math.sin on complex, imaginary part

Percentage Accurate: 54.9% → 99.9%

Time: 5.4s

Alternatives: 16

Speedup: 1.4×

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

Specification

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * cos(re)) * (exp((0.0 - 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 * cos(re)) * (exp((0.0d0 - im)) - exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.cos(re)) * (Math.exp((0.0 - im)) - Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.cos(re)) * (math.exp((0.0 - im)) - math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(0.0 - im)) - exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
end

Wolfram

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

Initial Program: 54.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * cos(re)) * (exp((0.0 - 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 * cos(re)) * (exp((0.0d0 - im)) - exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.cos(re)) * (Math.exp((0.0 - im)) - Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.cos(re)) * (math.exp((0.0 - im)) - math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(0.0 - im)) - exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
end

Wolfram

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

Alternative 1: 99.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	return ((2.0 * sinh(-im)) * cos(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)) * cos(re)) * 0.5d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.9%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

3. Applied rewrites 99.9%

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

Alternative 2: 86.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{0 - im}\\ \mathbf{if}\;im \leq -5.5:\\ \;\;\;\;\mathsf{fma}\left(re, -0.25, 0.5\right) \cdot \left(t\_0 - \left(1 + im\right)\right)\\ \mathbf{elif}\;im \leq 0.0102:\\ \;\;\;\;-1 \cdot \left(im \cdot \cos re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, -0.25, 0.5\right) \cdot \left(t\_0 - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (exp (- 0.0 im))))
   (if (<= im -5.5)
     (* (fma re -0.25 0.5) (- t_0 (+ 1.0 im)))
     (if (<= im 0.0102)
       (* -1.0 (* im (cos re)))
       (* (fma re -0.25 0.5) (- t_0 (exp im)))))))

C

double code(double re, double im) {
	double t_0 = exp((0.0 - im));
	double tmp;
	if (im <= -5.5) {
		tmp = fma(re, -0.25, 0.5) * (t_0 - (1.0 + im));
	} else if (im <= 0.0102) {
		tmp = -1.0 * (im * cos(re));
	} else {
		tmp = fma(re, -0.25, 0.5) * (t_0 - exp(im));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = exp(Float64(0.0 - im))
	tmp = 0.0
	if (im <= -5.5)
		tmp = Float64(fma(re, -0.25, 0.5) * Float64(t_0 - Float64(1.0 + im)));
	elseif (im <= 0.0102)
		tmp = Float64(-1.0 * Float64(im * cos(re)));
	else
		tmp = Float64(fma(re, -0.25, 0.5) * Float64(t_0 - exp(im)));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[im, -5.5], N[(N[(re * -0.25 + 0.5), $MachinePrecision] * N[(t$95$0 - N[(1.0 + im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 0.0102], N[(-1.0 * N[(im * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(re * -0.25 + 0.5), $MachinePrecision] * N[(t$95$0 - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < -5.5

   1. Initial program 54.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

   2. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \cdot \left(e^{0 - im} - e^{im}\right) \]

   4. Applied rewrites 40.8%

      \[\leadsto \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \cdot \left(e^{0 - im} - e^{im}\right) \]

   6. Applied rewrites 41.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, -0.25, 0.5\right)} \cdot \left(e^{0 - im} - e^{im}\right) \]

   7. Taylor expanded in im around 0

      \[\leadsto \mathsf{fma}\left(re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(e^{0 - im} - \color{blue}{\left(1 + im\right)}\right) \]

   9. Applied rewrites 26.1%

      \[\leadsto \mathsf{fma}\left(re, -0.25, 0.5\right) \cdot \left(e^{0 - im} - \color{blue}{\left(1 + im\right)}\right) \]

   if -5.5 < im < 0.010200000000000001

   1. Initial program 54.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 51.5%

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

   if 0.010200000000000001 < im

   1. Initial program 54.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

   2. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \cdot \left(e^{0 - im} - e^{im}\right) \]

   4. Applied rewrites 40.8%

      \[\leadsto \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \cdot \left(e^{0 - im} - e^{im}\right) \]

   6. Applied rewrites 41.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, -0.25, 0.5\right)} \cdot \left(e^{0 - im} - e^{im}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 76.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \cos re \leq -0.01:\\ \;\;\;\;\left(-0.25 \cdot re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \sinh \left(-im\right)\right) \cdot 1\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (cos re)) -0.01)
   (* (* -0.25 re) (- (exp (- 0.0 im)) (exp im)))
   (* (* (* 2.0 (sinh (- im))) 1.0) 0.5)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * cos(re)) <= -0.01) {
		tmp = (-0.25 * re) * (exp((0.0 - im)) - exp(im));
	} else {
		tmp = ((2.0 * sinh(-im)) * 1.0) * 0.5;
	}
	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 * cos(re)) <= (-0.01d0)) then
        tmp = ((-0.25d0) * re) * (exp((0.0d0 - im)) - exp(im))
    else
        tmp = ((2.0d0 * sinh(-im)) * 1.0d0) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((0.5 * Math.cos(re)) <= -0.01) {
		tmp = (-0.25 * re) * (Math.exp((0.0 - im)) - Math.exp(im));
	} else {
		tmp = ((2.0 * Math.sinh(-im)) * 1.0) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (0.5 * math.cos(re)) <= -0.01:
		tmp = (-0.25 * re) * (math.exp((0.0 - im)) - math.exp(im))
	else:
		tmp = ((2.0 * math.sinh(-im)) * 1.0) * 0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * cos(re)) <= -0.01)
		tmp = Float64(Float64(-0.25 * re) * Float64(exp(Float64(0.0 - im)) - exp(im)));
	else
		tmp = Float64(Float64(Float64(2.0 * sinh(Float64(-im))) * 1.0) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((0.5 * cos(re)) <= -0.01)
		tmp = (-0.25 * re) * (exp((0.0 - im)) - exp(im));
	else
		tmp = ((2.0 * sinh(-im)) * 1.0) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision], -0.01], N[(N[(-0.25 * re), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * N[Sinh[(-im)], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) < -0.0100000000000000002

   1. Initial program 54.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

   2. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \cdot \left(e^{0 - im} - e^{im}\right) \]

   4. Applied rewrites 40.8%

      \[\leadsto \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \cdot \left(e^{0 - im} - e^{im}\right) \]

   6. Applied rewrites 41.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, -0.25, 0.5\right)} \cdot \left(e^{0 - im} - e^{im}\right) \]

   7. Taylor expanded in re around inf

      \[\leadsto \left(\frac{-1}{4} \cdot \color{blue}{re}\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

   9. Applied rewrites 28.1%

      \[\leadsto \left(-0.25 \cdot \color{blue}{re}\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

   if -0.0100000000000000002 < (*.f64 #s(literal 1/2 binary64) (cos.f64 re))

   1. Initial program 54.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in re around 0

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

   6. Applied rewrites 65.8%

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

Alternative 4: 75.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \cos re \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(-1, im, 0.5 \cdot \left(im \cdot {\left(re \cdot re\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \sinh \left(-im\right)\right) \cdot 1\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (cos re)) -0.02)
   (fma -1.0 im (* 0.5 (* im (pow (* re re) 2.0))))
   (* (* (* 2.0 (sinh (- im))) 1.0) 0.5)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * cos(re)) <= -0.02) {
		tmp = fma(-1.0, im, (0.5 * (im * pow((re * re), 2.0))));
	} else {
		tmp = ((2.0 * sinh(-im)) * 1.0) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * cos(re)) <= -0.02)
		tmp = fma(-1.0, im, Float64(0.5 * Float64(im * (Float64(re * re) ^ 2.0))));
	else
		tmp = Float64(Float64(Float64(2.0 * sinh(Float64(-im))) * 1.0) * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision], -0.02], N[(-1.0 * im + N[(0.5 * N[(im * N[Power[N[(re * re), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * N[Sinh[(-im)], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 54.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 51.5%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 36.5%

      \[\leadsto \mathsf{fma}\left(-1, \color{blue}{im}, 0.5 \cdot \left(im \cdot {re}^{2}\right)\right) \]

   9. Applied rewrites 37.6%

      \[\leadsto \mathsf{fma}\left(-1, im, 0.5 \cdot \left(im \cdot {\left(re \cdot re\right)}^{2}\right)\right) \]

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

   1. Initial program 54.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in re around 0

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

   6. Applied rewrites 65.8%

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

Alternative 5: 72.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \cos re \leq -0.02:\\ \;\;\;\;im \cdot \left(0.5 \cdot {re}^{2} - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \sinh \left(-im\right)\right) \cdot 1\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (cos re)) -0.02)
   (* im (- (* 0.5 (pow re 2.0)) 1.0))
   (* (* (* 2.0 (sinh (- im))) 1.0) 0.5)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * cos(re)) <= -0.02) {
		tmp = im * ((0.5 * pow(re, 2.0)) - 1.0);
	} else {
		tmp = ((2.0 * sinh(-im)) * 1.0) * 0.5;
	}
	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 * cos(re)) <= (-0.02d0)) then
        tmp = im * ((0.5d0 * (re ** 2.0d0)) - 1.0d0)
    else
        tmp = ((2.0d0 * sinh(-im)) * 1.0d0) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((0.5 * Math.cos(re)) <= -0.02) {
		tmp = im * ((0.5 * Math.pow(re, 2.0)) - 1.0);
	} else {
		tmp = ((2.0 * Math.sinh(-im)) * 1.0) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (0.5 * math.cos(re)) <= -0.02:
		tmp = im * ((0.5 * math.pow(re, 2.0)) - 1.0)
	else:
		tmp = ((2.0 * math.sinh(-im)) * 1.0) * 0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * cos(re)) <= -0.02)
		tmp = Float64(im * Float64(Float64(0.5 * (re ^ 2.0)) - 1.0));
	else
		tmp = Float64(Float64(Float64(2.0 * sinh(Float64(-im))) * 1.0) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((0.5 * cos(re)) <= -0.02)
		tmp = im * ((0.5 * (re ^ 2.0)) - 1.0);
	else
		tmp = ((2.0 * sinh(-im)) * 1.0) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision], -0.02], N[(im * N[(N[(0.5 * N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * N[Sinh[(-im)], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 54.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 51.5%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 36.5%

      \[\leadsto \mathsf{fma}\left(-1, \color{blue}{im}, 0.5 \cdot \left(im \cdot {re}^{2}\right)\right) \]

   8. Taylor expanded in im around 0

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

   10. Applied rewrites 36.4%

       \[\leadsto im \cdot \left(0.5 \cdot {re}^{2} - \color{blue}{1}\right) \]

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

   1. Initial program 54.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in re around 0

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

   6. Applied rewrites 65.8%

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

Alternative 6: 69.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \cos re \leq -0.01:\\ \;\;\;\;0.5 \cdot \left(\left(1 + -1 \cdot im\right) - \left(1 + im \cdot \left(1 + 0.5 \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \sinh \left(-im\right)\right) \cdot 1\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (cos re)) -0.01)
   (* 0.5 (- (+ 1.0 (* -1.0 im)) (+ 1.0 (* im (+ 1.0 (* 0.5 im))))))
   (* (* (* 2.0 (sinh (- im))) 1.0) 0.5)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * cos(re)) <= -0.01) {
		tmp = 0.5 * ((1.0 + (-1.0 * im)) - (1.0 + (im * (1.0 + (0.5 * im)))));
	} else {
		tmp = ((2.0 * sinh(-im)) * 1.0) * 0.5;
	}
	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 * cos(re)) <= (-0.01d0)) then
        tmp = 0.5d0 * ((1.0d0 + ((-1.0d0) * im)) - (1.0d0 + (im * (1.0d0 + (0.5d0 * im)))))
    else
        tmp = ((2.0d0 * sinh(-im)) * 1.0d0) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((0.5 * Math.cos(re)) <= -0.01) {
		tmp = 0.5 * ((1.0 + (-1.0 * im)) - (1.0 + (im * (1.0 + (0.5 * im)))));
	} else {
		tmp = ((2.0 * Math.sinh(-im)) * 1.0) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (0.5 * math.cos(re)) <= -0.01:
		tmp = 0.5 * ((1.0 + (-1.0 * im)) - (1.0 + (im * (1.0 + (0.5 * im)))))
	else:
		tmp = ((2.0 * math.sinh(-im)) * 1.0) * 0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * cos(re)) <= -0.01)
		tmp = Float64(0.5 * Float64(Float64(1.0 + Float64(-1.0 * im)) - Float64(1.0 + Float64(im * Float64(1.0 + Float64(0.5 * im))))));
	else
		tmp = Float64(Float64(Float64(2.0 * sinh(Float64(-im))) * 1.0) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((0.5 * cos(re)) <= -0.01)
		tmp = 0.5 * ((1.0 + (-1.0 * im)) - (1.0 + (im * (1.0 + (0.5 * im)))));
	else
		tmp = ((2.0 * sinh(-im)) * 1.0) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision], -0.01], N[(0.5 * N[(N[(1.0 + N[(-1.0 * im), $MachinePrecision]), $MachinePrecision] - N[(1.0 + N[(im * N[(1.0 + N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * N[Sinh[(-im)], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) < -0.0100000000000000002

   1. Initial program 54.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 41.5%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 23.2%

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

   8. Taylor expanded in im around 0

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

   10. Applied rewrites 16.4%

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

   if -0.0100000000000000002 < (*.f64 #s(literal 1/2 binary64) (cos.f64 re))

   1. Initial program 54.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in re around 0

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

   6. Applied rewrites 65.8%

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

Alternative 7: 68.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \cos re \leq -0.09:\\ \;\;\;\;0.5 \cdot \left(2 - \left(1 + im \cdot \left(1 + 0.5 \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \sinh \left(-im\right)\right) \cdot 1\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (cos re)) -0.09)
   (* 0.5 (- 2.0 (+ 1.0 (* im (+ 1.0 (* 0.5 im))))))
   (* (* (* 2.0 (sinh (- im))) 1.0) 0.5)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * cos(re)) <= -0.09) {
		tmp = 0.5 * (2.0 - (1.0 + (im * (1.0 + (0.5 * im)))));
	} else {
		tmp = ((2.0 * sinh(-im)) * 1.0) * 0.5;
	}
	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 * cos(re)) <= (-0.09d0)) then
        tmp = 0.5d0 * (2.0d0 - (1.0d0 + (im * (1.0d0 + (0.5d0 * im)))))
    else
        tmp = ((2.0d0 * sinh(-im)) * 1.0d0) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((0.5 * Math.cos(re)) <= -0.09) {
		tmp = 0.5 * (2.0 - (1.0 + (im * (1.0 + (0.5 * im)))));
	} else {
		tmp = ((2.0 * Math.sinh(-im)) * 1.0) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (0.5 * math.cos(re)) <= -0.09:
		tmp = 0.5 * (2.0 - (1.0 + (im * (1.0 + (0.5 * im)))))
	else:
		tmp = ((2.0 * math.sinh(-im)) * 1.0) * 0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * cos(re)) <= -0.09)
		tmp = Float64(0.5 * Float64(2.0 - Float64(1.0 + Float64(im * Float64(1.0 + Float64(0.5 * im))))));
	else
		tmp = Float64(Float64(Float64(2.0 * sinh(Float64(-im))) * 1.0) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((0.5 * cos(re)) <= -0.09)
		tmp = 0.5 * (2.0 - (1.0 + (im * (1.0 + (0.5 * im)))));
	else
		tmp = ((2.0 * sinh(-im)) * 1.0) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision], -0.09], N[(0.5 * N[(2.0 - N[(1.0 + N[(im * N[(1.0 + N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * N[Sinh[(-im)], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) < -0.089999999999999997

   1. Initial program 54.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 41.5%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 23.2%

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

   8. Taylor expanded in im around 0

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

   10. Applied rewrites 16.4%

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

   12. Applied rewrites 15.0%

       \[\leadsto 0.5 \cdot \left(2 - \left(\color{blue}{1} + im \cdot \left(1 + 0.5 \cdot im\right)\right)\right) \]

   if -0.089999999999999997 < (*.f64 #s(literal 1/2 binary64) (cos.f64 re))

   1. Initial program 54.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in re around 0

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

   6. Applied rewrites 65.8%

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

Alternative 8: 65.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;0.5 \cdot \left(2 - e^{im}\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;-1 \cdot im\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - \left(1 + im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (* 0.5 (- 2.0 (exp im)))
     (if (<= t_0 2e-5) (* -1.0 im) (* 0.5 (- (exp (- im)) (+ 1.0 im)))))))

C

double code(double re, double im) {
	double t_0 = (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = 0.5 * (2.0 - exp(im));
	} else if (t_0 <= 2e-5) {
		tmp = -1.0 * im;
	} else {
		tmp = 0.5 * (exp(-im) - (1.0 + im));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = (0.5 * Math.cos(re)) * (Math.exp((0.0 - im)) - Math.exp(im));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = 0.5 * (2.0 - Math.exp(im));
	} else if (t_0 <= 2e-5) {
		tmp = -1.0 * im;
	} else {
		tmp = 0.5 * (Math.exp(-im) - (1.0 + im));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (0.5 * math.cos(re)) * (math.exp((0.0 - im)) - math.exp(im))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = 0.5 * (2.0 - math.exp(im))
	elif t_0 <= 2e-5:
		tmp = -1.0 * im
	else:
		tmp = 0.5 * (math.exp(-im) - (1.0 + im))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(0.0 - im)) - exp(im)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(0.5 * Float64(2.0 - exp(im)));
	elseif (t_0 <= 2e-5)
		tmp = Float64(-1.0 * im);
	else
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) - Float64(1.0 + im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = 0.5 * (2.0 - exp(im));
	elseif (t_0 <= 2e-5)
		tmp = -1.0 * im;
	else
		tmp = 0.5 * (exp(-im) - (1.0 + im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(0.5 * N[(2.0 - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-5], N[(-1.0 * im), $MachinePrecision], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] - N[(1.0 + im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -inf.0

   1. Initial program 54.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 41.5%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 23.2%

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

   9. Applied rewrites 21.4%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 2.00000000000000016e-5

   1. Initial program 54.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 51.5%

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

   5. Taylor expanded in re around 0

      \[\leadsto -1 \cdot im \]

   7. Applied rewrites 29.7%

      \[\leadsto -1 \cdot im \]

   if 2.00000000000000016e-5 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 54.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 41.5%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 23.3%

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

Alternative 9: 53.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;0.5 \cdot \left(2 - e^{im}\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;-1 \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, im, 0.5 \cdot \left(\left(2 \cdot re\right) \cdot \left(2 \cdot re\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (* 0.5 (- 2.0 (exp im)))
     (if (<= t_0 2e-5)
       (* -1.0 im)
       (fma -1.0 im (* 0.5 (* (* 2.0 re) (* 2.0 re))))))))

C

double code(double re, double im) {
	double t_0 = (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = 0.5 * (2.0 - exp(im));
	} else if (t_0 <= 2e-5) {
		tmp = -1.0 * im;
	} else {
		tmp = fma(-1.0, im, (0.5 * ((2.0 * re) * (2.0 * re))));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(0.0 - im)) - exp(im)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(0.5 * Float64(2.0 - exp(im)));
	elseif (t_0 <= 2e-5)
		tmp = Float64(-1.0 * im);
	else
		tmp = fma(-1.0, im, Float64(0.5 * Float64(Float64(2.0 * re) * Float64(2.0 * re))));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(0.5 * N[(2.0 - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-5], N[(-1.0 * im), $MachinePrecision], N[(-1.0 * im + N[(0.5 * N[(N[(2.0 * re), $MachinePrecision] * N[(2.0 * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -inf.0

   1. Initial program 54.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 41.5%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 23.2%

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

   9. Applied rewrites 21.4%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 2.00000000000000016e-5

   1. Initial program 54.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 51.5%

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

   5. Taylor expanded in re around 0

      \[\leadsto -1 \cdot im \]

   7. Applied rewrites 29.7%

      \[\leadsto -1 \cdot im \]

   if 2.00000000000000016e-5 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 54.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 51.5%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 36.5%

      \[\leadsto \mathsf{fma}\left(-1, \color{blue}{im}, 0.5 \cdot \left(im \cdot {re}^{2}\right)\right) \]

   9. Applied rewrites 27.2%

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

Alternative 10: 53.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;0.5 \cdot \left(2 - e^{im}\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;-1 \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot re\right) \cdot \left(re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (* 0.5 (- 2.0 (exp im)))
     (if (<= t_0 2e-5) (* -1.0 im) (* (* 2.0 re) (* re 0.5))))))

C

double code(double re, double im) {
	double t_0 = (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = 0.5 * (2.0 - exp(im));
	} else if (t_0 <= 2e-5) {
		tmp = -1.0 * im;
	} else {
		tmp = (2.0 * re) * (re * 0.5);
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = (0.5 * Math.cos(re)) * (Math.exp((0.0 - im)) - Math.exp(im));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = 0.5 * (2.0 - Math.exp(im));
	} else if (t_0 <= 2e-5) {
		tmp = -1.0 * im;
	} else {
		tmp = (2.0 * re) * (re * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (0.5 * math.cos(re)) * (math.exp((0.0 - im)) - math.exp(im))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = 0.5 * (2.0 - math.exp(im))
	elif t_0 <= 2e-5:
		tmp = -1.0 * im
	else:
		tmp = (2.0 * re) * (re * 0.5)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(0.0 - im)) - exp(im)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(0.5 * Float64(2.0 - exp(im)));
	elseif (t_0 <= 2e-5)
		tmp = Float64(-1.0 * im);
	else
		tmp = Float64(Float64(2.0 * re) * Float64(re * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = 0.5 * (2.0 - exp(im));
	elseif (t_0 <= 2e-5)
		tmp = -1.0 * im;
	else
		tmp = (2.0 * re) * (re * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(0.5 * N[(2.0 - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-5], N[(-1.0 * im), $MachinePrecision], N[(N[(2.0 * re), $MachinePrecision] * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -inf.0

   1. Initial program 54.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 41.5%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 23.2%

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

   9. Applied rewrites 21.4%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 2.00000000000000016e-5

   1. Initial program 54.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 51.5%

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

   5. Taylor expanded in re around 0

      \[\leadsto -1 \cdot im \]

   7. Applied rewrites 29.7%

      \[\leadsto -1 \cdot im \]

   if 2.00000000000000016e-5 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 54.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 51.5%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 36.5%

      \[\leadsto \mathsf{fma}\left(-1, \color{blue}{im}, 0.5 \cdot \left(im \cdot {re}^{2}\right)\right) \]

   9. Applied rewrites 8.8%

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

Alternative 11: 41.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-1, im, 0.5 \cdot \left(\left(2 \cdot re\right) \cdot \left(-re\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;-1 \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot re\right) \cdot \left(re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (fma -1.0 im (* 0.5 (* (* 2.0 re) (- re))))
     (if (<= t_0 2e-5) (* -1.0 im) (* (* 2.0 re) (* re 0.5))))))

C

double code(double re, double im) {
	double t_0 = (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(-1.0, im, (0.5 * ((2.0 * re) * -re)));
	} else if (t_0 <= 2e-5) {
		tmp = -1.0 * im;
	} else {
		tmp = (2.0 * re) * (re * 0.5);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(0.0 - im)) - exp(im)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = fma(-1.0, im, Float64(0.5 * Float64(Float64(2.0 * re) * Float64(-re))));
	elseif (t_0 <= 2e-5)
		tmp = Float64(-1.0 * im);
	else
		tmp = Float64(Float64(2.0 * re) * Float64(re * 0.5));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(-1.0 * im + N[(0.5 * N[(N[(2.0 * re), $MachinePrecision] * (-re)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-5], N[(-1.0 * im), $MachinePrecision], N[(N[(2.0 * re), $MachinePrecision] * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -inf.0

   1. Initial program 54.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 51.5%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 36.5%

      \[\leadsto \mathsf{fma}\left(-1, \color{blue}{im}, 0.5 \cdot \left(im \cdot {re}^{2}\right)\right) \]

   9. Applied rewrites 27.1%

      \[\leadsto \mathsf{fma}\left(-1, im, 0.5 \cdot \left(\left(2 \cdot re\right) \cdot \left(-re\right)\right)\right) \]

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 2.00000000000000016e-5

   1. Initial program 54.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 51.5%

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

   5. Taylor expanded in re around 0

      \[\leadsto -1 \cdot im \]

   7. Applied rewrites 29.7%

      \[\leadsto -1 \cdot im \]

   if 2.00000000000000016e-5 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 54.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 51.5%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 36.5%

      \[\leadsto \mathsf{fma}\left(-1, \color{blue}{im}, 0.5 \cdot \left(im \cdot {re}^{2}\right)\right) \]

   9. Applied rewrites 8.8%

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

Alternative 12: 40.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot 2\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;-1 \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot re\right) \cdot \left(re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (* (fma (* re re) -0.25 0.5) 2.0)
     (if (<= t_0 2e-5) (* -1.0 im) (* (* 2.0 re) (* re 0.5))))))

C

double code(double re, double im) {
	double t_0 = (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma((re * re), -0.25, 0.5) * 2.0;
	} else if (t_0 <= 2e-5) {
		tmp = -1.0 * im;
	} else {
		tmp = (2.0 * re) * (re * 0.5);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(0.0 - im)) - exp(im)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(Float64(re * re), -0.25, 0.5) * 2.0);
	elseif (t_0 <= 2e-5)
		tmp = Float64(-1.0 * im);
	else
		tmp = Float64(Float64(2.0 * re) * Float64(re * 0.5));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[t$95$0, 2e-5], N[(-1.0 * im), $MachinePrecision], N[(N[(2.0 * re), $MachinePrecision] * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -inf.0

   1. Initial program 54.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

   2. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \cdot \left(e^{0 - im} - e^{im}\right) \]

   4. Applied rewrites 40.8%

      \[\leadsto \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \cdot \left(e^{0 - im} - e^{im}\right) \]

   6. Applied rewrites 2.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, -0.25, 0.5\right) \cdot 2} \]

   8. Applied rewrites 8.4%

      \[\leadsto \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot 2 \]

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 2.00000000000000016e-5

   1. Initial program 54.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 51.5%

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

   5. Taylor expanded in re around 0

      \[\leadsto -1 \cdot im \]

   7. Applied rewrites 29.7%

      \[\leadsto -1 \cdot im \]

   if 2.00000000000000016e-5 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 54.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 51.5%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 36.5%

      \[\leadsto \mathsf{fma}\left(-1, \color{blue}{im}, 0.5 \cdot \left(im \cdot {re}^{2}\right)\right) \]

   9. Applied rewrites 8.8%

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

Alternative 13: 35.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \leq 2 \cdot 10^{-5}:\\ \;\;\;\;-1 \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot re\right) \cdot \left(re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))) 2e-5)
   (* -1.0 im)
   (* (* 2.0 re) (* re 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * cos(re)) * (exp((0.0 - im)) - exp(im))) <= 2e-5) {
		tmp = -1.0 * im;
	} else {
		tmp = (2.0 * re) * (re * 0.5);
	}
	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 * cos(re)) * (exp((0.0d0 - im)) - exp(im))) <= 2d-5) then
        tmp = (-1.0d0) * im
    else
        tmp = (2.0d0 * re) * (re * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (((0.5 * Math.cos(re)) * (Math.exp((0.0 - im)) - Math.exp(im))) <= 2e-5) {
		tmp = -1.0 * im;
	} else {
		tmp = (2.0 * re) * (re * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if ((0.5 * math.cos(re)) * (math.exp((0.0 - im)) - math.exp(im))) <= 2e-5:
		tmp = -1.0 * im
	else:
		tmp = (2.0 * re) * (re * 0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(0.0 - im)) - exp(im))) <= 2e-5)
		tmp = Float64(-1.0 * im);
	else
		tmp = Float64(Float64(2.0 * re) * Float64(re * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (((0.5 * cos(re)) * (exp((0.0 - im)) - exp(im))) <= 2e-5)
		tmp = -1.0 * im;
	else
		tmp = (2.0 * re) * (re * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-5], N[(-1.0 * im), $MachinePrecision], N[(N[(2.0 * re), $MachinePrecision] * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 2.00000000000000016e-5

   1. Initial program 54.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 51.5%

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

   5. Taylor expanded in re around 0

      \[\leadsto -1 \cdot im \]

   7. Applied rewrites 29.7%

      \[\leadsto -1 \cdot im \]

   if 2.00000000000000016e-5 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 54.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 51.5%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 36.5%

      \[\leadsto \mathsf{fma}\left(-1, \color{blue}{im}, 0.5 \cdot \left(im \cdot {re}^{2}\right)\right) \]

   9. Applied rewrites 8.8%

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

Alternative 14: 32.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \cos re \leq -0.01:\\ \;\;\;\;\left(im \cdot 1\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (cos re)) -0.01) (* (* im 1.0) 0.5) (* -1.0 im)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * cos(re)) <= -0.01) {
		tmp = (im * 1.0) * 0.5;
	} else {
		tmp = -1.0 * 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 ((0.5d0 * cos(re)) <= (-0.01d0)) then
        tmp = (im * 1.0d0) * 0.5d0
    else
        tmp = (-1.0d0) * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((0.5 * Math.cos(re)) <= -0.01) {
		tmp = (im * 1.0) * 0.5;
	} else {
		tmp = -1.0 * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (0.5 * math.cos(re)) <= -0.01:
		tmp = (im * 1.0) * 0.5
	else:
		tmp = -1.0 * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * cos(re)) <= -0.01)
		tmp = Float64(Float64(im * 1.0) * 0.5);
	else
		tmp = Float64(-1.0 * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((0.5 * cos(re)) <= -0.01)
		tmp = (im * 1.0) * 0.5;
	else
		tmp = -1.0 * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision], -0.01], N[(N[(im * 1.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(-1.0 * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) < -0.0100000000000000002

   1. Initial program 54.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in re around 0

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

   6. Applied rewrites 65.8%

      \[\leadsto \left(\left(2 \cdot \sinh \left(-im\right)\right) \cdot \color{blue}{1}\right) \cdot 0.5 \]

   8. Applied rewrites 4.6%

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

   if -0.0100000000000000002 < (*.f64 #s(literal 1/2 binary64) (cos.f64 re))

   1. Initial program 54.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 51.5%

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

   5. Taylor expanded in re around 0

      \[\leadsto -1 \cdot im \]

   7. Applied rewrites 29.7%

      \[\leadsto -1 \cdot im \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 29.7% accurate, 16.3× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	return -1.0 * 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 = (-1.0d0) * im
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.9%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

2. Taylor expanded in im around 0

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

4. Applied rewrites 51.5%

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

5. Taylor expanded in re around 0

   \[\leadsto -1 \cdot im \]

7. Applied rewrites 29.7%

   \[\leadsto -1 \cdot im \]
8. Add Preprocessing

Alternative 16: 3.0% accurate, 16.3× speedup

Math

\[\begin{array}{l} \\ re \cdot 0.5 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* re 0.5))

C

double code(double re, double im) {
	return 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 = re * 0.5d0
end function

Java

public static double code(double re, double im) {
	return re * 0.5;
}

Python

def code(re, im):
	return re * 0.5

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.9%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

2. Taylor expanded in im around 0

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

4. Applied rewrites 51.5%

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

5. Taylor expanded in re around 0

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

7. Applied rewrites 36.5%

   \[\leadsto \mathsf{fma}\left(-1, \color{blue}{im}, 0.5 \cdot \left(im \cdot {re}^{2}\right)\right) \]

9. Applied rewrites 3.0%

   \[\leadsto re \cdot 0.5 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025159 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"
  :precision binary64
  (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))