math.cos on complex, imaginary part

Percentage Accurate: 65.0% → 99.6%

Time: 8.3s

Alternatives: 27

Speedup: 2.6×

• 49.4% 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: 65.0% 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: 99.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ \mathbf{if}\;im \leq -2.95:\\ \;\;\;\;t\_0 \cdot \left(e^{-im} - 1\right)\\ \mathbf{elif}\;im \leq 3:\\ \;\;\;\;\mathsf{fma}\left(\sin re \cdot \mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right), im \cdot im, -\sin re\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(1 - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re))))
   (if (<= im -2.95)
     (* t_0 (- (exp (- im)) 1.0))
     (if (<= im 3.0)
       (*
        (fma
         (*
          (sin re)
          (fma -0.008333333333333333 (* im im) -0.16666666666666666))
         (* im im)
         (- (sin re)))
        im)
       (* t_0 (- 1.0 (exp im)))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sin(re);
	double tmp;
	if (im <= -2.95) {
		tmp = t_0 * (exp(-im) - 1.0);
	} else if (im <= 3.0) {
		tmp = fma((sin(re) * fma(-0.008333333333333333, (im * im), -0.16666666666666666)), (im * im), -sin(re)) * im;
	} else {
		tmp = t_0 * (1.0 - exp(im));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * sin(re))
	tmp = 0.0
	if (im <= -2.95)
		tmp = Float64(t_0 * Float64(exp(Float64(-im)) - 1.0));
	elseif (im <= 3.0)
		tmp = Float64(fma(Float64(sin(re) * fma(-0.008333333333333333, Float64(im * im), -0.16666666666666666)), Float64(im * im), Float64(-sin(re))) * im);
	else
		tmp = Float64(t_0 * Float64(1.0 - exp(im)));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -2.95], N[(t$95$0 * N[(N[Exp[(-im)], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 3.0], N[(N[(N[(N[Sin[re], $MachinePrecision] * N[(-0.008333333333333333 * N[(im * im), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision] + (-N[Sin[re], $MachinePrecision])), $MachinePrecision] * im), $MachinePrecision], N[(t$95$0 * N[(1.0 - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < -2.9500000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 100.0%

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

   if -2.9500000000000002 < im < 3

   1. Initial program 32.5%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot \color{blue}{im} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot \color{blue}{im} \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sin re \cdot \mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right), im \cdot im, -\sin re\right) \cdot im} \]

   if 3 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 100.0%

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

Alternative 2: 99.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ \mathbf{if}\;im \leq -2.95:\\ \;\;\;\;t\_0 \cdot \left(e^{-im} - 1\right)\\ \mathbf{elif}\;im \leq 3:\\ \;\;\;\;t\_0 \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(1 - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re))))
   (if (<= im -2.95)
     (* t_0 (- (exp (- im)) 1.0))
     (if (<= im 3.0)
       (*
        t_0
        (*
         (-
          (*
           (* (- (* -0.016666666666666666 (* im im)) 0.3333333333333333) im)
           im)
          2.0)
         im))
       (* t_0 (- 1.0 (exp im)))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sin(re);
	double tmp;
	if (im <= -2.95) {
		tmp = t_0 * (exp(-im) - 1.0);
	} else if (im <= 3.0) {
		tmp = t_0 * ((((((-0.016666666666666666 * (im * im)) - 0.3333333333333333) * im) * im) - 2.0) * im);
	} else {
		tmp = t_0 * (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) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * sin(re)
    if (im <= (-2.95d0)) then
        tmp = t_0 * (exp(-im) - 1.0d0)
    else if (im <= 3.0d0) then
        tmp = t_0 * (((((((-0.016666666666666666d0) * (im * im)) - 0.3333333333333333d0) * im) * im) - 2.0d0) * im)
    else
        tmp = t_0 * (1.0d0 - exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.sin(re);
	double tmp;
	if (im <= -2.95) {
		tmp = t_0 * (Math.exp(-im) - 1.0);
	} else if (im <= 3.0) {
		tmp = t_0 * ((((((-0.016666666666666666 * (im * im)) - 0.3333333333333333) * im) * im) - 2.0) * im);
	} else {
		tmp = t_0 * (1.0 - Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sin(re)
	tmp = 0
	if im <= -2.95:
		tmp = t_0 * (math.exp(-im) - 1.0)
	elif im <= 3.0:
		tmp = t_0 * ((((((-0.016666666666666666 * (im * im)) - 0.3333333333333333) * im) * im) - 2.0) * im)
	else:
		tmp = t_0 * (1.0 - math.exp(im))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sin(re))
	tmp = 0.0
	if (im <= -2.95)
		tmp = Float64(t_0 * Float64(exp(Float64(-im)) - 1.0));
	elseif (im <= 3.0)
		tmp = Float64(t_0 * Float64(Float64(Float64(Float64(Float64(Float64(-0.016666666666666666 * Float64(im * im)) - 0.3333333333333333) * im) * im) - 2.0) * im));
	else
		tmp = Float64(t_0 * Float64(1.0 - exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * sin(re);
	tmp = 0.0;
	if (im <= -2.95)
		tmp = t_0 * (exp(-im) - 1.0);
	elseif (im <= 3.0)
		tmp = t_0 * ((((((-0.016666666666666666 * (im * im)) - 0.3333333333333333) * im) * im) - 2.0) * im);
	else
		tmp = t_0 * (1.0 - exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -2.95], N[(t$95$0 * N[(N[Exp[(-im)], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 3.0], N[(t$95$0 * N[(N[(N[(N[(N[(N[(-0.016666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(1.0 - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < -2.9500000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 100.0%

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

   if -2.9500000000000002 < im < 3

   1. Initial program 32.5%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

      3. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right) \cdot im\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) \cdot {im}^{2} - 2\right) \cdot im\right) \]

      5. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      6. associate-*r* N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      9. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      11. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      12. lower-*.f64 99.8

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

   5. Applied rewrites 99.8%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left(\left(\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right)} \]

   if 3 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 100.0%

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

Alternative 3: 96.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ \mathbf{if}\;im \leq 3.75:\\ \;\;\;\;t\_0 \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(1 - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re))))
   (if (<= im 3.75)
     (*
      t_0
      (*
       (-
        (*
         (-
          (*
           (* (- (* -0.0003968253968253968 (* im im)) 0.016666666666666666) im)
           im)
          0.3333333333333333)
         (* im im))
        2.0)
       im))
     (* t_0 (- 1.0 (exp im))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sin(re);
	double tmp;
	if (im <= 3.75) {
		tmp = t_0 * ((((((((-0.0003968253968253968 * (im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	} else {
		tmp = t_0 * (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) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * sin(re)
    if (im <= 3.75d0) then
        tmp = t_0 * (((((((((-0.0003968253968253968d0) * (im * im)) - 0.016666666666666666d0) * im) * im) - 0.3333333333333333d0) * (im * im)) - 2.0d0) * im)
    else
        tmp = t_0 * (1.0d0 - exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.sin(re);
	double tmp;
	if (im <= 3.75) {
		tmp = t_0 * ((((((((-0.0003968253968253968 * (im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	} else {
		tmp = t_0 * (1.0 - Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sin(re)
	tmp = 0
	if im <= 3.75:
		tmp = t_0 * ((((((((-0.0003968253968253968 * (im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im)
	else:
		tmp = t_0 * (1.0 - math.exp(im))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sin(re))
	tmp = 0.0
	if (im <= 3.75)
		tmp = Float64(t_0 * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * Float64(im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * Float64(im * im)) - 2.0) * im));
	else
		tmp = Float64(t_0 * Float64(1.0 - exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * sin(re);
	tmp = 0.0;
	if (im <= 3.75)
		tmp = t_0 * ((((((((-0.0003968253968253968 * (im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	else
		tmp = t_0 * (1.0 - exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 3.75], N[(t$95$0 * N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.016666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(1.0 - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if im < 3.75

   1. Initial program 54.2%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

   5. Applied rewrites 95.8%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]

   if 3.75 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 100.0%

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

Alternative 4: 57.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re \cdot re, 0.004166666666666667\right) \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(im \cdot im\right) \cdot \left(\left(-0.0003968253968253968 \cdot im\right) \cdot im - 0.016666666666666666\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot im, -0.008333333333333333, 0.16666666666666666 \cdot im\right), re \cdot re, -im\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) 5e-7)
   (*
    (*
     (fma
      (-
       (* (fma -9.92063492063492e-5 (* re re) 0.004166666666666667) (* re re))
       0.08333333333333333)
      (* re re)
      0.5)
     re)
    (*
     (-
      (*
       (*
        (-
         (*
          (* im im)
          (- (* (* -0.0003968253968253968 im) im) 0.016666666666666666))
         0.3333333333333333)
        im)
       im)
      2.0)
     im))
   (*
    (fma
     (fma (* (* re re) im) -0.008333333333333333 (* 0.16666666666666666 im))
     (* re re)
     (- im))
    re)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= 5e-7) {
		tmp = (fma(((fma(-9.92063492063492e-5, (re * re), 0.004166666666666667) * (re * re)) - 0.08333333333333333), (re * re), 0.5) * re) * (((((((im * im) * (((-0.0003968253968253968 * im) * im) - 0.016666666666666666)) - 0.3333333333333333) * im) * im) - 2.0) * im);
	} else {
		tmp = fma(fma(((re * re) * im), -0.008333333333333333, (0.16666666666666666 * im)), (re * re), -im) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= 5e-7)
		tmp = Float64(Float64(fma(Float64(Float64(fma(-9.92063492063492e-5, Float64(re * re), 0.004166666666666667) * Float64(re * re)) - 0.08333333333333333), Float64(re * re), 0.5) * re) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(im * im) * Float64(Float64(Float64(-0.0003968253968253968 * im) * im) - 0.016666666666666666)) - 0.3333333333333333) * im) * im) - 2.0) * im));
	else
		tmp = Float64(fma(fma(Float64(Float64(re * re) * im), -0.008333333333333333, Float64(0.16666666666666666 * im)), Float64(re * re), Float64(-im)) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], 5e-7], N[(N[(N[(N[(N[(N[(-9.92063492063492e-5 * N[(re * re), $MachinePrecision] + 0.004166666666666667), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.08333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(im * im), $MachinePrecision] * N[(N[(N[(-0.0003968253968253968 * im), $MachinePrecision] * im), $MachinePrecision] - 0.016666666666666666), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(re * re), $MachinePrecision] * im), $MachinePrecision] * -0.008333333333333333 + N[(0.16666666666666666 * im), $MachinePrecision]), $MachinePrecision] * N[(re * re), $MachinePrecision] + (-im)), $MachinePrecision] * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < 4.99999999999999977e-7

   1. Initial program 68.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

   5. Applied rewrites 90.3%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right)\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right) \cdot \color{blue}{re}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right) \cdot \color{blue}{re}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   8. Applied rewrites 60.2%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re \cdot re, 0.004166666666666667\right) \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right)} \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{10080}, re \cdot re, \frac{1}{240}\right) \cdot \left(re \cdot re\right) - \frac{1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      2. lift--.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{10080}, re \cdot re, \frac{1}{240}\right) \cdot \left(re \cdot re\right) - \frac{1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{10080}, re \cdot re, \frac{1}{240}\right) \cdot \left(re \cdot re\right) - \frac{1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{10080}, re \cdot re, \frac{1}{240}\right) \cdot \left(re \cdot re\right) - \frac{1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      5. lift--.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{10080}, re \cdot re, \frac{1}{240}\right) \cdot \left(re \cdot re\right) - \frac{1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{10080}, re \cdot re, \frac{1}{240}\right) \cdot \left(re \cdot re\right) - \frac{1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{10080}, re \cdot re, \frac{1}{240}\right) \cdot \left(re \cdot re\right) - \frac{1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{10080}, re \cdot re, \frac{1}{240}\right) \cdot \left(re \cdot re\right) - \frac{1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      9. associate-*r* N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{10080}, re \cdot re, \frac{1}{240}\right) \cdot \left(re \cdot re\right) - \frac{1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{10080}, re \cdot re, \frac{1}{240}\right) \cdot \left(re \cdot re\right) - \frac{1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

   10. Applied rewrites 60.2%

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re \cdot re, 0.004166666666666667\right) \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(im \cdot im\right) \cdot \left(\left(-0.0003968253968253968 \cdot im\right) \cdot im - 0.016666666666666666\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

   if 4.99999999999999977e-7 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 63.8%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\sin re\right)\right) \cdot im \]

      5. lower-neg.f64 N/A

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

      6. lift-sin.f64 44.6

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

   5. Applied rewrites 44.6%

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

   6. Taylor expanded in re around 0

      \[\leadsto re \cdot \color{blue}{\left(-1 \cdot im + {re}^{2} \cdot \left(\frac{-1}{120} \cdot \left(im \cdot {re}^{2}\right) + \frac{1}{6} \cdot im\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-1 \cdot im + {re}^{2} \cdot \left(\frac{-1}{120} \cdot \left(im \cdot {re}^{2}\right) + \frac{1}{6} \cdot im\right)\right) \cdot re \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot im + {re}^{2} \cdot \left(\frac{-1}{120} \cdot \left(im \cdot {re}^{2}\right) + \frac{1}{6} \cdot im\right)\right) \cdot re \]

      3. +-commutative N/A

         \[\leadsto \left({re}^{2} \cdot \left(\frac{-1}{120} \cdot \left(im \cdot {re}^{2}\right) + \frac{1}{6} \cdot im\right) + -1 \cdot im\right) \cdot re \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\frac{-1}{120} \cdot \left(im \cdot {re}^{2}\right) + \frac{1}{6} \cdot im\right) \cdot {re}^{2} + -1 \cdot im\right) \cdot re \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{120} \cdot \left(im \cdot {re}^{2}\right) + \frac{1}{6} \cdot im, {re}^{2}, -1 \cdot im\right) \cdot re \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(im \cdot {re}^{2}\right) \cdot \frac{-1}{120} + \frac{1}{6} \cdot im, {re}^{2}, -1 \cdot im\right) \cdot re \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im \cdot {re}^{2}, \frac{-1}{120}, \frac{1}{6} \cdot im\right), {re}^{2}, -1 \cdot im\right) \cdot re \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left({re}^{2} \cdot im, \frac{-1}{120}, \frac{1}{6} \cdot im\right), {re}^{2}, -1 \cdot im\right) \cdot re \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left({re}^{2} \cdot im, \frac{-1}{120}, \frac{1}{6} \cdot im\right), {re}^{2}, -1 \cdot im\right) \cdot re \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot im, \frac{-1}{120}, \frac{1}{6} \cdot im\right), {re}^{2}, -1 \cdot im\right) \cdot re \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot im, \frac{-1}{120}, \frac{1}{6} \cdot im\right), {re}^{2}, -1 \cdot im\right) \cdot re \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot im, \frac{-1}{120}, \frac{1}{6} \cdot im\right), {re}^{2}, -1 \cdot im\right) \cdot re \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot im, \frac{-1}{120}, \frac{1}{6} \cdot im\right), re \cdot re, -1 \cdot im\right) \cdot re \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot im, \frac{-1}{120}, \frac{1}{6} \cdot im\right), re \cdot re, -1 \cdot im\right) \cdot re \]

      15. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot im, \frac{-1}{120}, \frac{1}{6} \cdot im\right), re \cdot re, \mathsf{neg}\left(im\right)\right) \cdot re \]

      16. lower-neg.f64 24.1

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot im, -0.008333333333333333, 0.16666666666666666 \cdot im\right), re \cdot re, -im\right) \cdot re \]

   8. Applied rewrites 24.1%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot im, -0.008333333333333333, 0.16666666666666666 \cdot im\right), re \cdot re, -im\right) \cdot \color{blue}{re} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 57.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(\left(re \cdot re\right) \cdot -9.92063492063492 \cdot 10^{-5}\right) \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot im, -0.008333333333333333, 0.16666666666666666 \cdot im\right), re \cdot re, -im\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) 5e-7)
   (*
    (*
     (fma
      (- (* (* (* re re) -9.92063492063492e-5) (* re re)) 0.08333333333333333)
      (* re re)
      0.5)
     re)
    (*
     (-
      (*
       (-
        (*
         (* (- (* -0.0003968253968253968 (* im im)) 0.016666666666666666) im)
         im)
        0.3333333333333333)
       (* im im))
      2.0)
     im))
   (*
    (fma
     (fma (* (* re re) im) -0.008333333333333333 (* 0.16666666666666666 im))
     (* re re)
     (- im))
    re)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= 5e-7) {
		tmp = (fma(((((re * re) * -9.92063492063492e-5) * (re * re)) - 0.08333333333333333), (re * re), 0.5) * re) * ((((((((-0.0003968253968253968 * (im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	} else {
		tmp = fma(fma(((re * re) * im), -0.008333333333333333, (0.16666666666666666 * im)), (re * re), -im) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= 5e-7)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(Float64(re * re) * -9.92063492063492e-5) * Float64(re * re)) - 0.08333333333333333), Float64(re * re), 0.5) * re) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * Float64(im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * Float64(im * im)) - 2.0) * im));
	else
		tmp = Float64(fma(fma(Float64(Float64(re * re) * im), -0.008333333333333333, Float64(0.16666666666666666 * im)), Float64(re * re), Float64(-im)) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], 5e-7], N[(N[(N[(N[(N[(N[(N[(re * re), $MachinePrecision] * -9.92063492063492e-5), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.08333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.016666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(re * re), $MachinePrecision] * im), $MachinePrecision] * -0.008333333333333333 + N[(0.16666666666666666 * im), $MachinePrecision]), $MachinePrecision] * N[(re * re), $MachinePrecision] + (-im)), $MachinePrecision] * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < 4.99999999999999977e-7

   1. Initial program 68.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

   5. Applied rewrites 90.3%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right)\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right) \cdot \color{blue}{re}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right) \cdot \color{blue}{re}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   8. Applied rewrites 60.2%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re \cdot re, 0.004166666666666667\right) \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right)} \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   9. Taylor expanded in re around inf

      \[\leadsto \left(\mathsf{fma}\left(\left(\frac{-1}{10080} \cdot {re}^{2}\right) \cdot \left(re \cdot re\right) - \frac{1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\left({re}^{2} \cdot \frac{-1}{10080}\right) \cdot \left(re \cdot re\right) - \frac{1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left({re}^{2} \cdot \frac{-1}{10080}\right) \cdot \left(re \cdot re\right) - \frac{1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       3. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(\left(re \cdot re\right) \cdot \frac{-1}{10080}\right) \cdot \left(re \cdot re\right) - \frac{1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       4. lift-*.f64 60.2

          \[\leadsto \left(\mathsf{fma}\left(\left(\left(re \cdot re\right) \cdot -9.92063492063492 \cdot 10^{-5}\right) \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   11. Applied rewrites 60.2%

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(re \cdot re\right) \cdot -9.92063492063492 \cdot 10^{-5}\right) \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   if 4.99999999999999977e-7 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 63.8%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\sin re\right)\right) \cdot im \]

      5. lower-neg.f64 N/A

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

      6. lift-sin.f64 44.6

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

   5. Applied rewrites 44.6%

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

   6. Taylor expanded in re around 0

      \[\leadsto re \cdot \color{blue}{\left(-1 \cdot im + {re}^{2} \cdot \left(\frac{-1}{120} \cdot \left(im \cdot {re}^{2}\right) + \frac{1}{6} \cdot im\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-1 \cdot im + {re}^{2} \cdot \left(\frac{-1}{120} \cdot \left(im \cdot {re}^{2}\right) + \frac{1}{6} \cdot im\right)\right) \cdot re \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot im + {re}^{2} \cdot \left(\frac{-1}{120} \cdot \left(im \cdot {re}^{2}\right) + \frac{1}{6} \cdot im\right)\right) \cdot re \]

      3. +-commutative N/A

         \[\leadsto \left({re}^{2} \cdot \left(\frac{-1}{120} \cdot \left(im \cdot {re}^{2}\right) + \frac{1}{6} \cdot im\right) + -1 \cdot im\right) \cdot re \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\frac{-1}{120} \cdot \left(im \cdot {re}^{2}\right) + \frac{1}{6} \cdot im\right) \cdot {re}^{2} + -1 \cdot im\right) \cdot re \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{120} \cdot \left(im \cdot {re}^{2}\right) + \frac{1}{6} \cdot im, {re}^{2}, -1 \cdot im\right) \cdot re \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(im \cdot {re}^{2}\right) \cdot \frac{-1}{120} + \frac{1}{6} \cdot im, {re}^{2}, -1 \cdot im\right) \cdot re \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im \cdot {re}^{2}, \frac{-1}{120}, \frac{1}{6} \cdot im\right), {re}^{2}, -1 \cdot im\right) \cdot re \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left({re}^{2} \cdot im, \frac{-1}{120}, \frac{1}{6} \cdot im\right), {re}^{2}, -1 \cdot im\right) \cdot re \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left({re}^{2} \cdot im, \frac{-1}{120}, \frac{1}{6} \cdot im\right), {re}^{2}, -1 \cdot im\right) \cdot re \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot im, \frac{-1}{120}, \frac{1}{6} \cdot im\right), {re}^{2}, -1 \cdot im\right) \cdot re \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot im, \frac{-1}{120}, \frac{1}{6} \cdot im\right), {re}^{2}, -1 \cdot im\right) \cdot re \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot im, \frac{-1}{120}, \frac{1}{6} \cdot im\right), {re}^{2}, -1 \cdot im\right) \cdot re \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot im, \frac{-1}{120}, \frac{1}{6} \cdot im\right), re \cdot re, -1 \cdot im\right) \cdot re \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot im, \frac{-1}{120}, \frac{1}{6} \cdot im\right), re \cdot re, -1 \cdot im\right) \cdot re \]

      15. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot im, \frac{-1}{120}, \frac{1}{6} \cdot im\right), re \cdot re, \mathsf{neg}\left(im\right)\right) \cdot re \]

      16. lower-neg.f64 24.1

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot im, -0.008333333333333333, 0.16666666666666666 \cdot im\right), re \cdot re, -im\right) \cdot re \]

   8. Applied rewrites 24.1%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot im, -0.008333333333333333, 0.16666666666666666 \cdot im\right), re \cdot re, -im\right) \cdot \color{blue}{re} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 57.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot im, -0.008333333333333333, 0.16666666666666666 \cdot im\right), re \cdot re, -im\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) 5e-7)
   (*
    (* (fma (* re re) -0.08333333333333333 0.5) re)
    (*
     (-
      (*
       (-
        (*
         (* (- (* -0.0003968253968253968 (* im im)) 0.016666666666666666) im)
         im)
        0.3333333333333333)
       (* im im))
      2.0)
     im))
   (*
    (fma
     (fma (* (* re re) im) -0.008333333333333333 (* 0.16666666666666666 im))
     (* re re)
     (- im))
    re)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= 5e-7) {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * re) * ((((((((-0.0003968253968253968 * (im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	} else {
		tmp = fma(fma(((re * re) * im), -0.008333333333333333, (0.16666666666666666 * im)), (re * re), -im) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= 5e-7)
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * Float64(im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * Float64(im * im)) - 2.0) * im));
	else
		tmp = Float64(fma(fma(Float64(Float64(re * re) * im), -0.008333333333333333, Float64(0.16666666666666666 * im)), Float64(re * re), Float64(-im)) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], 5e-7], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.016666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(re * re), $MachinePrecision] * im), $MachinePrecision] * -0.008333333333333333 + N[(0.16666666666666666 * im), $MachinePrecision]), $MachinePrecision] * N[(re * re), $MachinePrecision] + (-im)), $MachinePrecision] * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < 4.99999999999999977e-7

   1. Initial program 68.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

   5. Applied rewrites 90.3%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot \color{blue}{re}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot \color{blue}{re}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\frac{-1}{12} \cdot {re}^{2} + \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\left({re}^{2} \cdot \frac{-1}{12} + \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left({re}^{2}, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      6. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      7. lower-*.f64 59.9

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   8. Applied rewrites 59.9%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)} \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   if 4.99999999999999977e-7 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 63.8%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\sin re\right)\right) \cdot im \]

      5. lower-neg.f64 N/A

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

      6. lift-sin.f64 44.6

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

   5. Applied rewrites 44.6%

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

   6. Taylor expanded in re around 0

      \[\leadsto re \cdot \color{blue}{\left(-1 \cdot im + {re}^{2} \cdot \left(\frac{-1}{120} \cdot \left(im \cdot {re}^{2}\right) + \frac{1}{6} \cdot im\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-1 \cdot im + {re}^{2} \cdot \left(\frac{-1}{120} \cdot \left(im \cdot {re}^{2}\right) + \frac{1}{6} \cdot im\right)\right) \cdot re \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot im + {re}^{2} \cdot \left(\frac{-1}{120} \cdot \left(im \cdot {re}^{2}\right) + \frac{1}{6} \cdot im\right)\right) \cdot re \]

      3. +-commutative N/A

         \[\leadsto \left({re}^{2} \cdot \left(\frac{-1}{120} \cdot \left(im \cdot {re}^{2}\right) + \frac{1}{6} \cdot im\right) + -1 \cdot im\right) \cdot re \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\frac{-1}{120} \cdot \left(im \cdot {re}^{2}\right) + \frac{1}{6} \cdot im\right) \cdot {re}^{2} + -1 \cdot im\right) \cdot re \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{120} \cdot \left(im \cdot {re}^{2}\right) + \frac{1}{6} \cdot im, {re}^{2}, -1 \cdot im\right) \cdot re \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(im \cdot {re}^{2}\right) \cdot \frac{-1}{120} + \frac{1}{6} \cdot im, {re}^{2}, -1 \cdot im\right) \cdot re \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im \cdot {re}^{2}, \frac{-1}{120}, \frac{1}{6} \cdot im\right), {re}^{2}, -1 \cdot im\right) \cdot re \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left({re}^{2} \cdot im, \frac{-1}{120}, \frac{1}{6} \cdot im\right), {re}^{2}, -1 \cdot im\right) \cdot re \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left({re}^{2} \cdot im, \frac{-1}{120}, \frac{1}{6} \cdot im\right), {re}^{2}, -1 \cdot im\right) \cdot re \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot im, \frac{-1}{120}, \frac{1}{6} \cdot im\right), {re}^{2}, -1 \cdot im\right) \cdot re \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot im, \frac{-1}{120}, \frac{1}{6} \cdot im\right), {re}^{2}, -1 \cdot im\right) \cdot re \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot im, \frac{-1}{120}, \frac{1}{6} \cdot im\right), {re}^{2}, -1 \cdot im\right) \cdot re \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot im, \frac{-1}{120}, \frac{1}{6} \cdot im\right), re \cdot re, -1 \cdot im\right) \cdot re \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot im, \frac{-1}{120}, \frac{1}{6} \cdot im\right), re \cdot re, -1 \cdot im\right) \cdot re \]

      15. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot im, \frac{-1}{120}, \frac{1}{6} \cdot im\right), re \cdot re, \mathsf{neg}\left(im\right)\right) \cdot re \]

      16. lower-neg.f64 24.1

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot im, -0.008333333333333333, 0.16666666666666666 \cdot im\right), re \cdot re, -im\right) \cdot re \]

   8. Applied rewrites 24.1%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot im, -0.008333333333333333, 0.16666666666666666 \cdot im\right), re \cdot re, -im\right) \cdot \color{blue}{re} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 57.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot im, -0.008333333333333333, 0.16666666666666666 \cdot im\right), re \cdot re, -im\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) 5e-7)
   (*
    (* (fma (* re re) -0.08333333333333333 0.5) re)
    (*
     (-
      (*
       (-
        (* (* (* (* -0.0003968253968253968 im) im) im) im)
        0.3333333333333333)
       (* im im))
      2.0)
     im))
   (*
    (fma
     (fma (* (* re re) im) -0.008333333333333333 (* 0.16666666666666666 im))
     (* re re)
     (- im))
    re)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= 5e-7) {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * re) * ((((((((-0.0003968253968253968 * im) * im) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	} else {
		tmp = fma(fma(((re * re) * im), -0.008333333333333333, (0.16666666666666666 * im)), (re * re), -im) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= 5e-7)
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * im) * im) * im) * im) - 0.3333333333333333) * Float64(im * im)) - 2.0) * im));
	else
		tmp = Float64(fma(fma(Float64(Float64(re * re) * im), -0.008333333333333333, Float64(0.16666666666666666 * im)), Float64(re * re), Float64(-im)) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], 5e-7], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(re * re), $MachinePrecision] * im), $MachinePrecision] * -0.008333333333333333 + N[(0.16666666666666666 * im), $MachinePrecision]), $MachinePrecision] * N[(re * re), $MachinePrecision] + (-im)), $MachinePrecision] * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < 4.99999999999999977e-7

   1. Initial program 68.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

   5. Applied rewrites 90.3%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]

   6. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 86.9%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot \color{blue}{re}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot \color{blue}{re}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       3. +-commutative N/A

          \[\leadsto \left(\left(\frac{-1}{12} \cdot {re}^{2} + \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       4. *-commutative N/A

          \[\leadsto \left(\left({re}^{2} \cdot \frac{-1}{12} + \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       5. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left({re}^{2}, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       6. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       7. lift-*.f64 57.9

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   11. Applied rewrites 57.9%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)} \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   12. Taylor expanded in im around inf

       \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot {im}^{2}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   13. Step-by-step derivation

       1. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right)\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       2. associate-*r* N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       3. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       4. lower-*.f64 59.9

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   14. Applied rewrites 59.9%

       \[\leadsto \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   if 4.99999999999999977e-7 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 63.8%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\sin re\right)\right) \cdot im \]

      5. lower-neg.f64 N/A

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

      6. lift-sin.f64 44.6

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

   5. Applied rewrites 44.6%

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

   6. Taylor expanded in re around 0

      \[\leadsto re \cdot \color{blue}{\left(-1 \cdot im + {re}^{2} \cdot \left(\frac{-1}{120} \cdot \left(im \cdot {re}^{2}\right) + \frac{1}{6} \cdot im\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-1 \cdot im + {re}^{2} \cdot \left(\frac{-1}{120} \cdot \left(im \cdot {re}^{2}\right) + \frac{1}{6} \cdot im\right)\right) \cdot re \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot im + {re}^{2} \cdot \left(\frac{-1}{120} \cdot \left(im \cdot {re}^{2}\right) + \frac{1}{6} \cdot im\right)\right) \cdot re \]

      3. +-commutative N/A

         \[\leadsto \left({re}^{2} \cdot \left(\frac{-1}{120} \cdot \left(im \cdot {re}^{2}\right) + \frac{1}{6} \cdot im\right) + -1 \cdot im\right) \cdot re \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\frac{-1}{120} \cdot \left(im \cdot {re}^{2}\right) + \frac{1}{6} \cdot im\right) \cdot {re}^{2} + -1 \cdot im\right) \cdot re \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{120} \cdot \left(im \cdot {re}^{2}\right) + \frac{1}{6} \cdot im, {re}^{2}, -1 \cdot im\right) \cdot re \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(im \cdot {re}^{2}\right) \cdot \frac{-1}{120} + \frac{1}{6} \cdot im, {re}^{2}, -1 \cdot im\right) \cdot re \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im \cdot {re}^{2}, \frac{-1}{120}, \frac{1}{6} \cdot im\right), {re}^{2}, -1 \cdot im\right) \cdot re \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left({re}^{2} \cdot im, \frac{-1}{120}, \frac{1}{6} \cdot im\right), {re}^{2}, -1 \cdot im\right) \cdot re \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left({re}^{2} \cdot im, \frac{-1}{120}, \frac{1}{6} \cdot im\right), {re}^{2}, -1 \cdot im\right) \cdot re \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot im, \frac{-1}{120}, \frac{1}{6} \cdot im\right), {re}^{2}, -1 \cdot im\right) \cdot re \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot im, \frac{-1}{120}, \frac{1}{6} \cdot im\right), {re}^{2}, -1 \cdot im\right) \cdot re \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot im, \frac{-1}{120}, \frac{1}{6} \cdot im\right), {re}^{2}, -1 \cdot im\right) \cdot re \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot im, \frac{-1}{120}, \frac{1}{6} \cdot im\right), re \cdot re, -1 \cdot im\right) \cdot re \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot im, \frac{-1}{120}, \frac{1}{6} \cdot im\right), re \cdot re, -1 \cdot im\right) \cdot re \]

      15. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot im, \frac{-1}{120}, \frac{1}{6} \cdot im\right), re \cdot re, \mathsf{neg}\left(im\right)\right) \cdot re \]

      16. lower-neg.f64 24.1

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot im, -0.008333333333333333, 0.16666666666666666 \cdot im\right), re \cdot re, -im\right) \cdot re \]

   8. Applied rewrites 24.1%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot im, -0.008333333333333333, 0.16666666666666666 \cdot im\right), re \cdot re, -im\right) \cdot \color{blue}{re} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 95.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 7.8 \lor \neg \left(im \leq 3.35 \cdot 10^{+44}\right):\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(1 - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 7.8) (not (<= im 3.35e+44)))
   (*
    (* 0.5 (sin re))
    (*
     (-
      (*
       (-
        (*
         (* (- (* -0.0003968253968253968 (* im im)) 0.016666666666666666) im)
         im)
        0.3333333333333333)
       (* im im))
      2.0)
     im))
   (* (* 0.5 re) (- 1.0 (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 7.8) || !(im <= 3.35e+44)) {
		tmp = (0.5 * sin(re)) * ((((((((-0.0003968253968253968 * (im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	} 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 <= 7.8d0) .or. (.not. (im <= 3.35d+44))) then
        tmp = (0.5d0 * sin(re)) * (((((((((-0.0003968253968253968d0) * (im * im)) - 0.016666666666666666d0) * im) * im) - 0.3333333333333333d0) * (im * im)) - 2.0d0) * im)
    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 <= 7.8) || !(im <= 3.35e+44)) {
		tmp = (0.5 * Math.sin(re)) * ((((((((-0.0003968253968253968 * (im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	} else {
		tmp = (0.5 * re) * (1.0 - Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= 7.8) or not (im <= 3.35e+44):
		tmp = (0.5 * math.sin(re)) * ((((((((-0.0003968253968253968 * (im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im)
	else:
		tmp = (0.5 * re) * (1.0 - math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 7.8) || !(im <= 3.35e+44))
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * Float64(im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * Float64(im * im)) - 2.0) * im));
	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 <= 7.8) || ~((im <= 3.35e+44)))
		tmp = (0.5 * sin(re)) * ((((((((-0.0003968253968253968 * (im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	else
		tmp = (0.5 * re) * (1.0 - exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 7.8], N[Not[LessEqual[im, 3.35e+44]], $MachinePrecision]], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.016666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $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 < 7.79999999999999982 or 3.35000000000000018e44 < im

   1. Initial program 64.6%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

   5. Applied rewrites 96.7%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]

   if 7.79999999999999982 < im < 3.35000000000000018e44

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 94.4%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 7.8 \lor \neg \left(im \leq 3.35 \cdot 10^{+44}\right):\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(1 - e^{im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right)\\ \mathbf{if}\;im \leq -1.04 \cdot 10^{+65}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im \leq -10:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(e^{-im} - 1\right)\\ \mathbf{elif}\;im \leq 7.2 \lor \neg \left(im \leq 9.2 \cdot 10^{+61}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(1 - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (*
          (* 0.5 (sin re))
          (*
           (-
            (*
             (* (- (* -0.016666666666666666 (* im im)) 0.3333333333333333) im)
             im)
            2.0)
           im))))
   (if (<= im -1.04e+65)
     t_0
     (if (<= im -10.0)
       (* (* re 0.5) (- (exp (- im)) 1.0))
       (if (or (<= im 7.2) (not (<= im 9.2e+61)))
         t_0
         (* (* 0.5 re) (- 1.0 (exp im))))))))

C

double code(double re, double im) {
	double t_0 = (0.5 * sin(re)) * ((((((-0.016666666666666666 * (im * im)) - 0.3333333333333333) * im) * im) - 2.0) * im);
	double tmp;
	if (im <= -1.04e+65) {
		tmp = t_0;
	} else if (im <= -10.0) {
		tmp = (re * 0.5) * (exp(-im) - 1.0);
	} else if ((im <= 7.2) || !(im <= 9.2e+61)) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = (0.5d0 * sin(re)) * (((((((-0.016666666666666666d0) * (im * im)) - 0.3333333333333333d0) * im) * im) - 2.0d0) * im)
    if (im <= (-1.04d+65)) then
        tmp = t_0
    else if (im <= (-10.0d0)) then
        tmp = (re * 0.5d0) * (exp(-im) - 1.0d0)
    else if ((im <= 7.2d0) .or. (.not. (im <= 9.2d+61))) then
        tmp = t_0
    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 t_0 = (0.5 * Math.sin(re)) * ((((((-0.016666666666666666 * (im * im)) - 0.3333333333333333) * im) * im) - 2.0) * im);
	double tmp;
	if (im <= -1.04e+65) {
		tmp = t_0;
	} else if (im <= -10.0) {
		tmp = (re * 0.5) * (Math.exp(-im) - 1.0);
	} else if ((im <= 7.2) || !(im <= 9.2e+61)) {
		tmp = t_0;
	} else {
		tmp = (0.5 * re) * (1.0 - Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (0.5 * math.sin(re)) * ((((((-0.016666666666666666 * (im * im)) - 0.3333333333333333) * im) * im) - 2.0) * im)
	tmp = 0
	if im <= -1.04e+65:
		tmp = t_0
	elif im <= -10.0:
		tmp = (re * 0.5) * (math.exp(-im) - 1.0)
	elif (im <= 7.2) or not (im <= 9.2e+61):
		tmp = t_0
	else:
		tmp = (0.5 * re) * (1.0 - math.exp(im))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(Float64(Float64(Float64(Float64(Float64(-0.016666666666666666 * Float64(im * im)) - 0.3333333333333333) * im) * im) - 2.0) * im))
	tmp = 0.0
	if (im <= -1.04e+65)
		tmp = t_0;
	elseif (im <= -10.0)
		tmp = Float64(Float64(re * 0.5) * Float64(exp(Float64(-im)) - 1.0));
	elseif ((im <= 7.2) || !(im <= 9.2e+61))
		tmp = t_0;
	else
		tmp = Float64(Float64(0.5 * re) * Float64(1.0 - exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (0.5 * sin(re)) * ((((((-0.016666666666666666 * (im * im)) - 0.3333333333333333) * im) * im) - 2.0) * im);
	tmp = 0.0;
	if (im <= -1.04e+65)
		tmp = t_0;
	elseif (im <= -10.0)
		tmp = (re * 0.5) * (exp(-im) - 1.0);
	elseif ((im <= 7.2) || ~((im <= 9.2e+61)))
		tmp = t_0;
	else
		tmp = (0.5 * re) * (1.0 - exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(-0.016666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.04e+65], t$95$0, If[LessEqual[im, -10.0], N[(N[(re * 0.5), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[im, 7.2], N[Not[LessEqual[im, 9.2e+61]], $MachinePrecision]], t$95$0, 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 < -1.03999999999999999e65 or -10 < im < 7.20000000000000018 or 9.1999999999999998e61 < im

   1. Initial program 61.8%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

      3. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right) \cdot im\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) \cdot {im}^{2} - 2\right) \cdot im\right) \]

      5. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      6. associate-*r* N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      9. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      11. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      12. lower-*.f64 99.5

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

   5. Applied rewrites 99.5%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left(\left(\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right)} \]

   if -1.03999999999999999e65 < im < -10

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 76.9

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

   8. Applied rewrites 76.9%

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

   if 7.20000000000000018 < im < 9.1999999999999998e61

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 90.9%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.04 \cdot 10^{+65}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right)\\ \mathbf{elif}\;im \leq -10:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(e^{-im} - 1\right)\\ \mathbf{elif}\;im \leq 7.2 \lor \neg \left(im \leq 9.2 \cdot 10^{+61}\right):\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(1 - e^{im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 58.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.005:\\ \;\;\;\;\left(\mathsf{fma}\left(re, re \cdot -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.005)
   (*
    (* (fma re (* re -0.08333333333333333) 0.5) re)
    (*
     (-
      (* (- (* (* -0.016666666666666666 im) im) 0.3333333333333333) (* im im))
      2.0)
     im))
   (*
    (* 0.5 re)
    (*
     (-
      (*
       (-
        (*
         (* (- (* -0.0003968253968253968 (* im im)) 0.016666666666666666) im)
         im)
        0.3333333333333333)
       (* im im))
      2.0)
     im))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.005) {
		tmp = (fma(re, (re * -0.08333333333333333), 0.5) * re) * ((((((-0.016666666666666666 * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	} else {
		tmp = (0.5 * re) * ((((((((-0.0003968253968253968 * (im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.005)
		tmp = Float64(Float64(fma(re, Float64(re * -0.08333333333333333), 0.5) * re) * Float64(Float64(Float64(Float64(Float64(Float64(-0.016666666666666666 * im) * im) - 0.3333333333333333) * Float64(im * im)) - 2.0) * im));
	else
		tmp = Float64(Float64(0.5 * re) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * Float64(im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * Float64(im * im)) - 2.0) * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.005], N[(N[(N[(re * N[(re * -0.08333333333333333), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(N[(N[(N[(-0.016666666666666666 * im), $MachinePrecision] * im), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.016666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < -0.0050000000000000001

   1. Initial program 48.3%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

   5. Applied rewrites 91.5%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]

   6. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 87.9%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot \color{blue}{re}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot \color{blue}{re}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       3. +-commutative N/A

          \[\leadsto \left(\left(\frac{-1}{12} \cdot {re}^{2} + \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       4. *-commutative N/A

          \[\leadsto \left(\left({re}^{2} \cdot \frac{-1}{12} + \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       5. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left({re}^{2}, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       6. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       7. lift-*.f64 15.7

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   11. Applied rewrites 15.7%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)} \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   12. Step-by-step derivation

       1. lift-fma.f64 N/A

          \[\leadsto \left(\left(\left(re \cdot re\right) \cdot \frac{-1}{12} + \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(re \cdot re\right) \cdot \frac{-1}{12} + \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       3. associate-*l* N/A

          \[\leadsto \left(\left(re \cdot \left(re \cdot \frac{-1}{12}\right) + \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       4. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(re, re \cdot \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       5. lower-*.f64 15.7

          \[\leadsto \left(\mathsf{fma}\left(re, re \cdot -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   13. Applied rewrites 15.7%

       \[\leadsto \left(\mathsf{fma}\left(re, re \cdot -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   if -0.0050000000000000001 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 75.4%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

   5. Applied rewrites 89.7%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{re}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 67.1%

      \[\leadsto \left(0.5 \cdot \color{blue}{re}\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 56.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\left(\mathsf{fma}\left(re, re \cdot -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot im, -0.008333333333333333, 0.16666666666666666 \cdot im\right), re \cdot re, -im\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) 5e-7)
   (*
    (* (fma re (* re -0.08333333333333333) 0.5) re)
    (*
     (-
      (* (- (* (* -0.016666666666666666 im) im) 0.3333333333333333) (* im im))
      2.0)
     im))
   (*
    (fma
     (fma (* (* re re) im) -0.008333333333333333 (* 0.16666666666666666 im))
     (* re re)
     (- im))
    re)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= 5e-7) {
		tmp = (fma(re, (re * -0.08333333333333333), 0.5) * re) * ((((((-0.016666666666666666 * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	} else {
		tmp = fma(fma(((re * re) * im), -0.008333333333333333, (0.16666666666666666 * im)), (re * re), -im) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= 5e-7)
		tmp = Float64(Float64(fma(re, Float64(re * -0.08333333333333333), 0.5) * re) * Float64(Float64(Float64(Float64(Float64(Float64(-0.016666666666666666 * im) * im) - 0.3333333333333333) * Float64(im * im)) - 2.0) * im));
	else
		tmp = Float64(fma(fma(Float64(Float64(re * re) * im), -0.008333333333333333, Float64(0.16666666666666666 * im)), Float64(re * re), Float64(-im)) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], 5e-7], N[(N[(N[(re * N[(re * -0.08333333333333333), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(N[(N[(N[(-0.016666666666666666 * im), $MachinePrecision] * im), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(re * re), $MachinePrecision] * im), $MachinePrecision] * -0.008333333333333333 + N[(0.16666666666666666 * im), $MachinePrecision]), $MachinePrecision] * N[(re * re), $MachinePrecision] + (-im)), $MachinePrecision] * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < 4.99999999999999977e-7

   1. Initial program 68.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

   5. Applied rewrites 90.3%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]

   6. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 86.9%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot \color{blue}{re}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot \color{blue}{re}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       3. +-commutative N/A

          \[\leadsto \left(\left(\frac{-1}{12} \cdot {re}^{2} + \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       4. *-commutative N/A

          \[\leadsto \left(\left({re}^{2} \cdot \frac{-1}{12} + \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       5. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left({re}^{2}, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       6. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       7. lift-*.f64 57.9

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   11. Applied rewrites 57.9%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)} \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   12. Step-by-step derivation

       1. lift-fma.f64 N/A

          \[\leadsto \left(\left(\left(re \cdot re\right) \cdot \frac{-1}{12} + \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(re \cdot re\right) \cdot \frac{-1}{12} + \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       3. associate-*l* N/A

          \[\leadsto \left(\left(re \cdot \left(re \cdot \frac{-1}{12}\right) + \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       4. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(re, re \cdot \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       5. lower-*.f64 57.9

          \[\leadsto \left(\mathsf{fma}\left(re, re \cdot -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   13. Applied rewrites 57.9%

       \[\leadsto \left(\mathsf{fma}\left(re, re \cdot -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   if 4.99999999999999977e-7 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 63.8%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\sin re\right)\right) \cdot im \]

      5. lower-neg.f64 N/A

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

      6. lift-sin.f64 44.6

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

   5. Applied rewrites 44.6%

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

   6. Taylor expanded in re around 0

      \[\leadsto re \cdot \color{blue}{\left(-1 \cdot im + {re}^{2} \cdot \left(\frac{-1}{120} \cdot \left(im \cdot {re}^{2}\right) + \frac{1}{6} \cdot im\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-1 \cdot im + {re}^{2} \cdot \left(\frac{-1}{120} \cdot \left(im \cdot {re}^{2}\right) + \frac{1}{6} \cdot im\right)\right) \cdot re \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot im + {re}^{2} \cdot \left(\frac{-1}{120} \cdot \left(im \cdot {re}^{2}\right) + \frac{1}{6} \cdot im\right)\right) \cdot re \]

      3. +-commutative N/A

         \[\leadsto \left({re}^{2} \cdot \left(\frac{-1}{120} \cdot \left(im \cdot {re}^{2}\right) + \frac{1}{6} \cdot im\right) + -1 \cdot im\right) \cdot re \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\frac{-1}{120} \cdot \left(im \cdot {re}^{2}\right) + \frac{1}{6} \cdot im\right) \cdot {re}^{2} + -1 \cdot im\right) \cdot re \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{120} \cdot \left(im \cdot {re}^{2}\right) + \frac{1}{6} \cdot im, {re}^{2}, -1 \cdot im\right) \cdot re \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(im \cdot {re}^{2}\right) \cdot \frac{-1}{120} + \frac{1}{6} \cdot im, {re}^{2}, -1 \cdot im\right) \cdot re \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im \cdot {re}^{2}, \frac{-1}{120}, \frac{1}{6} \cdot im\right), {re}^{2}, -1 \cdot im\right) \cdot re \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left({re}^{2} \cdot im, \frac{-1}{120}, \frac{1}{6} \cdot im\right), {re}^{2}, -1 \cdot im\right) \cdot re \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left({re}^{2} \cdot im, \frac{-1}{120}, \frac{1}{6} \cdot im\right), {re}^{2}, -1 \cdot im\right) \cdot re \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot im, \frac{-1}{120}, \frac{1}{6} \cdot im\right), {re}^{2}, -1 \cdot im\right) \cdot re \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot im, \frac{-1}{120}, \frac{1}{6} \cdot im\right), {re}^{2}, -1 \cdot im\right) \cdot re \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot im, \frac{-1}{120}, \frac{1}{6} \cdot im\right), {re}^{2}, -1 \cdot im\right) \cdot re \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot im, \frac{-1}{120}, \frac{1}{6} \cdot im\right), re \cdot re, -1 \cdot im\right) \cdot re \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot im, \frac{-1}{120}, \frac{1}{6} \cdot im\right), re \cdot re, -1 \cdot im\right) \cdot re \]

      15. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot im, \frac{-1}{120}, \frac{1}{6} \cdot im\right), re \cdot re, \mathsf{neg}\left(im\right)\right) \cdot re \]

      16. lower-neg.f64 24.1

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot im, -0.008333333333333333, 0.16666666666666666 \cdot im\right), re \cdot re, -im\right) \cdot re \]

   8. Applied rewrites 24.1%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot im, -0.008333333333333333, 0.16666666666666666 \cdot im\right), re \cdot re, -im\right) \cdot \color{blue}{re} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 56.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\left(-0.016666666666666666 \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\\ \mathbf{if}\;0.5 \cdot \sin re \leq -0.005:\\ \;\;\;\;\left(\left(\left(re \cdot re\right) \cdot -0.08333333333333333\right) \cdot re\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (*
          (-
           (*
            (- (* (* -0.016666666666666666 im) im) 0.3333333333333333)
            (* im im))
           2.0)
          im)))
   (if (<= (* 0.5 (sin re)) -0.005)
     (* (* (* (* re re) -0.08333333333333333) re) t_0)
     (* (* re 0.5) t_0))))

C

double code(double re, double im) {
	double t_0 = (((((-0.016666666666666666 * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im;
	double tmp;
	if ((0.5 * sin(re)) <= -0.005) {
		tmp = (((re * re) * -0.08333333333333333) * re) * t_0;
	} else {
		tmp = (re * 0.5) * t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = ((((((-0.016666666666666666d0) * im) * im) - 0.3333333333333333d0) * (im * im)) - 2.0d0) * im
    if ((0.5d0 * sin(re)) <= (-0.005d0)) then
        tmp = (((re * re) * (-0.08333333333333333d0)) * re) * t_0
    else
        tmp = (re * 0.5d0) * t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (((((-0.016666666666666666 * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im;
	double tmp;
	if ((0.5 * Math.sin(re)) <= -0.005) {
		tmp = (((re * re) * -0.08333333333333333) * re) * t_0;
	} else {
		tmp = (re * 0.5) * t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (((((-0.016666666666666666 * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im
	tmp = 0
	if (0.5 * math.sin(re)) <= -0.005:
		tmp = (((re * re) * -0.08333333333333333) * re) * t_0
	else:
		tmp = (re * 0.5) * t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(Float64(Float64(Float64(Float64(-0.016666666666666666 * im) * im) - 0.3333333333333333) * Float64(im * im)) - 2.0) * im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.005)
		tmp = Float64(Float64(Float64(Float64(re * re) * -0.08333333333333333) * re) * t_0);
	else
		tmp = Float64(Float64(re * 0.5) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (((((-0.016666666666666666 * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im;
	tmp = 0.0;
	if ((0.5 * sin(re)) <= -0.005)
		tmp = (((re * re) * -0.08333333333333333) * re) * t_0;
	else
		tmp = (re * 0.5) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[(N[(N[(N[(-0.016666666666666666 * im), $MachinePrecision] * im), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.005], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333), $MachinePrecision] * re), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(re * 0.5), $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < -0.0050000000000000001

   1. Initial program 48.3%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

   5. Applied rewrites 91.5%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]

   6. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 87.9%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot \color{blue}{re}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot \color{blue}{re}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       3. +-commutative N/A

          \[\leadsto \left(\left(\frac{-1}{12} \cdot {re}^{2} + \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       4. *-commutative N/A

          \[\leadsto \left(\left({re}^{2} \cdot \frac{-1}{12} + \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       5. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left({re}^{2}, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       6. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       7. lift-*.f64 15.7

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   11. Applied rewrites 15.7%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)} \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   12. Taylor expanded in re around inf

       \[\leadsto \left(\left(\frac{-1}{12} \cdot {re}^{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left({re}^{2} \cdot \frac{-1}{12}\right) \cdot re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left({re}^{2} \cdot \frac{-1}{12}\right) \cdot re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       3. pow2 N/A

          \[\leadsto \left(\left(\left(re \cdot re\right) \cdot \frac{-1}{12}\right) \cdot re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       4. lift-*.f64 15.7

          \[\leadsto \left(\left(\left(re \cdot re\right) \cdot -0.08333333333333333\right) \cdot re\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   14. Applied rewrites 15.7%

       \[\leadsto \left(\left(\left(re \cdot re\right) \cdot -0.08333333333333333\right) \cdot re\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   if -0.0050000000000000001 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 75.4%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

   5. Applied rewrites 89.7%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]

   6. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 86.5%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot re\right)} \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(re \cdot \color{blue}{\frac{1}{2}}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       2. lower-*.f64 64.9

          \[\leadsto \left(re \cdot \color{blue}{0.5}\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   11. Applied rewrites 64.9%

       \[\leadsto \color{blue}{\left(re \cdot 0.5\right)} \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 95.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(\left(-0.3333333333333333 \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \mathbf{if}\;im \leq -4.8 \cdot 10^{+110}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im \leq -6.1:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(e^{-im} - 1\right)\\ \mathbf{elif}\;im \leq 4.7:\\ \;\;\;\;\left(\sin re \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\right) \cdot im\\ \mathbf{elif}\;im \leq 8 \cdot 10^{+102}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(1 - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (*
          (* 0.5 (sin re))
          (* (- (* -0.3333333333333333 (* im im)) 2.0) im))))
   (if (<= im -4.8e+110)
     t_0
     (if (<= im -6.1)
       (* (* re 0.5) (- (exp (- im)) 1.0))
       (if (<= im 4.7)
         (* (* (sin re) (fma (* -0.16666666666666666 im) im -1.0)) im)
         (if (<= im 8e+102) (* (* 0.5 re) (- 1.0 (exp im))) t_0))))))

C

double code(double re, double im) {
	double t_0 = (0.5 * sin(re)) * (((-0.3333333333333333 * (im * im)) - 2.0) * im);
	double tmp;
	if (im <= -4.8e+110) {
		tmp = t_0;
	} else if (im <= -6.1) {
		tmp = (re * 0.5) * (exp(-im) - 1.0);
	} else if (im <= 4.7) {
		tmp = (sin(re) * fma((-0.16666666666666666 * im), im, -1.0)) * im;
	} else if (im <= 8e+102) {
		tmp = (0.5 * re) * (1.0 - exp(im));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(Float64(Float64(-0.3333333333333333 * Float64(im * im)) - 2.0) * im))
	tmp = 0.0
	if (im <= -4.8e+110)
		tmp = t_0;
	elseif (im <= -6.1)
		tmp = Float64(Float64(re * 0.5) * Float64(exp(Float64(-im)) - 1.0));
	elseif (im <= 4.7)
		tmp = Float64(Float64(sin(re) * fma(Float64(-0.16666666666666666 * im), im, -1.0)) * im);
	elseif (im <= 8e+102)
		tmp = Float64(Float64(0.5 * re) * Float64(1.0 - exp(im)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(-0.3333333333333333 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -4.8e+110], t$95$0, If[LessEqual[im, -6.1], N[(N[(re * 0.5), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 4.7], N[(N[(N[Sin[re], $MachinePrecision] * N[(N[(-0.16666666666666666 * im), $MachinePrecision] * im + -1.0), $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision], If[LessEqual[im, 8e+102], N[(N[(0.5 * re), $MachinePrecision] * N[(1.0 - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -4.80000000000000025e110 or 7.99999999999999982e102 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 2\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(\frac{-1}{3} \cdot {im}^{2} - 2\right) \cdot \color{blue}{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(\frac{-1}{3} \cdot {im}^{2} - 2\right) \cdot \color{blue}{im}\right) \]

      3. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(\frac{-1}{3} \cdot {im}^{2} - 2\right) \cdot im\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(\frac{-1}{3} \cdot {im}^{2} - 2\right) \cdot im\right) \]

      5. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(\frac{-1}{3} \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      6. lower-*.f64 100.0

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\left(-0.3333333333333333 \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   5. Applied rewrites 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left(-0.3333333333333333 \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]

   if -4.80000000000000025e110 < im < -6.0999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 78.9

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

   8. Applied rewrites 78.9%

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

   if -6.0999999999999996 < im < 4.70000000000000018

   1. Initial program 32.5%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if 4.70000000000000018 < im < 7.99999999999999982e102

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 84.4%

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

Alternative 14: 56.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.005:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(-0.3333333333333333 \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.005)
   (*
    (* (fma (* re re) -0.08333333333333333 0.5) re)
    (* (- (* -0.3333333333333333 (* im im)) 2.0) im))
   (*
    (* re 0.5)
    (*
     (-
      (* (- (* (* -0.016666666666666666 im) im) 0.3333333333333333) (* im im))
      2.0)
     im))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.005) {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * re) * (((-0.3333333333333333 * (im * im)) - 2.0) * im);
	} else {
		tmp = (re * 0.5) * ((((((-0.016666666666666666 * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.005)
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re) * Float64(Float64(Float64(-0.3333333333333333 * Float64(im * im)) - 2.0) * im));
	else
		tmp = Float64(Float64(re * 0.5) * Float64(Float64(Float64(Float64(Float64(Float64(-0.016666666666666666 * im) * im) - 0.3333333333333333) * Float64(im * im)) - 2.0) * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.005], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(-0.3333333333333333 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[(re * 0.5), $MachinePrecision] * N[(N[(N[(N[(N[(N[(-0.016666666666666666 * im), $MachinePrecision] * im), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < -0.0050000000000000001

   1. Initial program 48.3%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 2\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(\frac{-1}{3} \cdot {im}^{2} - 2\right) \cdot \color{blue}{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(\frac{-1}{3} \cdot {im}^{2} - 2\right) \cdot \color{blue}{im}\right) \]

      3. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(\frac{-1}{3} \cdot {im}^{2} - 2\right) \cdot im\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(\frac{-1}{3} \cdot {im}^{2} - 2\right) \cdot im\right) \]

      5. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(\frac{-1}{3} \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      6. lower-*.f64 83.1

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\left(-0.3333333333333333 \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   5. Applied rewrites 83.1%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left(-0.3333333333333333 \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(\left(\frac{-1}{3} \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot \color{blue}{re}\right) \cdot \left(\left(\frac{-1}{3} \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot \color{blue}{re}\right) \cdot \left(\left(\frac{-1}{3} \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\frac{-1}{12} \cdot {re}^{2} + \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\frac{-1}{3} \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\left({re}^{2} \cdot \frac{-1}{12} + \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\frac{-1}{3} \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left({re}^{2}, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\frac{-1}{3} \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      6. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\frac{-1}{3} \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      7. lower-*.f64 15.7

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(-0.3333333333333333 \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   8. Applied rewrites 15.7%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)} \cdot \left(\left(-0.3333333333333333 \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   if -0.0050000000000000001 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 75.4%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

   5. Applied rewrites 89.7%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]

   6. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 86.5%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot re\right)} \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(re \cdot \color{blue}{\frac{1}{2}}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       2. lower-*.f64 64.9

          \[\leadsto \left(re \cdot \color{blue}{0.5}\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   11. Applied rewrites 64.9%

       \[\leadsto \color{blue}{\left(re \cdot 0.5\right)} \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 53.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(-0.3333333333333333 \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot im, -0.008333333333333333, 0.16666666666666666 \cdot im\right), re \cdot re, -im\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) 5e-7)
   (*
    (* (fma (* re re) -0.08333333333333333 0.5) re)
    (* (- (* -0.3333333333333333 (* im im)) 2.0) im))
   (*
    (fma
     (fma (* (* re re) im) -0.008333333333333333 (* 0.16666666666666666 im))
     (* re re)
     (- im))
    re)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= 5e-7) {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * re) * (((-0.3333333333333333 * (im * im)) - 2.0) * im);
	} else {
		tmp = fma(fma(((re * re) * im), -0.008333333333333333, (0.16666666666666666 * im)), (re * re), -im) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= 5e-7)
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re) * Float64(Float64(Float64(-0.3333333333333333 * Float64(im * im)) - 2.0) * im));
	else
		tmp = Float64(fma(fma(Float64(Float64(re * re) * im), -0.008333333333333333, Float64(0.16666666666666666 * im)), Float64(re * re), Float64(-im)) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], 5e-7], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(-0.3333333333333333 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(re * re), $MachinePrecision] * im), $MachinePrecision] * -0.008333333333333333 + N[(0.16666666666666666 * im), $MachinePrecision]), $MachinePrecision] * N[(re * re), $MachinePrecision] + (-im)), $MachinePrecision] * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < 4.99999999999999977e-7

   1. Initial program 68.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 2\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(\frac{-1}{3} \cdot {im}^{2} - 2\right) \cdot \color{blue}{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(\frac{-1}{3} \cdot {im}^{2} - 2\right) \cdot \color{blue}{im}\right) \]

      3. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(\frac{-1}{3} \cdot {im}^{2} - 2\right) \cdot im\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(\frac{-1}{3} \cdot {im}^{2} - 2\right) \cdot im\right) \]

      5. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(\frac{-1}{3} \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      6. lower-*.f64 81.9

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\left(-0.3333333333333333 \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   5. Applied rewrites 81.9%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left(-0.3333333333333333 \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(\left(\frac{-1}{3} \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot \color{blue}{re}\right) \cdot \left(\left(\frac{-1}{3} \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot \color{blue}{re}\right) \cdot \left(\left(\frac{-1}{3} \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\frac{-1}{12} \cdot {re}^{2} + \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\frac{-1}{3} \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\left({re}^{2} \cdot \frac{-1}{12} + \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\frac{-1}{3} \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left({re}^{2}, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\frac{-1}{3} \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      6. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\frac{-1}{3} \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      7. lower-*.f64 54.9

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(-0.3333333333333333 \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   8. Applied rewrites 54.9%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)} \cdot \left(\left(-0.3333333333333333 \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   if 4.99999999999999977e-7 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 63.8%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\sin re\right)\right) \cdot im \]

      5. lower-neg.f64 N/A

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

      6. lift-sin.f64 44.6

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

   5. Applied rewrites 44.6%

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

   6. Taylor expanded in re around 0

      \[\leadsto re \cdot \color{blue}{\left(-1 \cdot im + {re}^{2} \cdot \left(\frac{-1}{120} \cdot \left(im \cdot {re}^{2}\right) + \frac{1}{6} \cdot im\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-1 \cdot im + {re}^{2} \cdot \left(\frac{-1}{120} \cdot \left(im \cdot {re}^{2}\right) + \frac{1}{6} \cdot im\right)\right) \cdot re \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot im + {re}^{2} \cdot \left(\frac{-1}{120} \cdot \left(im \cdot {re}^{2}\right) + \frac{1}{6} \cdot im\right)\right) \cdot re \]

      3. +-commutative N/A

         \[\leadsto \left({re}^{2} \cdot \left(\frac{-1}{120} \cdot \left(im \cdot {re}^{2}\right) + \frac{1}{6} \cdot im\right) + -1 \cdot im\right) \cdot re \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\frac{-1}{120} \cdot \left(im \cdot {re}^{2}\right) + \frac{1}{6} \cdot im\right) \cdot {re}^{2} + -1 \cdot im\right) \cdot re \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{120} \cdot \left(im \cdot {re}^{2}\right) + \frac{1}{6} \cdot im, {re}^{2}, -1 \cdot im\right) \cdot re \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(im \cdot {re}^{2}\right) \cdot \frac{-1}{120} + \frac{1}{6} \cdot im, {re}^{2}, -1 \cdot im\right) \cdot re \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im \cdot {re}^{2}, \frac{-1}{120}, \frac{1}{6} \cdot im\right), {re}^{2}, -1 \cdot im\right) \cdot re \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left({re}^{2} \cdot im, \frac{-1}{120}, \frac{1}{6} \cdot im\right), {re}^{2}, -1 \cdot im\right) \cdot re \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left({re}^{2} \cdot im, \frac{-1}{120}, \frac{1}{6} \cdot im\right), {re}^{2}, -1 \cdot im\right) \cdot re \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot im, \frac{-1}{120}, \frac{1}{6} \cdot im\right), {re}^{2}, -1 \cdot im\right) \cdot re \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot im, \frac{-1}{120}, \frac{1}{6} \cdot im\right), {re}^{2}, -1 \cdot im\right) \cdot re \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot im, \frac{-1}{120}, \frac{1}{6} \cdot im\right), {re}^{2}, -1 \cdot im\right) \cdot re \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot im, \frac{-1}{120}, \frac{1}{6} \cdot im\right), re \cdot re, -1 \cdot im\right) \cdot re \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot im, \frac{-1}{120}, \frac{1}{6} \cdot im\right), re \cdot re, -1 \cdot im\right) \cdot re \]

      15. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot im, \frac{-1}{120}, \frac{1}{6} \cdot im\right), re \cdot re, \mathsf{neg}\left(im\right)\right) \cdot re \]

      16. lower-neg.f64 24.1

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot im, -0.008333333333333333, 0.16666666666666666 \cdot im\right), re \cdot re, -im\right) \cdot re \]

   8. Applied rewrites 24.1%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot im, -0.008333333333333333, 0.16666666666666666 \cdot im\right), re \cdot re, -im\right) \cdot \color{blue}{re} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 55.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(re \cdot \left(im \cdot re\right), 0.16666666666666666, -im\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot re\right) \cdot \left(\left(\left(\left(im \cdot im\right) \cdot -0.008333333333333333 - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.005)
   (* (fma (* re (* im re)) 0.16666666666666666 (- im)) re)
   (*
    (* im re)
    (-
     (* (* (- (* (* im im) -0.008333333333333333) 0.16666666666666666) im) im)
     1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.005) {
		tmp = fma((re * (im * re)), 0.16666666666666666, -im) * re;
	} else {
		tmp = (im * re) * ((((((im * im) * -0.008333333333333333) - 0.16666666666666666) * im) * im) - 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.005)
		tmp = Float64(fma(Float64(re * Float64(im * re)), 0.16666666666666666, Float64(-im)) * re);
	else
		tmp = Float64(Float64(im * re) * Float64(Float64(Float64(Float64(Float64(Float64(im * im) * -0.008333333333333333) - 0.16666666666666666) * im) * im) - 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.005], N[(N[(N[(re * N[(im * re), $MachinePrecision]), $MachinePrecision] * 0.16666666666666666 + (-im)), $MachinePrecision] * re), $MachinePrecision], N[(N[(im * re), $MachinePrecision] * N[(N[(N[(N[(N[(N[(im * im), $MachinePrecision] * -0.008333333333333333), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < -0.0050000000000000001

   1. Initial program 48.3%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\sin re\right)\right) \cdot im \]

      5. lower-neg.f64 N/A

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

      6. lift-sin.f64 57.6

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

   5. Applied rewrites 57.6%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 13.3

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

   8. Applied rewrites 13.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 13.3

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

   10. Applied rewrites 13.3%

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

   if -0.0050000000000000001 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 75.4%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot \color{blue}{im} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot \color{blue}{im} \]

   5. Applied rewrites 82.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sin re \cdot \mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right), im \cdot im, -\sin re\right) \cdot im} \]

   6. Taylor expanded in re around 0

      \[\leadsto \left(re \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right)\right) \cdot im \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right) \cdot re\right) \cdot im \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right) \cdot re\right) \cdot im \]

      3. lower--.f64 N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right) \cdot re\right) \cdot im \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) \cdot {im}^{2} - 1\right) \cdot re\right) \cdot im \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) \cdot {im}^{2} - 1\right) \cdot re\right) \cdot im \]

      6. lower--.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) \cdot {im}^{2} - 1\right) \cdot re\right) \cdot im \]

      7. *-commutative N/A

         \[\leadsto \left(\left(\left({im}^{2} \cdot \frac{-1}{120} - \frac{1}{6}\right) \cdot {im}^{2} - 1\right) \cdot re\right) \cdot im \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\left(\left({im}^{2} \cdot \frac{-1}{120} - \frac{1}{6}\right) \cdot {im}^{2} - 1\right) \cdot re\right) \cdot im \]

      9. pow2 N/A

         \[\leadsto \left(\left(\left(\left(im \cdot im\right) \cdot \frac{-1}{120} - \frac{1}{6}\right) \cdot {im}^{2} - 1\right) \cdot re\right) \cdot im \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(\left(im \cdot im\right) \cdot \frac{-1}{120} - \frac{1}{6}\right) \cdot {im}^{2} - 1\right) \cdot re\right) \cdot im \]

      11. pow2 N/A

          \[\leadsto \left(\left(\left(\left(im \cdot im\right) \cdot \frac{-1}{120} - \frac{1}{6}\right) \cdot \left(im \cdot im\right) - 1\right) \cdot re\right) \cdot im \]

      12. lift-*.f64 62.8

          \[\leadsto \left(\left(\left(\left(im \cdot im\right) \cdot -0.008333333333333333 - 0.16666666666666666\right) \cdot \left(im \cdot im\right) - 1\right) \cdot re\right) \cdot im \]

   8. Applied rewrites 62.8%

      \[\leadsto \left(\left(\left(\left(im \cdot im\right) \cdot -0.008333333333333333 - 0.16666666666666666\right) \cdot \left(im \cdot im\right) - 1\right) \cdot re\right) \cdot im \]

   9. Taylor expanded in re around 0

      \[\leadsto im \cdot \color{blue}{\left(re \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(im \cdot re\right) \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - \color{blue}{1}\right) \]

       2. *-commutative N/A

          \[\leadsto \left(im \cdot re\right) \cdot \left(\left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) \cdot {im}^{2} - 1\right) \]

       3. *-commutative N/A

          \[\leadsto \left(im \cdot re\right) \cdot \left(\left({im}^{2} \cdot \frac{-1}{120} - \frac{1}{6}\right) \cdot {im}^{2} - 1\right) \]

       4. pow2 N/A

          \[\leadsto \left(im \cdot re\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \frac{-1}{120} - \frac{1}{6}\right) \cdot {im}^{2} - 1\right) \]

       5. pow2 N/A

          \[\leadsto \left(im \cdot re\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \frac{-1}{120} - \frac{1}{6}\right) \cdot \left(im \cdot im\right) - 1\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \left(im \cdot re\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \frac{-1}{120} - \frac{1}{6}\right) \cdot \left(im \cdot im\right) - \color{blue}{1}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \left(im \cdot re\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \frac{-1}{120} - \frac{1}{6}\right) \cdot \left(im \cdot im\right) - 1\right) \]

       8. lift-*.f64 N/A

          \[\leadsto \left(im \cdot re\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \frac{-1}{120} - \frac{1}{6}\right) \cdot \left(im \cdot im\right) - 1\right) \]

       9. lift-*.f64 N/A

          \[\leadsto \left(im \cdot re\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \frac{-1}{120} - \frac{1}{6}\right) \cdot \left(im \cdot im\right) - 1\right) \]

       10. lift--.f64 N/A

           \[\leadsto \left(im \cdot re\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \frac{-1}{120} - \frac{1}{6}\right) \cdot \left(im \cdot im\right) - 1\right) \]

       11. lift-*.f64 N/A

           \[\leadsto \left(im \cdot re\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \frac{-1}{120} - \frac{1}{6}\right) \cdot \left(im \cdot im\right) - 1\right) \]

       12. lift-*.f64 N/A

           \[\leadsto \left(im \cdot re\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \frac{-1}{120} - \frac{1}{6}\right) \cdot \left(im \cdot im\right) - 1\right) \]

       13. lift--.f64 62.8

           \[\leadsto \left(im \cdot re\right) \cdot \left(\left(\left(im \cdot im\right) \cdot -0.008333333333333333 - 0.16666666666666666\right) \cdot \left(im \cdot im\right) - 1\right) \]

   11. Applied rewrites 62.8%

       \[\leadsto \left(im \cdot re\right) \cdot \color{blue}{\left(\left(\left(\left(im \cdot im\right) \cdot -0.008333333333333333 - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 92.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\sin re \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\right) \cdot im\\ \mathbf{if}\;im \leq -8.2 \cdot 10^{+113}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im \leq -6.1:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(e^{-im} - 1\right)\\ \mathbf{elif}\;im \leq 4.7 \lor \neg \left(im \leq 1.9 \cdot 10^{+129}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(1 - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* (sin re) (fma (* -0.16666666666666666 im) im -1.0)) im)))
   (if (<= im -8.2e+113)
     t_0
     (if (<= im -6.1)
       (* (* re 0.5) (- (exp (- im)) 1.0))
       (if (or (<= im 4.7) (not (<= im 1.9e+129)))
         t_0
         (* (* 0.5 re) (- 1.0 (exp im))))))))

C

double code(double re, double im) {
	double t_0 = (sin(re) * fma((-0.16666666666666666 * im), im, -1.0)) * im;
	double tmp;
	if (im <= -8.2e+113) {
		tmp = t_0;
	} else if (im <= -6.1) {
		tmp = (re * 0.5) * (exp(-im) - 1.0);
	} else if ((im <= 4.7) || !(im <= 1.9e+129)) {
		tmp = t_0;
	} else {
		tmp = (0.5 * re) * (1.0 - exp(im));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(sin(re) * fma(Float64(-0.16666666666666666 * im), im, -1.0)) * im)
	tmp = 0.0
	if (im <= -8.2e+113)
		tmp = t_0;
	elseif (im <= -6.1)
		tmp = Float64(Float64(re * 0.5) * Float64(exp(Float64(-im)) - 1.0));
	elseif ((im <= 4.7) || !(im <= 1.9e+129))
		tmp = t_0;
	else
		tmp = Float64(Float64(0.5 * re) * Float64(1.0 - exp(im)));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Sin[re], $MachinePrecision] * N[(N[(-0.16666666666666666 * im), $MachinePrecision] * im + -1.0), $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[im, -8.2e+113], t$95$0, If[LessEqual[im, -6.1], N[(N[(re * 0.5), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[im, 4.7], N[Not[LessEqual[im, 1.9e+129]], $MachinePrecision]], t$95$0, 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 < -8.19999999999999985e113 or -6.0999999999999996 < im < 4.70000000000000018 or 1.90000000000000003e129 < im

   1. Initial program 57.8%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 98.5

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

   5. Applied rewrites 98.5%

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

   if -8.19999999999999985e113 < im < -6.0999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 78.9

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

   8. Applied rewrites 78.9%

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

   if 4.70000000000000018 < im < 1.90000000000000003e129

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 86.5%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -8.2 \cdot 10^{+113}:\\ \;\;\;\;\left(\sin re \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\right) \cdot im\\ \mathbf{elif}\;im \leq -6.1:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(e^{-im} - 1\right)\\ \mathbf{elif}\;im \leq 4.7 \lor \neg \left(im \leq 1.9 \cdot 10^{+129}\right):\\ \;\;\;\;\left(\sin re \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(1 - e^{im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 54.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(re \cdot \left(im \cdot re\right), 0.16666666666666666, -im\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot re\right) \cdot \left(\left(\left(\left(-0.008333333333333333 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im - 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.005)
   (* (fma (* re (* im re)) 0.16666666666666666 (- im)) re)
   (* (* im re) (- (* (* (* (* -0.008333333333333333 im) im) im) im) 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.005) {
		tmp = fma((re * (im * re)), 0.16666666666666666, -im) * re;
	} else {
		tmp = (im * re) * (((((-0.008333333333333333 * im) * im) * im) * im) - 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.005)
		tmp = Float64(fma(Float64(re * Float64(im * re)), 0.16666666666666666, Float64(-im)) * re);
	else
		tmp = Float64(Float64(im * re) * Float64(Float64(Float64(Float64(Float64(-0.008333333333333333 * im) * im) * im) * im) - 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.005], N[(N[(N[(re * N[(im * re), $MachinePrecision]), $MachinePrecision] * 0.16666666666666666 + (-im)), $MachinePrecision] * re), $MachinePrecision], N[(N[(im * re), $MachinePrecision] * N[(N[(N[(N[(N[(-0.008333333333333333 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < -0.0050000000000000001

   1. Initial program 48.3%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\sin re\right)\right) \cdot im \]

      5. lower-neg.f64 N/A

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

      6. lift-sin.f64 57.6

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

   5. Applied rewrites 57.6%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 13.3

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

   8. Applied rewrites 13.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 13.3

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

   10. Applied rewrites 13.3%

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

   if -0.0050000000000000001 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 75.4%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot \color{blue}{im} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot \color{blue}{im} \]

   5. Applied rewrites 82.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sin re \cdot \mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right), im \cdot im, -\sin re\right) \cdot im} \]

   6. Taylor expanded in re around 0

      \[\leadsto \left(re \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right)\right) \cdot im \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right) \cdot re\right) \cdot im \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right) \cdot re\right) \cdot im \]

      3. lower--.f64 N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right) \cdot re\right) \cdot im \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) \cdot {im}^{2} - 1\right) \cdot re\right) \cdot im \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) \cdot {im}^{2} - 1\right) \cdot re\right) \cdot im \]

      6. lower--.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) \cdot {im}^{2} - 1\right) \cdot re\right) \cdot im \]

      7. *-commutative N/A

         \[\leadsto \left(\left(\left({im}^{2} \cdot \frac{-1}{120} - \frac{1}{6}\right) \cdot {im}^{2} - 1\right) \cdot re\right) \cdot im \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\left(\left({im}^{2} \cdot \frac{-1}{120} - \frac{1}{6}\right) \cdot {im}^{2} - 1\right) \cdot re\right) \cdot im \]

      9. pow2 N/A

         \[\leadsto \left(\left(\left(\left(im \cdot im\right) \cdot \frac{-1}{120} - \frac{1}{6}\right) \cdot {im}^{2} - 1\right) \cdot re\right) \cdot im \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(\left(im \cdot im\right) \cdot \frac{-1}{120} - \frac{1}{6}\right) \cdot {im}^{2} - 1\right) \cdot re\right) \cdot im \]

      11. pow2 N/A

          \[\leadsto \left(\left(\left(\left(im \cdot im\right) \cdot \frac{-1}{120} - \frac{1}{6}\right) \cdot \left(im \cdot im\right) - 1\right) \cdot re\right) \cdot im \]

      12. lift-*.f64 62.8

          \[\leadsto \left(\left(\left(\left(im \cdot im\right) \cdot -0.008333333333333333 - 0.16666666666666666\right) \cdot \left(im \cdot im\right) - 1\right) \cdot re\right) \cdot im \]

   8. Applied rewrites 62.8%

      \[\leadsto \left(\left(\left(\left(im \cdot im\right) \cdot -0.008333333333333333 - 0.16666666666666666\right) \cdot \left(im \cdot im\right) - 1\right) \cdot re\right) \cdot im \]

   9. Taylor expanded in re around 0

      \[\leadsto im \cdot \color{blue}{\left(re \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(im \cdot re\right) \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - \color{blue}{1}\right) \]

       2. *-commutative N/A

          \[\leadsto \left(im \cdot re\right) \cdot \left(\left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) \cdot {im}^{2} - 1\right) \]

       3. *-commutative N/A

          \[\leadsto \left(im \cdot re\right) \cdot \left(\left({im}^{2} \cdot \frac{-1}{120} - \frac{1}{6}\right) \cdot {im}^{2} - 1\right) \]

       4. pow2 N/A

          \[\leadsto \left(im \cdot re\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \frac{-1}{120} - \frac{1}{6}\right) \cdot {im}^{2} - 1\right) \]

       5. pow2 N/A

          \[\leadsto \left(im \cdot re\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \frac{-1}{120} - \frac{1}{6}\right) \cdot \left(im \cdot im\right) - 1\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \left(im \cdot re\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \frac{-1}{120} - \frac{1}{6}\right) \cdot \left(im \cdot im\right) - \color{blue}{1}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \left(im \cdot re\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \frac{-1}{120} - \frac{1}{6}\right) \cdot \left(im \cdot im\right) - 1\right) \]

       8. lift-*.f64 N/A

          \[\leadsto \left(im \cdot re\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \frac{-1}{120} - \frac{1}{6}\right) \cdot \left(im \cdot im\right) - 1\right) \]

       9. lift-*.f64 N/A

          \[\leadsto \left(im \cdot re\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \frac{-1}{120} - \frac{1}{6}\right) \cdot \left(im \cdot im\right) - 1\right) \]

       10. lift--.f64 N/A

           \[\leadsto \left(im \cdot re\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \frac{-1}{120} - \frac{1}{6}\right) \cdot \left(im \cdot im\right) - 1\right) \]

       11. lift-*.f64 N/A

           \[\leadsto \left(im \cdot re\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \frac{-1}{120} - \frac{1}{6}\right) \cdot \left(im \cdot im\right) - 1\right) \]

       12. lift-*.f64 N/A

           \[\leadsto \left(im \cdot re\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \frac{-1}{120} - \frac{1}{6}\right) \cdot \left(im \cdot im\right) - 1\right) \]

       13. lift--.f64 62.8

           \[\leadsto \left(im \cdot re\right) \cdot \left(\left(\left(im \cdot im\right) \cdot -0.008333333333333333 - 0.16666666666666666\right) \cdot \left(im \cdot im\right) - 1\right) \]

   11. Applied rewrites 62.8%

       \[\leadsto \left(im \cdot re\right) \cdot \color{blue}{\left(\left(\left(\left(im \cdot im\right) \cdot -0.008333333333333333 - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right)} \]

   12. Taylor expanded in im around inf

       \[\leadsto \left(im \cdot re\right) \cdot \left(\left(\left(\frac{-1}{120} \cdot {im}^{2}\right) \cdot im\right) \cdot im - 1\right) \]
   13. Step-by-step derivation

       1. pow2 N/A

          \[\leadsto \left(im \cdot re\right) \cdot \left(\left(\left(\frac{-1}{120} \cdot \left(im \cdot im\right)\right) \cdot im\right) \cdot im - 1\right) \]

       2. associate-*r* N/A

          \[\leadsto \left(im \cdot re\right) \cdot \left(\left(\left(\left(\frac{-1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im - 1\right) \]

       3. lower-*.f64 N/A

          \[\leadsto \left(im \cdot re\right) \cdot \left(\left(\left(\left(\frac{-1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im - 1\right) \]

       4. lower-*.f64 62.8

          \[\leadsto \left(im \cdot re\right) \cdot \left(\left(\left(\left(-0.008333333333333333 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im - 1\right) \]

   14. Applied rewrites 62.8%

       \[\leadsto \left(im \cdot re\right) \cdot \left(\left(\left(\left(-0.008333333333333333 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im - 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 52.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(re \cdot \left(im \cdot re\right), 0.16666666666666666, -im\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(\left(-0.3333333333333333 \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.005)
   (* (fma (* re (* im re)) 0.16666666666666666 (- im)) re)
   (* (* 0.5 re) (* (- (* -0.3333333333333333 (* im im)) 2.0) im))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.005) {
		tmp = fma((re * (im * re)), 0.16666666666666666, -im) * re;
	} else {
		tmp = (0.5 * re) * (((-0.3333333333333333 * (im * im)) - 2.0) * im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.005)
		tmp = Float64(fma(Float64(re * Float64(im * re)), 0.16666666666666666, Float64(-im)) * re);
	else
		tmp = Float64(Float64(0.5 * re) * Float64(Float64(Float64(-0.3333333333333333 * Float64(im * im)) - 2.0) * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.005], N[(N[(N[(re * N[(im * re), $MachinePrecision]), $MachinePrecision] * 0.16666666666666666 + (-im)), $MachinePrecision] * re), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(N[(N[(-0.3333333333333333 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < -0.0050000000000000001

   1. Initial program 48.3%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\sin re\right)\right) \cdot im \]

      5. lower-neg.f64 N/A

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

      6. lift-sin.f64 57.6

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

   5. Applied rewrites 57.6%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 13.3

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

   8. Applied rewrites 13.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 13.3

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

   10. Applied rewrites 13.3%

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

   if -0.0050000000000000001 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 75.4%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 2\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(\frac{-1}{3} \cdot {im}^{2} - 2\right) \cdot \color{blue}{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(\frac{-1}{3} \cdot {im}^{2} - 2\right) \cdot \color{blue}{im}\right) \]

      3. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(\frac{-1}{3} \cdot {im}^{2} - 2\right) \cdot im\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(\frac{-1}{3} \cdot {im}^{2} - 2\right) \cdot im\right) \]

      5. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(\frac{-1}{3} \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      6. lower-*.f64 80.4

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\left(-0.3333333333333333 \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   5. Applied rewrites 80.4%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left(-0.3333333333333333 \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{re}\right) \cdot \left(\left(\frac{-1}{3} \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 61.6%

      \[\leadsto \left(0.5 \cdot \color{blue}{re}\right) \cdot \left(\left(-0.3333333333333333 \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 49.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(re \cdot \left(im \cdot re\right), 0.16666666666666666, -im\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot re\right) \cdot \left(\left(-0.16666666666666666 \cdot im\right) \cdot im - 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.005)
   (* (fma (* re (* im re)) 0.16666666666666666 (- im)) re)
   (* (* im re) (- (* (* -0.16666666666666666 im) im) 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.005) {
		tmp = fma((re * (im * re)), 0.16666666666666666, -im) * re;
	} else {
		tmp = (im * re) * (((-0.16666666666666666 * im) * im) - 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.005)
		tmp = Float64(fma(Float64(re * Float64(im * re)), 0.16666666666666666, Float64(-im)) * re);
	else
		tmp = Float64(Float64(im * re) * Float64(Float64(Float64(-0.16666666666666666 * im) * im) - 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.005], N[(N[(N[(re * N[(im * re), $MachinePrecision]), $MachinePrecision] * 0.16666666666666666 + (-im)), $MachinePrecision] * re), $MachinePrecision], N[(N[(im * re), $MachinePrecision] * N[(N[(N[(-0.16666666666666666 * im), $MachinePrecision] * im), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < -0.0050000000000000001

   1. Initial program 48.3%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\sin re\right)\right) \cdot im \]

      5. lower-neg.f64 N/A

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

      6. lift-sin.f64 57.6

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

   5. Applied rewrites 57.6%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 13.3

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

   8. Applied rewrites 13.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 13.3

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

   10. Applied rewrites 13.3%

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

   if -0.0050000000000000001 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 75.4%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot \color{blue}{im} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot \color{blue}{im} \]

   5. Applied rewrites 82.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sin re \cdot \mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right), im \cdot im, -\sin re\right) \cdot im} \]

   6. Taylor expanded in re around 0

      \[\leadsto \left(re \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right)\right) \cdot im \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right) \cdot re\right) \cdot im \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right) \cdot re\right) \cdot im \]

      3. lower--.f64 N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right) \cdot re\right) \cdot im \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) \cdot {im}^{2} - 1\right) \cdot re\right) \cdot im \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) \cdot {im}^{2} - 1\right) \cdot re\right) \cdot im \]

      6. lower--.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) \cdot {im}^{2} - 1\right) \cdot re\right) \cdot im \]

      7. *-commutative N/A

         \[\leadsto \left(\left(\left({im}^{2} \cdot \frac{-1}{120} - \frac{1}{6}\right) \cdot {im}^{2} - 1\right) \cdot re\right) \cdot im \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\left(\left({im}^{2} \cdot \frac{-1}{120} - \frac{1}{6}\right) \cdot {im}^{2} - 1\right) \cdot re\right) \cdot im \]

      9. pow2 N/A

         \[\leadsto \left(\left(\left(\left(im \cdot im\right) \cdot \frac{-1}{120} - \frac{1}{6}\right) \cdot {im}^{2} - 1\right) \cdot re\right) \cdot im \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(\left(im \cdot im\right) \cdot \frac{-1}{120} - \frac{1}{6}\right) \cdot {im}^{2} - 1\right) \cdot re\right) \cdot im \]

      11. pow2 N/A

          \[\leadsto \left(\left(\left(\left(im \cdot im\right) \cdot \frac{-1}{120} - \frac{1}{6}\right) \cdot \left(im \cdot im\right) - 1\right) \cdot re\right) \cdot im \]

      12. lift-*.f64 62.8

          \[\leadsto \left(\left(\left(\left(im \cdot im\right) \cdot -0.008333333333333333 - 0.16666666666666666\right) \cdot \left(im \cdot im\right) - 1\right) \cdot re\right) \cdot im \]

   8. Applied rewrites 62.8%

      \[\leadsto \left(\left(\left(\left(im \cdot im\right) \cdot -0.008333333333333333 - 0.16666666666666666\right) \cdot \left(im \cdot im\right) - 1\right) \cdot re\right) \cdot im \]

   9. Taylor expanded in re around 0

      \[\leadsto im \cdot \color{blue}{\left(re \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(im \cdot re\right) \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - \color{blue}{1}\right) \]

       2. *-commutative N/A

          \[\leadsto \left(im \cdot re\right) \cdot \left(\left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) \cdot {im}^{2} - 1\right) \]

       3. *-commutative N/A

          \[\leadsto \left(im \cdot re\right) \cdot \left(\left({im}^{2} \cdot \frac{-1}{120} - \frac{1}{6}\right) \cdot {im}^{2} - 1\right) \]

       4. pow2 N/A

          \[\leadsto \left(im \cdot re\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \frac{-1}{120} - \frac{1}{6}\right) \cdot {im}^{2} - 1\right) \]

       5. pow2 N/A

          \[\leadsto \left(im \cdot re\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \frac{-1}{120} - \frac{1}{6}\right) \cdot \left(im \cdot im\right) - 1\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \left(im \cdot re\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \frac{-1}{120} - \frac{1}{6}\right) \cdot \left(im \cdot im\right) - \color{blue}{1}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \left(im \cdot re\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \frac{-1}{120} - \frac{1}{6}\right) \cdot \left(im \cdot im\right) - 1\right) \]

       8. lift-*.f64 N/A

          \[\leadsto \left(im \cdot re\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \frac{-1}{120} - \frac{1}{6}\right) \cdot \left(im \cdot im\right) - 1\right) \]

       9. lift-*.f64 N/A

          \[\leadsto \left(im \cdot re\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \frac{-1}{120} - \frac{1}{6}\right) \cdot \left(im \cdot im\right) - 1\right) \]

       10. lift--.f64 N/A

           \[\leadsto \left(im \cdot re\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \frac{-1}{120} - \frac{1}{6}\right) \cdot \left(im \cdot im\right) - 1\right) \]

       11. lift-*.f64 N/A

           \[\leadsto \left(im \cdot re\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \frac{-1}{120} - \frac{1}{6}\right) \cdot \left(im \cdot im\right) - 1\right) \]

       12. lift-*.f64 N/A

           \[\leadsto \left(im \cdot re\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \frac{-1}{120} - \frac{1}{6}\right) \cdot \left(im \cdot im\right) - 1\right) \]

       13. lift--.f64 62.8

           \[\leadsto \left(im \cdot re\right) \cdot \left(\left(\left(im \cdot im\right) \cdot -0.008333333333333333 - 0.16666666666666666\right) \cdot \left(im \cdot im\right) - 1\right) \]

   11. Applied rewrites 62.8%

       \[\leadsto \left(im \cdot re\right) \cdot \color{blue}{\left(\left(\left(\left(im \cdot im\right) \cdot -0.008333333333333333 - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right)} \]

   12. Taylor expanded in im around 0

       \[\leadsto \left(im \cdot re\right) \cdot \left(\left(\frac{-1}{6} \cdot im\right) \cdot im - 1\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 57.4%

       \[\leadsto \left(im \cdot re\right) \cdot \left(\left(-0.16666666666666666 \cdot im\right) \cdot im - 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 34.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(re \cdot \left(im \cdot re\right), 0.16666666666666666, -im\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(-re\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.005)
   (* (fma (* re (* im re)) 0.16666666666666666 (- im)) re)
   (* (- re) im)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.005) {
		tmp = fma((re * (im * re)), 0.16666666666666666, -im) * re;
	} else {
		tmp = -re * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.005)
		tmp = Float64(fma(Float64(re * Float64(im * re)), 0.16666666666666666, Float64(-im)) * re);
	else
		tmp = Float64(Float64(-re) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.005], N[(N[(N[(re * N[(im * re), $MachinePrecision]), $MachinePrecision] * 0.16666666666666666 + (-im)), $MachinePrecision] * re), $MachinePrecision], N[((-re) * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < -0.0050000000000000001

   1. Initial program 48.3%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\sin re\right)\right) \cdot im \]

      5. lower-neg.f64 N/A

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

      6. lift-sin.f64 57.6

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

   5. Applied rewrites 57.6%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 13.3

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

   8. Applied rewrites 13.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 13.3

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

   10. Applied rewrites 13.3%

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

   if -0.0050000000000000001 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 75.4%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\sin re\right)\right) \cdot im \]

      5. lower-neg.f64 N/A

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

      6. lift-sin.f64 47.7

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

   5. Applied rewrites 47.7%

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

   6. Taylor expanded in re around 0

      \[\leadsto \left(-re\right) \cdot im \]
   7. Step-by-step derivation

   8. Applied rewrites 37.2%

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

Alternative 22: 34.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.005:\\ \;\;\;\;\left(\left(\left(re \cdot re\right) \cdot 0.16666666666666666 - 1\right) \cdot im\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(-re\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.005)
   (* (* (- (* (* re re) 0.16666666666666666) 1.0) im) re)
   (* (- re) im)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.005) {
		tmp = ((((re * re) * 0.16666666666666666) - 1.0) * im) * re;
	} else {
		tmp = -re * 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 * sin(re)) <= (-0.005d0)) then
        tmp = ((((re * re) * 0.16666666666666666d0) - 1.0d0) * im) * re
    else
        tmp = -re * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((0.5 * Math.sin(re)) <= -0.005) {
		tmp = ((((re * re) * 0.16666666666666666) - 1.0) * im) * re;
	} else {
		tmp = -re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (0.5 * math.sin(re)) <= -0.005:
		tmp = ((((re * re) * 0.16666666666666666) - 1.0) * im) * re
	else:
		tmp = -re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.005)
		tmp = Float64(Float64(Float64(Float64(Float64(re * re) * 0.16666666666666666) - 1.0) * im) * re);
	else
		tmp = Float64(Float64(-re) * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((0.5 * sin(re)) <= -0.005)
		tmp = ((((re * re) * 0.16666666666666666) - 1.0) * im) * re;
	else
		tmp = -re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.005], N[(N[(N[(N[(N[(re * re), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] - 1.0), $MachinePrecision] * im), $MachinePrecision] * re), $MachinePrecision], N[((-re) * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < -0.0050000000000000001

   1. Initial program 48.3%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\sin re\right)\right) \cdot im \]

      5. lower-neg.f64 N/A

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

      6. lift-sin.f64 57.6

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

   5. Applied rewrites 57.6%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 13.3

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

   8. Applied rewrites 13.3%

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

   9. Taylor expanded in im around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower--.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 13.3

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

   11. Applied rewrites 13.3%

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

   if -0.0050000000000000001 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 75.4%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\sin re\right)\right) \cdot im \]

      5. lower-neg.f64 N/A

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

      6. lift-sin.f64 47.7

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

   5. Applied rewrites 47.7%

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

   6. Taylor expanded in re around 0

      \[\leadsto \left(-re\right) \cdot im \]
   7. Step-by-step derivation

   8. Applied rewrites 37.2%

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

Alternative 23: 86.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -6.1:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(e^{-im} - 1\right)\\ \mathbf{elif}\;im \leq 3.15:\\ \;\;\;\;\left(-\sin re\right) \cdot im\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+135}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(1 - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -6.1)
   (* (* re 0.5) (- (exp (- im)) 1.0))
   (if (<= im 3.15)
     (* (- (sin re)) im)
     (if (<= im 2e+135)
       (* (* 0.5 re) (- 1.0 (exp im)))
       (*
        (* (fma (* re re) -0.08333333333333333 0.5) re)
        (*
         (-
          (*
           (-
            (* (* (* (* -0.0003968253968253968 im) im) im) im)
            0.3333333333333333)
           (* im im))
          2.0)
         im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -6.1) {
		tmp = (re * 0.5) * (exp(-im) - 1.0);
	} else if (im <= 3.15) {
		tmp = -sin(re) * im;
	} else if (im <= 2e+135) {
		tmp = (0.5 * re) * (1.0 - exp(im));
	} else {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * re) * ((((((((-0.0003968253968253968 * im) * im) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -6.1)
		tmp = Float64(Float64(re * 0.5) * Float64(exp(Float64(-im)) - 1.0));
	elseif (im <= 3.15)
		tmp = Float64(Float64(-sin(re)) * im);
	elseif (im <= 2e+135)
		tmp = Float64(Float64(0.5 * re) * Float64(1.0 - exp(im)));
	else
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * im) * im) * im) * im) - 0.3333333333333333) * Float64(im * im)) - 2.0) * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, -6.1], N[(N[(re * 0.5), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 3.15], N[((-N[Sin[re], $MachinePrecision]) * im), $MachinePrecision], If[LessEqual[im, 2e+135], N[(N[(0.5 * re), $MachinePrecision] * N[(1.0 - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < -6.0999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 71.2

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

   8. Applied rewrites 71.2%

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

   if -6.0999999999999996 < im < 3.14999999999999991

   1. Initial program 32.5%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\sin re\right)\right) \cdot im \]

      5. lower-neg.f64 N/A

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

      6. lift-sin.f64 99.3

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

   5. Applied rewrites 99.3%

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

   if 3.14999999999999991 < im < 1.99999999999999992e135

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 87.2%

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

   if 1.99999999999999992e135 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

   5. Applied rewrites 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]

   6. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot \color{blue}{re}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot \color{blue}{re}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       3. +-commutative N/A

          \[\leadsto \left(\left(\frac{-1}{12} \cdot {re}^{2} + \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       4. *-commutative N/A

          \[\leadsto \left(\left({re}^{2} \cdot \frac{-1}{12} + \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       5. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left({re}^{2}, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       6. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       7. lift-*.f64 84.8

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   11. Applied rewrites 84.8%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)} \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   12. Taylor expanded in im around inf

       \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot {im}^{2}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   13. Step-by-step derivation

       1. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right)\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       2. associate-*r* N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       3. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       4. lower-*.f64 84.8

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   14. Applied rewrites 84.8%

       \[\leadsto \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 24: 84.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1360:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \mathbf{elif}\;im \leq 3.15:\\ \;\;\;\;\left(-\sin re\right) \cdot im\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+135}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(1 - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -1360.0)
   (*
    (* 0.5 re)
    (*
     (-
      (*
       (-
        (*
         (* (- (* -0.0003968253968253968 (* im im)) 0.016666666666666666) im)
         im)
        0.3333333333333333)
       (* im im))
      2.0)
     im))
   (if (<= im 3.15)
     (* (- (sin re)) im)
     (if (<= im 2e+135)
       (* (* 0.5 re) (- 1.0 (exp im)))
       (*
        (* (fma (* re re) -0.08333333333333333 0.5) re)
        (*
         (-
          (*
           (-
            (* (* (* (* -0.0003968253968253968 im) im) im) im)
            0.3333333333333333)
           (* im im))
          2.0)
         im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -1360.0) {
		tmp = (0.5 * re) * ((((((((-0.0003968253968253968 * (im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	} else if (im <= 3.15) {
		tmp = -sin(re) * im;
	} else if (im <= 2e+135) {
		tmp = (0.5 * re) * (1.0 - exp(im));
	} else {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * re) * ((((((((-0.0003968253968253968 * im) * im) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -1360.0)
		tmp = Float64(Float64(0.5 * re) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * Float64(im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * Float64(im * im)) - 2.0) * im));
	elseif (im <= 3.15)
		tmp = Float64(Float64(-sin(re)) * im);
	elseif (im <= 2e+135)
		tmp = Float64(Float64(0.5 * re) * Float64(1.0 - exp(im)));
	else
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * im) * im) * im) * im) - 0.3333333333333333) * Float64(im * im)) - 2.0) * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, -1360.0], N[(N[(0.5 * re), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.016666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 3.15], N[((-N[Sin[re], $MachinePrecision]) * im), $MachinePrecision], If[LessEqual[im, 2e+135], N[(N[(0.5 * re), $MachinePrecision] * N[(1.0 - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < -1360

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

   5. Applied rewrites 87.2%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{re}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 67.9%

      \[\leadsto \left(0.5 \cdot \color{blue}{re}\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   if -1360 < im < 3.14999999999999991

   1. Initial program 32.5%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\sin re\right)\right) \cdot im \]

      5. lower-neg.f64 N/A

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

      6. lift-sin.f64 99.3

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

   5. Applied rewrites 99.3%

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

   if 3.14999999999999991 < im < 1.99999999999999992e135

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 87.2%

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

   if 1.99999999999999992e135 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

   5. Applied rewrites 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]

   6. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot \color{blue}{re}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot \color{blue}{re}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       3. +-commutative N/A

          \[\leadsto \left(\left(\frac{-1}{12} \cdot {re}^{2} + \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       4. *-commutative N/A

          \[\leadsto \left(\left({re}^{2} \cdot \frac{-1}{12} + \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       5. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left({re}^{2}, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       6. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       7. lift-*.f64 84.8

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   11. Applied rewrites 84.8%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)} \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   12. Taylor expanded in im around inf

       \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot {im}^{2}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   13. Step-by-step derivation

       1. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right)\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       2. associate-*r* N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       3. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       4. lower-*.f64 84.8

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   14. Applied rewrites 84.8%

       \[\leadsto \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 25: 34.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.005:\\ \;\;\;\;\left(\left(\left(re \cdot re\right) \cdot im\right) \cdot 0.16666666666666666\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(-re\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.005)
   (* (* (* (* re re) im) 0.16666666666666666) re)
   (* (- re) im)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.005) {
		tmp = (((re * re) * im) * 0.16666666666666666) * re;
	} else {
		tmp = -re * 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 * sin(re)) <= (-0.005d0)) then
        tmp = (((re * re) * im) * 0.16666666666666666d0) * re
    else
        tmp = -re * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((0.5 * Math.sin(re)) <= -0.005) {
		tmp = (((re * re) * im) * 0.16666666666666666) * re;
	} else {
		tmp = -re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (0.5 * math.sin(re)) <= -0.005:
		tmp = (((re * re) * im) * 0.16666666666666666) * re
	else:
		tmp = -re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.005)
		tmp = Float64(Float64(Float64(Float64(re * re) * im) * 0.16666666666666666) * re);
	else
		tmp = Float64(Float64(-re) * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((0.5 * sin(re)) <= -0.005)
		tmp = (((re * re) * im) * 0.16666666666666666) * re;
	else
		tmp = -re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.005], N[(N[(N[(N[(re * re), $MachinePrecision] * im), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * re), $MachinePrecision], N[((-re) * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < -0.0050000000000000001

   1. Initial program 48.3%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\sin re\right)\right) \cdot im \]

      5. lower-neg.f64 N/A

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

      6. lift-sin.f64 57.6

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

   5. Applied rewrites 57.6%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 13.3

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

   8. Applied rewrites 13.3%

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

   9. Taylor expanded in re around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 13.3

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

   11. Applied rewrites 13.3%

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

   if -0.0050000000000000001 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 75.4%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\sin re\right)\right) \cdot im \]

      5. lower-neg.f64 N/A

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

      6. lift-sin.f64 47.7

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

   5. Applied rewrites 47.7%

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

   6. Taylor expanded in re around 0

      \[\leadsto \left(-re\right) \cdot im \]
   7. Step-by-step derivation

   8. Applied rewrites 37.2%

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

Alternative 26: 82.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot re\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \mathbf{if}\;im \leq -1360:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im \leq 85:\\ \;\;\;\;\left(-\sin re\right) \cdot im\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+135}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (*
          (* 0.5 re)
          (*
           (-
            (*
             (-
              (*
               (*
                (- (* -0.0003968253968253968 (* im im)) 0.016666666666666666)
                im)
               im)
              0.3333333333333333)
             (* im im))
            2.0)
           im))))
   (if (<= im -1360.0)
     t_0
     (if (<= im 85.0)
       (* (- (sin re)) im)
       (if (<= im 2e+135)
         t_0
         (*
          (* (fma (* re re) -0.08333333333333333 0.5) re)
          (*
           (-
            (*
             (-
              (* (* (* (* -0.0003968253968253968 im) im) im) im)
              0.3333333333333333)
             (* im im))
            2.0)
           im)))))))

C

double code(double re, double im) {
	double t_0 = (0.5 * re) * ((((((((-0.0003968253968253968 * (im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	double tmp;
	if (im <= -1360.0) {
		tmp = t_0;
	} else if (im <= 85.0) {
		tmp = -sin(re) * im;
	} else if (im <= 2e+135) {
		tmp = t_0;
	} else {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * re) * ((((((((-0.0003968253968253968 * im) * im) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * re) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * Float64(im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * Float64(im * im)) - 2.0) * im))
	tmp = 0.0
	if (im <= -1360.0)
		tmp = t_0;
	elseif (im <= 85.0)
		tmp = Float64(Float64(-sin(re)) * im);
	elseif (im <= 2e+135)
		tmp = t_0;
	else
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * im) * im) * im) * im) - 0.3333333333333333) * Float64(im * im)) - 2.0) * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * re), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.016666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1360.0], t$95$0, If[LessEqual[im, 85.0], N[((-N[Sin[re], $MachinePrecision]) * im), $MachinePrecision], If[LessEqual[im, 2e+135], t$95$0, N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -1360 or 85 < im < 1.99999999999999992e135

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

   5. Applied rewrites 74.8%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{re}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 62.9%

      \[\leadsto \left(0.5 \cdot \color{blue}{re}\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   if -1360 < im < 85

   1. Initial program 32.5%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\sin re\right)\right) \cdot im \]

      5. lower-neg.f64 N/A

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

      6. lift-sin.f64 99.3

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

   5. Applied rewrites 99.3%

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

   if 1.99999999999999992e135 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

   5. Applied rewrites 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]

   6. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot \color{blue}{re}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot \color{blue}{re}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       3. +-commutative N/A

          \[\leadsto \left(\left(\frac{-1}{12} \cdot {re}^{2} + \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       4. *-commutative N/A

          \[\leadsto \left(\left({re}^{2} \cdot \frac{-1}{12} + \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       5. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left({re}^{2}, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       6. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       7. lift-*.f64 84.8

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   11. Applied rewrites 84.8%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)} \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   12. Taylor expanded in im around inf

       \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot {im}^{2}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   13. Step-by-step derivation

       1. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right)\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       2. associate-*r* N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       3. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       4. lower-*.f64 84.8

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   14. Applied rewrites 84.8%

       \[\leadsto \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 27: 31.9% accurate, 39.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.1%

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

3. Taylor expanded in im around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. mul-1-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(\sin re\right)\right) \cdot im \]

   5. lower-neg.f64 N/A

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

   6. lift-sin.f64 50.8

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

5. Applied rewrites 50.8%

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

6. Taylor expanded in re around 0

   \[\leadsto \left(-re\right) \cdot im \]
7. Step-by-step derivation

8. Applied rewrites 30.9%

   \[\leadsto \left(-re\right) \cdot im \]
9. Add Preprocessing

Developer Target 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|im\right| < 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(0.16666666666666666 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (< (fabs im) 1.0)
   (-
    (*
     (sin re)
     (+
      (+ im (* (* (* 0.16666666666666666 im) im) im))
      (* (* (* (* (* 0.008333333333333333 im) im) im) im) im))))
   (* (* 0.5 (sin re)) (- (exp (- im)) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if (fabs(im) < 1.0) {
		tmp = -(sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * sin(re)) * (exp(-im) - 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 (abs(im) < 1.0d0) then
        tmp = -(sin(re) * ((im + (((0.16666666666666666d0 * im) * im) * im)) + (((((0.008333333333333333d0 * im) * im) * im) * im) * im)))
    else
        tmp = (0.5d0 * sin(re)) * (exp(-im) - exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.abs(im) < 1.0) {
		tmp = -(Math.sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * Math.sin(re)) * (Math.exp(-im) - Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.fabs(im) < 1.0:
		tmp = -(math.sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)))
	else:
		tmp = (0.5 * math.sin(re)) * (math.exp(-im) - math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (abs(im) < 1.0)
		tmp = Float64(-Float64(sin(re) * Float64(Float64(im + Float64(Float64(Float64(0.16666666666666666 * im) * im) * im)) + Float64(Float64(Float64(Float64(Float64(0.008333333333333333 * im) * im) * im) * im) * im))));
	else
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (abs(im) < 1.0)
		tmp = -(sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	else
		tmp = (0.5 * sin(re)) * (exp(-im) - exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Less[N[Abs[im], $MachinePrecision], 1.0], (-N[(N[Sin[re], $MachinePrecision] * N[(N[(im + N[(N[(N[(0.16666666666666666 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[(0.008333333333333333 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2025056 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (fabs im) 1) (- (* (sin re) (+ im (* 1/6 im im im) (* 1/120 im im im im im)))) (* (* 1/2 (sin re)) (- (exp (- im)) (exp im)))))

  (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))