - e^{im}\right) \]

   9. Applied rewrites 38.3%

      \[\leadsto \left(\mathsf{fma}\left(re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\color{blue}{\left(1 + -1 \cdot im\right)} - e^{im}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 63.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(re, -0.08333333333333333, 0.5\right) \cdot re\\ \mathbf{if}\;0.5 \cdot \sin re \leq 5 \cdot 10^{-5}:\\ \;\;\;\;t\_0 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (fma re -0.08333333333333333 0.5) re)))
   (if (<= (* 0.5 (sin re)) 5e-5)
     (* t_0 (- (exp (- im)) (exp im)))
     (* t_0 im))))

C

double code(double re, double im) {
	double t_0 = fma(re, -0.08333333333333333, 0.5) * re;
	double tmp;
	if ((0.5 * sin(re)) <= 5e-5) {
		tmp = t_0 * (exp(-im) - exp(im));
	} else {
		tmp = t_0 * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(fma(re, -0.08333333333333333, 0.5) * re)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= 5e-5)
		tmp = Float64(t_0 * Float64(exp(Float64(-im)) - exp(im)));
	else
		tmp = Float64(t_0 * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(re * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision]}, If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], 5e-5], N[(t$95$0 * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * im), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < 5.00000000000000024e-5

   1. Initial program 66.3%

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

   2. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(e^{-im} - e^{im}\right) \]

   4. Applied rewrites 52.2%

      \[\leadsto \color{blue}{\left(re \cdot \left(0.5 + -0.08333333333333333 \cdot {re}^{2}\right)\right)} \cdot \left(e^{-im} - e^{im}\right) \]

   6. Applied rewrites 52.5%

      \[\leadsto \left(\mathsf{fma}\left(re, -0.08333333333333333, 0.5\right) \cdot \color{blue}{re}\right) \cdot \left(e^{-im} - e^{im}\right) \]

   if 5.00000000000000024e-5 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 66.3%

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

   2. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(e^{-im} - e^{im}\right) \]

   4. Applied rewrites 52.2%

      \[\leadsto \color{blue}{\left(re \cdot \left(0.5 + -0.08333333333333333 \cdot {re}^{2}\right)\right)} \cdot \left(e^{-im} - e^{im}\right) \]

   6. Applied rewrites 52.5%

      \[\leadsto \left(\mathsf{fma}\left(re, -0.08333333333333333, 0.5\right) \cdot \color{blue}{re}\right) \cdot \left(e^{-im} - e^{im}\right) \]

   8. Applied rewrites 23.0%

      \[\leadsto \left(\mathsf{fma}\left(re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \color{blue}{im} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 63.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1.66:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{-im} - 2\right)\\ \mathbf{elif}\;im \leq 0.96:\\ \;\;\;\;-1 \cdot \left(im \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(1 + -1 \cdot im\right) - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -1.66)
   (* (* 0.5 re) (- (exp (- im)) 2.0))
   (if (<= im 0.96)
     (* -1.0 (* im re))
     (*
      (* (fma re -0.08333333333333333 0.5) re)
      (- (+ 1.0 (* -1.0 im)) (exp im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -1.66) {
		tmp = (0.5 * re) * (exp(-im) - 2.0);
	} else if (im <= 0.96) {
		tmp = -1.0 * (im * re);
	} else {
		tmp = (fma(re, -0.08333333333333333, 0.5) * re) * ((1.0 + (-1.0 * im)) - exp(im));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -1.66)
		tmp = Float64(Float64(0.5 * re) * Float64(exp(Float64(-im)) - 2.0));
	elseif (im <= 0.96)
		tmp = Float64(-1.0 * Float64(im * re));
	else
		tmp = Float64(Float64(fma(re, -0.08333333333333333, 0.5) * re) * Float64(Float64(1.0 + Float64(-1.0 * im)) - exp(im)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, -1.66], N[(N[(0.5 * re), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 0.96], N[(-1.0 * N[(im * re), $MachinePrecision]), $MachinePrecision], N[(N[(N[(re * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(1.0 + N[(-1.0 * im), $MachinePrecision]), $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.65999999999999992

   1. Initial program 66.3%

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

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 52.7%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 37.3%

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

   9. Applied rewrites 21.8%

      \[\leadsto \left(0.5 \cdot re\right) \cdot \left(e^{-im} - \color{blue}{2}\right) \]

   if -1.65999999999999992 < im < 0.95999999999999996

   1. Initial program 66.3%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 51.0%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 33.2%

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

   if 0.95999999999999996 < im

   1. Initial program 66.3%

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

   2. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(e^{-im} - e^{im}\right) \]

   4. Applied rewrites 52.2%

      \[\leadsto \color{blue}{\left(re \cdot \left(0.5 + -0.08333333333333333 \cdot {re}^{2}\right)\right)} \cdot \left(e^{-im} - e^{im}\right) \]

   6. Applied rewrites 52.5%

      \[\leadsto \left(\mathsf{fma}\left(re, -0.08333333333333333, 0.5\right) \cdot \color{blue}{re}\right) \cdot \left(e^{-im} - e^{im}\right) \]

   7. Taylor expanded in im around 0

      \[\leadsto \left(\mathsf{fma}\left(re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\color{blue}{\left(1 + -1 \cdot im\right)} - e^{im}\right) \]

   9. Applied rewrites 38.3%

      \[\leadsto \left(\mathsf{fma}\left(re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\color{blue}{\left(1 + -1 \cdot im\right)} - e^{im}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 61.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1.66:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{-im} - 2\right)\\ \mathbf{elif}\;im \leq 1.25:\\ \;\;\;\;-1 \cdot \left(im \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(1 - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -1.66)
   (* (* 0.5 re) (- (exp (- im)) 2.0))
   (if (<= im 1.25) (* -1.0 (* im re)) (* (* 0.5 re) (- 1.0 (exp im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -1.66) {
		tmp = (0.5 * re) * (exp(-im) - 2.0);
	} else if (im <= 1.25) {
		tmp = -1.0 * (im * re);
	} else {
		tmp = (0.5 * re) * (1.0 - exp(im));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= (-1.66d0)) then
        tmp = (0.5d0 * re) * (exp(-im) - 2.0d0)
    else if (im <= 1.25d0) then
        tmp = (-1.0d0) * (im * re)
    else
        tmp = (0.5d0 * re) * (1.0d0 - exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -1.66) {
		tmp = (0.5 * re) * (Math.exp(-im) - 2.0);
	} else if (im <= 1.25) {
		tmp = -1.0 * (im * re);
	} else {
		tmp = (0.5 * re) * (1.0 - Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -1.66:
		tmp = (0.5 * re) * (math.exp(-im) - 2.0)
	elif im <= 1.25:
		tmp = -1.0 * (im * re)
	else:
		tmp = (0.5 * re) * (1.0 - math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -1.66)
		tmp = Float64(Float64(0.5 * re) * Float64(exp(Float64(-im)) - 2.0));
	elseif (im <= 1.25)
		tmp = Float64(-1.0 * Float64(im * re));
	else
		tmp = Float64(Float64(0.5 * re) * Float64(1.0 - exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -1.66)
		tmp = (0.5 * re) * (exp(-im) - 2.0);
	elseif (im <= 1.25)
		tmp = -1.0 * (im * re);
	else
		tmp = (0.5 * re) * (1.0 - exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -1.66], N[(N[(0.5 * re), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.25], N[(-1.0 * N[(im * re), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(1.0 - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.65999999999999992

   1. Initial program 66.3%

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

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 52.7%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 37.3%

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

   9. Applied rewrites 21.8%

      \[\leadsto \left(0.5 \cdot re\right) \cdot \left(e^{-im} - \color{blue}{2}\right) \]

   if -1.65999999999999992 < im < 1.25

   1. Initial program 66.3%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 51.0%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 33.2%

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

   if 1.25 < im

   1. Initial program 66.3%

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

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 52.7%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 33.6%

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

Alternative 8: 48.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 31000000000000:\\ \;\;\;\;-1 \cdot \left(im \cdot \left(re \cdot \left(1 + \left(-0.16666666666666666 \cdot re\right) \cdot re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(1 - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 31000000000000.0)
   (* -1.0 (* im (* re (+ 1.0 (* (* -0.16666666666666666 re) re)))))
   (* (* 0.5 re) (- 1.0 (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 31000000000000.0) {
		tmp = -1.0 * (im * (re * (1.0 + ((-0.16666666666666666 * re) * re))));
	} else {
		tmp = (0.5 * re) * (1.0 - exp(im));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 31000000000000.0d0) then
        tmp = (-1.0d0) * (im * (re * (1.0d0 + (((-0.16666666666666666d0) * re) * re))))
    else
        tmp = (0.5d0 * re) * (1.0d0 - exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 31000000000000.0) {
		tmp = -1.0 * (im * (re * (1.0 + ((-0.16666666666666666 * re) * re))));
	} else {
		tmp = (0.5 * re) * (1.0 - Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 31000000000000.0:
		tmp = -1.0 * (im * (re * (1.0 + ((-0.16666666666666666 * re) * re))))
	else:
		tmp = (0.5 * re) * (1.0 - math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 31000000000000.0)
		tmp = Float64(-1.0 * Float64(im * Float64(re * Float64(1.0 + Float64(Float64(-0.16666666666666666 * re) * re)))));
	else
		tmp = Float64(Float64(0.5 * re) * Float64(1.0 - exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 31000000000000.0)
		tmp = -1.0 * (im * (re * (1.0 + ((-0.16666666666666666 * re) * re))));
	else
		tmp = (0.5 * re) * (1.0 - exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 31000000000000.0], N[(-1.0 * N[(im * N[(re * N[(1.0 + N[(N[(-0.16666666666666666 * re), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(1.0 - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 3.1e13

   1. Initial program 66.3%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 51.0%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 36.8%

      \[\leadsto -1 \cdot \left(im \cdot \left(re \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot {re}^{2}\right)}\right)\right) \]

   9. Applied rewrites 36.8%

      \[\leadsto -1 \cdot \left(im \cdot \left(re \cdot \left(1 + \left(-0.16666666666666666 \cdot re\right) \cdot re\right)\right)\right) \]

   if 3.1e13 < im

   1. Initial program 66.3%

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

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 52.7%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 33.6%

      \[\leadsto \left(0.5 \cdot re\right) \cdot \left(\color{blue}{1} - e^{im}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 45.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ \mathbf{if}\;t\_0 \leq -0.03:\\ \;\;\;\;-1 \cdot \left(im \cdot \left(re \cdot \left(\mathsf{fma}\left(re, -0.16666666666666666, 2\right) \cdot 2\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;-1 \cdot \left(im \cdot \left(re \cdot \left(1 + \left(-0.16666666666666666 \cdot re\right) \cdot re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re))))
   (if (<= t_0 -0.03)
     (* -1.0 (* im (* re (* (fma re -0.16666666666666666 2.0) 2.0))))
     (if (<= t_0 5e-5)
       (* -1.0 (* im (* re (+ 1.0 (* (* -0.16666666666666666 re) re)))))
       (* (* (fma re -0.08333333333333333 0.5) re) im)))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sin(re);
	double tmp;
	if (t_0 <= -0.03) {
		tmp = -1.0 * (im * (re * (fma(re, -0.16666666666666666, 2.0) * 2.0)));
	} else if (t_0 <= 5e-5) {
		tmp = -1.0 * (im * (re * (1.0 + ((-0.16666666666666666 * re) * re))));
	} else {
		tmp = (fma(re, -0.08333333333333333, 0.5) * re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * sin(re))
	tmp = 0.0
	if (t_0 <= -0.03)
		tmp = Float64(-1.0 * Float64(im * Float64(re * Float64(fma(re, -0.16666666666666666, 2.0) * 2.0))));
	elseif (t_0 <= 5e-5)
		tmp = Float64(-1.0 * Float64(im * Float64(re * Float64(1.0 + Float64(Float64(-0.16666666666666666 * re) * re)))));
	else
		tmp = Float64(Float64(fma(re, -0.08333333333333333, 0.5) * re) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.03], N[(-1.0 * N[(im * N[(re * N[(N[(re * -0.16666666666666666 + 2.0), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-5], N[(-1.0 * N[(im * N[(re * N[(1.0 + N[(N[(-0.16666666666666666 * re), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(re * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * im), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < -0.029999999999999999

   1. Initial program 66.3%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 51.0%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 36.8%

      \[\leadsto -1 \cdot \left(im \cdot \left(re \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot {re}^{2}\right)}\right)\right) \]

   9. Applied rewrites 25.4%

      \[\leadsto -1 \cdot \left(im \cdot \left(re \cdot \left(\mathsf{fma}\left(re, -0.16666666666666666, 2\right) \cdot 2\right)\right)\right) \]

   if -0.029999999999999999 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < 5.00000000000000024e-5

   1. Initial program 66.3%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 51.0%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 36.8%

      \[\leadsto -1 \cdot \left(im \cdot \left(re \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot {re}^{2}\right)}\right)\right) \]

   9. Applied rewrites 36.8%

      \[\leadsto -1 \cdot \left(im \cdot \left(re \cdot \left(1 + \left(-0.16666666666666666 \cdot re\right) \cdot re\right)\right)\right) \]

   if 5.00000000000000024e-5 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 66.3%

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

   2. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(e^{-im} - e^{im}\right) \]

   4. Applied rewrites 52.2%

      \[\leadsto \color{blue}{\left(re \cdot \left(0.5 + -0.08333333333333333 \cdot {re}^{2}\right)\right)} \cdot \left(e^{-im} - e^{im}\right) \]

   6. Applied rewrites 52.5%

      \[\leadsto \left(\mathsf{fma}\left(re, -0.08333333333333333, 0.5\right) \cdot \color{blue}{re}\right) \cdot \left(e^{-im} - e^{im}\right) \]

   8. Applied rewrites 23.0%

      \[\leadsto \left(\mathsf{fma}\left(re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \color{blue}{im} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 44.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-26}:\\ \;\;\;\;-1 \cdot \left(im \cdot \left(re \cdot \left(\mathsf{fma}\left(re, -0.16666666666666666, 2\right) \cdot 2\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0.054:\\ \;\;\;\;-1 \cdot \left(im \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re))))
   (if (<= t_0 -5e-26)
     (* -1.0 (* im (* re (* (fma re -0.16666666666666666 2.0) 2.0))))
     (if (<= t_0 0.054)
       (* -1.0 (* im re))
       (* (* (fma re -0.08333333333333333 0.5) re) im)))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sin(re);
	double tmp;
	if (t_0 <= -5e-26) {
		tmp = -1.0 * (im * (re * (fma(re, -0.16666666666666666, 2.0) * 2.0)));
	} else if (t_0 <= 0.054) {
		tmp = -1.0 * (im * re);
	} else {
		tmp = (fma(re, -0.08333333333333333, 0.5) * re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * sin(re))
	tmp = 0.0
	if (t_0 <= -5e-26)
		tmp = Float64(-1.0 * Float64(im * Float64(re * Float64(fma(re, -0.16666666666666666, 2.0) * 2.0))));
	elseif (t_0 <= 0.054)
		tmp = Float64(-1.0 * Float64(im * re));
	else
		tmp = Float64(Float64(fma(re, -0.08333333333333333, 0.5) * re) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-26], N[(-1.0 * N[(im * N[(re * N[(N[(re * -0.16666666666666666 + 2.0), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.054], N[(-1.0 * N[(im * re), $MachinePrecision]), $MachinePrecision], N[(N[(N[(re * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * im), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < -5.00000000000000019e-26

   1. Initial program 66.3%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 51.0%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 36.8%

      \[\leadsto -1 \cdot \left(im \cdot \left(re \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot {re}^{2}\right)}\right)\right) \]

   9. Applied rewrites 25.4%

      \[\leadsto -1 \cdot \left(im \cdot \left(re \cdot \left(\mathsf{fma}\left(re, -0.16666666666666666, 2\right) \cdot 2\right)\right)\right) \]

   if -5.00000000000000019e-26 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < 0.0539999999999999994

   1. Initial program 66.3%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 51.0%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 33.2%

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

   if 0.0539999999999999994 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 66.3%

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

   2. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(e^{-im} - e^{im}\right) \]

   4. Applied rewrites 52.2%

      \[\leadsto \color{blue}{\left(re \cdot \left(0.5 + -0.08333333333333333 \cdot {re}^{2}\right)\right)} \cdot \left(e^{-im} - e^{im}\right) \]

   6. Applied rewrites 52.5%

      \[\leadsto \left(\mathsf{fma}\left(re, -0.08333333333333333, 0.5\right) \cdot \color{blue}{re}\right) \cdot \left(e^{-im} - e^{im}\right) \]

   8. Applied rewrites 23.0%

      \[\leadsto \left(\mathsf{fma}\left(re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \color{blue}{im} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 44.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-26}:\\ \;\;\;\;-1 \cdot \left(im \cdot \left(re \cdot \mathsf{fma}\left(re, -0.16666666666666666, 2\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0.054:\\ \;\;\;\;-1 \cdot \left(im \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re))))
   (if (<= t_0 -5e-26)
     (* -1.0 (* im (* re (fma re -0.16666666666666666 2.0))))
     (if (<= t_0 0.054)
       (* -1.0 (* im re))
       (* (* (fma re -0.08333333333333333 0.5) re) im)))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sin(re);
	double tmp;
	if (t_0 <= -5e-26) {
		tmp = -1.0 * (im * (re * fma(re, -0.16666666666666666, 2.0)));
	} else if (t_0 <= 0.054) {
		tmp = -1.0 * (im * re);
	} else {
		tmp = (fma(re, -0.08333333333333333, 0.5) * re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * sin(re))
	tmp = 0.0
	if (t_0 <= -5e-26)
		tmp = Float64(-1.0 * Float64(im * Float64(re * fma(re, -0.16666666666666666, 2.0))));
	elseif (t_0 <= 0.054)
		tmp = Float64(-1.0 * Float64(im * re));
	else
		tmp = Float64(Float64(fma(re, -0.08333333333333333, 0.5) * re) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-26], N[(-1.0 * N[(im * N[(re * N[(re * -0.16666666666666666 + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.054], N[(-1.0 * N[(im * re), $MachinePrecision]), $MachinePrecision], N[(N[(N[(re * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * im), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < -5.00000000000000019e-26

   1. Initial program 66.3%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 51.0%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 36.8%

      \[\leadsto -1 \cdot \left(im \cdot \left(re \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot {re}^{2}\right)}\right)\right) \]

   9. Applied rewrites 25.6%

      \[\leadsto -1 \cdot \left(im \cdot \left(re \cdot \mathsf{fma}\left(re, \color{blue}{-0.16666666666666666}, 2\right)\right)\right) \]

   if -5.00000000000000019e-26 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < 0.0539999999999999994

   1. Initial program 66.3%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 51.0%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 33.2%

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

   if 0.0539999999999999994 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 66.3%

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

   2. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(e^{-im} - e^{im}\right) \]

   4. Applied rewrites 52.2%

      \[\leadsto \color{blue}{\left(re \cdot \left(0.5 + -0.08333333333333333 \cdot {re}^{2}\right)\right)} \cdot \left(e^{-im} - e^{im}\right) \]

   6. Applied rewrites 52.5%

      \[\leadsto \left(\mathsf{fma}\left(re, -0.08333333333333333, 0.5\right) \cdot \color{blue}{re}\right) \cdot \left(e^{-im} - e^{im}\right) \]

   8. Applied rewrites 23.0%

      \[\leadsto \left(\mathsf{fma}\left(re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \color{blue}{im} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 44.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ \mathbf{if}\;t\_0 \leq -0.02:\\ \;\;\;\;-1 \cdot \left(im \cdot \left(re \cdot \left(-0.16666666666666666 \cdot re\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0.054:\\ \;\;\;\;-1 \cdot \left(im \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re))))
   (if (<= t_0 -0.02)
     (* -1.0 (* im (* re (* -0.16666666666666666 re))))
     (if (<= t_0 0.054)
       (* -1.0 (* im re))
       (* (* (fma re -0.08333333333333333 0.5) re) im)))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sin(re);
	double tmp;
	if (t_0 <= -0.02) {
		tmp = -1.0 * (im * (re * (-0.16666666666666666 * re)));
	} else if (t_0 <= 0.054) {
		tmp = -1.0 * (im * re);
	} else {
		tmp = (fma(re, -0.08333333333333333, 0.5) * re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * sin(re))
	tmp = 0.0
	if (t_0 <= -0.02)
		tmp = Float64(-1.0 * Float64(im * Float64(re * Float64(-0.16666666666666666 * re))));
	elseif (t_0 <= 0.054)
		tmp = Float64(-1.0 * Float64(im * re));
	else
		tmp = Float64(Float64(fma(re, -0.08333333333333333, 0.5) * re) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.02], N[(-1.0 * N[(im * N[(re * N[(-0.16666666666666666 * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.054], N[(-1.0 * N[(im * re), $MachinePrecision]), $MachinePrecision], N[(N[(N[(re * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * im), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < -0.0200000000000000004

   1. Initial program 66.3%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 51.0%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 36.8%

      \[\leadsto -1 \cdot \left(im \cdot \left(re \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot {re}^{2}\right)}\right)\right) \]

   9. Applied rewrites 25.6%

      \[\leadsto -1 \cdot \left(im \cdot \left(re \cdot \mathsf{fma}\left(re, \color{blue}{-0.16666666666666666}, 2\right)\right)\right) \]

   10. Taylor expanded in re around inf

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

   12. Applied rewrites 23.7%

       \[\leadsto -1 \cdot \left(im \cdot \left(re \cdot \left(-0.16666666666666666 \cdot re\right)\right)\right) \]

   if -0.0200000000000000004 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < 0.0539999999999999994

   1. Initial program 66.3%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 51.0%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 33.2%

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

   if 0.0539999999999999994 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 66.3%

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

   2. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(e^{-im} - e^{im}\right) \]

   4. Applied rewrites 52.2%

      \[\leadsto \color{blue}{\left(re \cdot \left(0.5 + -0.08333333333333333 \cdot {re}^{2}\right)\right)} \cdot \left(e^{-im} - e^{im}\right) \]

   6. Applied rewrites 52.5%

      \[\leadsto \left(\mathsf{fma}\left(re, -0.08333333333333333, 0.5\right) \cdot \color{blue}{re}\right) \cdot \left(e^{-im} - e^{im}\right) \]

   8. Applied rewrites 23.0%

      \[\leadsto \left(\mathsf{fma}\left(re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \color{blue}{im} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 38.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq 0.054:\\ \;\;\;\;-1 \cdot \left(im \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) 0.054)
   (* -1.0 (* im re))
   (* (* (fma re -0.08333333333333333 0.5) re) im)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= 0.054) {
		tmp = -1.0 * (im * re);
	} else {
		tmp = (fma(re, -0.08333333333333333, 0.5) * re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= 0.054)
		tmp = Float64(-1.0 * Float64(im * re));
	else
		tmp = Float64(Float64(fma(re, -0.08333333333333333, 0.5) * re) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], 0.054], N[(-1.0 * N[(im * re), $MachinePrecision]), $MachinePrecision], N[(N[(N[(re * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < 0.0539999999999999994

   1. Initial program 66.3%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 51.0%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 33.2%

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

   if 0.0539999999999999994 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 66.3%

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

   2. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(e^{-im} - e^{im}\right) \]

   4. Applied rewrites 52.2%

      \[\leadsto \color{blue}{\left(re \cdot \left(0.5 + -0.08333333333333333 \cdot {re}^{2}\right)\right)} \cdot \left(e^{-im} - e^{im}\right) \]

   6. Applied rewrites 52.5%

      \[\leadsto \left(\mathsf{fma}\left(re, -0.08333333333333333, 0.5\right) \cdot \color{blue}{re}\right) \cdot \left(e^{-im} - e^{im}\right) \]

   8. Applied rewrites 23.0%

      \[\leadsto \left(\mathsf{fma}\left(re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \color{blue}{im} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 38.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq 0.054:\\ \;\;\;\;-1 \cdot \left(im \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(im \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) 0.054) (* -1.0 (* im re)) (* -1.0 (* im (* re re)))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= 0.054) {
		tmp = -1.0 * (im * re);
	} else {
		tmp = -1.0 * (im * (re * re));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((0.5d0 * sin(re)) <= 0.054d0) then
        tmp = (-1.0d0) * (im * re)
    else
        tmp = (-1.0d0) * (im * (re * re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((0.5 * Math.sin(re)) <= 0.054) {
		tmp = -1.0 * (im * re);
	} else {
		tmp = -1.0 * (im * (re * re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (0.5 * math.sin(re)) <= 0.054:
		tmp = -1.0 * (im * re)
	else:
		tmp = -1.0 * (im * (re * re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= 0.054)
		tmp = Float64(-1.0 * Float64(im * re));
	else
		tmp = Float64(-1.0 * Float64(im * Float64(re * re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((0.5 * sin(re)) <= 0.054)
		tmp = -1.0 * (im * re);
	else
		tmp = -1.0 * (im * (re * re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], 0.054], N[(-1.0 * N[(im * re), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(im * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < 0.0539999999999999994

   1. Initial program 66.3%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 51.0%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 33.2%

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

   if 0.0539999999999999994 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 66.3%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 51.0%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 33.2%

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

   9. Applied rewrites 23.6%

      \[\leadsto -1 \cdot \left(im \cdot \left(re \cdot re\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 33.2% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ -1 \cdot \left(im \cdot re\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* -1.0 (* im re)))

C

double code(double re, double im) {
	return -1.0 * (im * 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) * (im * re)
end function

Java

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

Python

def code(re, im):
	return -1.0 * (im * re)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.3%

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

2. Taylor expanded in im around 0

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

4. Applied rewrites 51.0%

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

5. Taylor expanded in re around 0

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

7. Applied rewrites 33.2%

   \[\leadsto -1 \cdot \left(im \cdot \color{blue}{re}\right) \]
8. Add Preprocessing

Alternative 16: 14.9% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ -1 \cdot \left(re - re\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* -1.0 (- re re)))

C

double code(double re, double im) {
	return -1.0 * (re - 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 - re)
end function

Java

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

Python

def code(re, im):
	return -1.0 * (re - re)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.3%

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

2. Taylor expanded in im around 0

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

4. Applied rewrites 51.0%

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

5. Taylor expanded in re around 0

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

7. Applied rewrites 33.2%

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

9. Applied rewrites 3.2%

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

11. Applied rewrites 14.9%

    \[\leadsto -1 \cdot \left(re - re\right) \]
12. Add Preprocessing

Alternative 17: 3.2% accurate, 15.8× speedup

Math

\[\begin{array}{l} \\ -1 \cdot re \end{array} \]

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 66.3%

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

2. Taylor expanded in im around 0

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

4. Applied rewrites 51.0%

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

5. Taylor expanded in re around 0

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

7. Applied rewrites 33.2%

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

9. Applied rewrites 3.2%

   \[\leadsto -1 \cdot \color{blue}{re} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025159 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"
  :precision binary64
  (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))