math.cos on complex, imaginary part

Percentage Accurate: 65.6% → 99.9%

Time: 4.9s

Alternatives: 12

Speedup: 1.5×

• 50.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 65.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.6%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lift--.f64 N/A

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

   6. sub-negate-rev N/A

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

   7. distribute-lft-neg-out N/A

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

   8. metadata-eval N/A

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

   9. mult-flip N/A

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

   10. lift-exp.f64 N/A

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

   11. lift-exp.f64 N/A

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

   12. lift-neg.f64 N/A

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

   13. sinh-def N/A

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

   14. sinh-neg N/A

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

   15. lift-neg.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. lower-sinh.f64 99.9%

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

3. Applied rewrites 99.9%

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

Alternative 2: 93.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \left|im\right| \cdot \left|im\right|\\ t_1 := \left(\left(t\_0 \cdot -0.016666666666666666 - 0.3333333333333333\right) \cdot \left|im\right|\right) \cdot \left|im\right| - 1\\ \mathsf{copysign}\left(1, im\right) \cdot \begin{array}{l} \mathbf{if}\;\left|im\right| \leq 23000000000:\\ \;\;\;\;\left(\sin re \cdot 0.5\right) \cdot \left(\left(1 - \frac{1}{t\_1}\right) \cdot \left(t\_1 \cdot \left|im\right|\right)\right)\\ \mathbf{elif}\;\left|im\right| \leq 1.02 \cdot 10^{+62}:\\ \;\;\;\;-1 \cdot \left(\left|im\right| \cdot \left(\left(re \cdot \left(\left(\left(1 - \frac{6}{re \cdot re}\right) \cdot re\right) \cdot re\right)\right) \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(\left|im\right| \cdot \left(\left(\left(-0.016666666666666666 \cdot t\_0 - 0.3333333333333333\right) \cdot \left|im\right|\right) \cdot \left|im\right| - 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0 (* (fabs im) (fabs im)))
       (t_1
        (-
         (*
          (*
           (- (* t_0 -0.016666666666666666) 0.3333333333333333)
           (fabs im))
          (fabs im))
         1.0)))
  (*
   (copysign 1.0 im)
   (if (<= (fabs im) 23000000000.0)
     (* (* (sin re) 0.5) (* (- 1.0 (/ 1.0 t_1)) (* t_1 (fabs im))))
     (if (<= (fabs im) 1.02e+62)
       (*
        -1.0
        (*
         (fabs im)
         (*
          (* re (* (* (- 1.0 (/ 6.0 (* re re))) re) re))
          -0.16666666666666666)))
       (*
        (* 0.5 (sin re))
        (*
         (fabs im)
         (-
          (*
           (*
            (- (* -0.016666666666666666 t_0) 0.3333333333333333)
            (fabs im))
           (fabs im))
          2.0))))))))

C

double code(double re, double im) {
	double t_0 = fabs(im) * fabs(im);
	double t_1 = ((((t_0 * -0.016666666666666666) - 0.3333333333333333) * fabs(im)) * fabs(im)) - 1.0;
	double tmp;
	if (fabs(im) <= 23000000000.0) {
		tmp = (sin(re) * 0.5) * ((1.0 - (1.0 / t_1)) * (t_1 * fabs(im)));
	} else if (fabs(im) <= 1.02e+62) {
		tmp = -1.0 * (fabs(im) * ((re * (((1.0 - (6.0 / (re * re))) * re) * re)) * -0.16666666666666666));
	} else {
		tmp = (0.5 * sin(re)) * (fabs(im) * (((((-0.016666666666666666 * t_0) - 0.3333333333333333) * fabs(im)) * fabs(im)) - 2.0));
	}
	return copysign(1.0, im) * tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = Math.abs(im) * Math.abs(im);
	double t_1 = ((((t_0 * -0.016666666666666666) - 0.3333333333333333) * Math.abs(im)) * Math.abs(im)) - 1.0;
	double tmp;
	if (Math.abs(im) <= 23000000000.0) {
		tmp = (Math.sin(re) * 0.5) * ((1.0 - (1.0 / t_1)) * (t_1 * Math.abs(im)));
	} else if (Math.abs(im) <= 1.02e+62) {
		tmp = -1.0 * (Math.abs(im) * ((re * (((1.0 - (6.0 / (re * re))) * re) * re)) * -0.16666666666666666));
	} else {
		tmp = (0.5 * Math.sin(re)) * (Math.abs(im) * (((((-0.016666666666666666 * t_0) - 0.3333333333333333) * Math.abs(im)) * Math.abs(im)) - 2.0));
	}
	return Math.copySign(1.0, im) * tmp;
}

Python

def code(re, im):
	t_0 = math.fabs(im) * math.fabs(im)
	t_1 = ((((t_0 * -0.016666666666666666) - 0.3333333333333333) * math.fabs(im)) * math.fabs(im)) - 1.0
	tmp = 0
	if math.fabs(im) <= 23000000000.0:
		tmp = (math.sin(re) * 0.5) * ((1.0 - (1.0 / t_1)) * (t_1 * math.fabs(im)))
	elif math.fabs(im) <= 1.02e+62:
		tmp = -1.0 * (math.fabs(im) * ((re * (((1.0 - (6.0 / (re * re))) * re) * re)) * -0.16666666666666666))
	else:
		tmp = (0.5 * math.sin(re)) * (math.fabs(im) * (((((-0.016666666666666666 * t_0) - 0.3333333333333333) * math.fabs(im)) * math.fabs(im)) - 2.0))
	return math.copysign(1.0, im) * tmp

Julia

function code(re, im)
	t_0 = Float64(abs(im) * abs(im))
	t_1 = Float64(Float64(Float64(Float64(Float64(t_0 * -0.016666666666666666) - 0.3333333333333333) * abs(im)) * abs(im)) - 1.0)
	tmp = 0.0
	if (abs(im) <= 23000000000.0)
		tmp = Float64(Float64(sin(re) * 0.5) * Float64(Float64(1.0 - Float64(1.0 / t_1)) * Float64(t_1 * abs(im))));
	elseif (abs(im) <= 1.02e+62)
		tmp = Float64(-1.0 * Float64(abs(im) * Float64(Float64(re * Float64(Float64(Float64(1.0 - Float64(6.0 / Float64(re * re))) * re) * re)) * -0.16666666666666666)));
	else
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(abs(im) * Float64(Float64(Float64(Float64(Float64(-0.016666666666666666 * t_0) - 0.3333333333333333) * abs(im)) * abs(im)) - 2.0)));
	end
	return Float64(copysign(1.0, im) * tmp)
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = abs(im) * abs(im);
	t_1 = ((((t_0 * -0.016666666666666666) - 0.3333333333333333) * abs(im)) * abs(im)) - 1.0;
	tmp = 0.0;
	if (abs(im) <= 23000000000.0)
		tmp = (sin(re) * 0.5) * ((1.0 - (1.0 / t_1)) * (t_1 * abs(im)));
	elseif (abs(im) <= 1.02e+62)
		tmp = -1.0 * (abs(im) * ((re * (((1.0 - (6.0 / (re * re))) * re) * re)) * -0.16666666666666666));
	else
		tmp = (0.5 * sin(re)) * (abs(im) * (((((-0.016666666666666666 * t_0) - 0.3333333333333333) * abs(im)) * abs(im)) - 2.0));
	end
	tmp_2 = (sign(im) * abs(1.0)) * tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Abs[im], $MachinePrecision] * N[Abs[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(t$95$0 * -0.016666666666666666), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[Abs[im], $MachinePrecision]), $MachinePrecision] * N[Abs[im], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[im], $MachinePrecision], 23000000000.0], N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[(1.0 - N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[Abs[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[im], $MachinePrecision], 1.02e+62], N[(-1.0 * N[(N[Abs[im], $MachinePrecision] * N[(N[(re * N[(N[(N[(1.0 - N[(6.0 / N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[im], $MachinePrecision] * N[(N[(N[(N[(N[(-0.016666666666666666 * t$95$0), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[Abs[im], $MachinePrecision]), $MachinePrecision] * N[Abs[im], $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 2.3e10

   1. Initial program 65.6%

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

   2. Taylor expanded in im around 0

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 90.8%

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

   4. Applied rewrites 90.8%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(-0.016666666666666666 \cdot {im}^{2} - 0.3333333333333333\right) - 2\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 90.8%

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 90.8%

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

   6. Applied rewrites 90.8%

      \[\leadsto \color{blue}{\left(\sin re \cdot 0.5\right) \cdot \left(\left(-2 - \left(0.3333333333333333 - -0.016666666666666666 \cdot \left(im \cdot im\right)\right) \cdot \left(im \cdot im\right)\right) \cdot im\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

   8. Applied rewrites 90.8%

      \[\leadsto \left(\sin re \cdot 0.5\right) \cdot \left(\left(1 - \frac{1}{\left(\left(\left(im \cdot im\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) \cdot im\right) \cdot im - 1}\right) \cdot \color{blue}{\left(\left(\left(\left(\left(im \cdot im\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) \cdot im\right) \cdot im - 1\right) \cdot im\right)}\right) \]

   if 2.3e10 < im < 1.02e62

   1. Initial program 65.6%

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

   2. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 52.0%

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

   4. Applied rewrites 52.0%

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

   5. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 35.5%

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

   7. Applied rewrites 35.5%

      \[\leadsto -1 \cdot \left(im \cdot \left(re \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot {re}^{2}\right)}\right)\right) \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. metadata-eval N/A

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

      5. sub-to-mult N/A

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

      6. lower-unsound-*.f64 N/A

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

      7. lower-unsound--.f64 N/A

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

      8. lower-unsound-/.f64 23.5%

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 23.5%

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 23.5%

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 23.5%

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

      18. lift-pow.f64 N/A

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

      19. unpow2 N/A

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

      20. lower-*.f64 23.5%

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

   9. Applied rewrites 23.5%

      \[\leadsto -1 \cdot \left(im \cdot \left(re \cdot \left(\left(1 - \frac{-1}{\left(re \cdot re\right) \cdot -0.16666666666666666}\right) \cdot \left(\left(re \cdot re\right) \cdot \color{blue}{-0.16666666666666666}\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*r* N/A

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

       5. associate-*r* N/A

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

       6. lower-*.f64 N/A

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

   11. Applied rewrites 35.6%

       \[\leadsto -1 \cdot \left(im \cdot \left(\left(re \cdot \left(\left(\left(1 - \frac{6}{re \cdot re}\right) \cdot re\right) \cdot re\right)\right) \cdot -0.16666666666666666\right)\right) \]

   if 1.02e62 < im

   1. Initial program 65.6%

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

   2. Taylor expanded in im around 0

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 90.8%

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

   4. Applied rewrites 90.8%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(-0.016666666666666666 \cdot {im}^{2} - 0.3333333333333333\right) - 2\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 90.8%

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 90.8%

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

   6. Applied rewrites 90.8%

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

Alternative 3: 93.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(-0.016666666666666666 \cdot \left(\left|im\right| \cdot \left|im\right|\right) - 0.3333333333333333\right) \cdot \left|im\right|\right) \cdot \left|im\right|\\ t_1 := t\_0 - 1\\ t_2 := 0.5 \cdot \sin re\\ \mathsf{copysign}\left(1, im\right) \cdot \begin{array}{l} \mathbf{if}\;\left|im\right| \leq 23000000000:\\ \;\;\;\;t\_2 \cdot \left(\left|im\right| \cdot \left(\left(1 - \frac{1}{t\_1}\right) \cdot t\_1\right)\right)\\ \mathbf{elif}\;\left|im\right| \leq 1.02 \cdot 10^{+62}:\\ \;\;\;\;-1 \cdot \left(\left|im\right| \cdot \left(\left(re \cdot \left(\left(\left(1 - \frac{6}{re \cdot re}\right) \cdot re\right) \cdot re\right)\right) \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \left(\left|im\right| \cdot \left(t\_0 - 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0
        (*
         (*
          (-
           (* -0.016666666666666666 (* (fabs im) (fabs im)))
           0.3333333333333333)
          (fabs im))
         (fabs im)))
       (t_1 (- t_0 1.0))
       (t_2 (* 0.5 (sin re))))
  (*
   (copysign 1.0 im)
   (if (<= (fabs im) 23000000000.0)
     (* t_2 (* (fabs im) (* (- 1.0 (/ 1.0 t_1)) t_1)))
     (if (<= (fabs im) 1.02e+62)
       (*
        -1.0
        (*
         (fabs im)
         (*
          (* re (* (* (- 1.0 (/ 6.0 (* re re))) re) re))
          -0.16666666666666666)))
       (* t_2 (* (fabs im) (- t_0 2.0))))))))

C

double code(double re, double im) {
	double t_0 = (((-0.016666666666666666 * (fabs(im) * fabs(im))) - 0.3333333333333333) * fabs(im)) * fabs(im);
	double t_1 = t_0 - 1.0;
	double t_2 = 0.5 * sin(re);
	double tmp;
	if (fabs(im) <= 23000000000.0) {
		tmp = t_2 * (fabs(im) * ((1.0 - (1.0 / t_1)) * t_1));
	} else if (fabs(im) <= 1.02e+62) {
		tmp = -1.0 * (fabs(im) * ((re * (((1.0 - (6.0 / (re * re))) * re) * re)) * -0.16666666666666666));
	} else {
		tmp = t_2 * (fabs(im) * (t_0 - 2.0));
	}
	return copysign(1.0, im) * tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = (((-0.016666666666666666 * (Math.abs(im) * Math.abs(im))) - 0.3333333333333333) * Math.abs(im)) * Math.abs(im);
	double t_1 = t_0 - 1.0;
	double t_2 = 0.5 * Math.sin(re);
	double tmp;
	if (Math.abs(im) <= 23000000000.0) {
		tmp = t_2 * (Math.abs(im) * ((1.0 - (1.0 / t_1)) * t_1));
	} else if (Math.abs(im) <= 1.02e+62) {
		tmp = -1.0 * (Math.abs(im) * ((re * (((1.0 - (6.0 / (re * re))) * re) * re)) * -0.16666666666666666));
	} else {
		tmp = t_2 * (Math.abs(im) * (t_0 - 2.0));
	}
	return Math.copySign(1.0, im) * tmp;
}

Python

def code(re, im):
	t_0 = (((-0.016666666666666666 * (math.fabs(im) * math.fabs(im))) - 0.3333333333333333) * math.fabs(im)) * math.fabs(im)
	t_1 = t_0 - 1.0
	t_2 = 0.5 * math.sin(re)
	tmp = 0
	if math.fabs(im) <= 23000000000.0:
		tmp = t_2 * (math.fabs(im) * ((1.0 - (1.0 / t_1)) * t_1))
	elif math.fabs(im) <= 1.02e+62:
		tmp = -1.0 * (math.fabs(im) * ((re * (((1.0 - (6.0 / (re * re))) * re) * re)) * -0.16666666666666666))
	else:
		tmp = t_2 * (math.fabs(im) * (t_0 - 2.0))
	return math.copysign(1.0, im) * tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(Float64(Float64(-0.016666666666666666 * Float64(abs(im) * abs(im))) - 0.3333333333333333) * abs(im)) * abs(im))
	t_1 = Float64(t_0 - 1.0)
	t_2 = Float64(0.5 * sin(re))
	tmp = 0.0
	if (abs(im) <= 23000000000.0)
		tmp = Float64(t_2 * Float64(abs(im) * Float64(Float64(1.0 - Float64(1.0 / t_1)) * t_1)));
	elseif (abs(im) <= 1.02e+62)
		tmp = Float64(-1.0 * Float64(abs(im) * Float64(Float64(re * Float64(Float64(Float64(1.0 - Float64(6.0 / Float64(re * re))) * re) * re)) * -0.16666666666666666)));
	else
		tmp = Float64(t_2 * Float64(abs(im) * Float64(t_0 - 2.0)));
	end
	return Float64(copysign(1.0, im) * tmp)
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (((-0.016666666666666666 * (abs(im) * abs(im))) - 0.3333333333333333) * abs(im)) * abs(im);
	t_1 = t_0 - 1.0;
	t_2 = 0.5 * sin(re);
	tmp = 0.0;
	if (abs(im) <= 23000000000.0)
		tmp = t_2 * (abs(im) * ((1.0 - (1.0 / t_1)) * t_1));
	elseif (abs(im) <= 1.02e+62)
		tmp = -1.0 * (abs(im) * ((re * (((1.0 - (6.0 / (re * re))) * re) * re)) * -0.16666666666666666));
	else
		tmp = t_2 * (abs(im) * (t_0 - 2.0));
	end
	tmp_2 = (sign(im) * abs(1.0)) * tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[(N[(-0.016666666666666666 * N[(N[Abs[im], $MachinePrecision] * N[Abs[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[Abs[im], $MachinePrecision]), $MachinePrecision] * N[Abs[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[im], $MachinePrecision], 23000000000.0], N[(t$95$2 * N[(N[Abs[im], $MachinePrecision] * N[(N[(1.0 - N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[im], $MachinePrecision], 1.02e+62], N[(-1.0 * N[(N[Abs[im], $MachinePrecision] * N[(N[(re * N[(N[(N[(1.0 - N[(6.0 / N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[(N[Abs[im], $MachinePrecision] * N[(t$95$0 - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < 2.3e10

   1. Initial program 65.6%

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

   2. Taylor expanded in im around 0

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 90.8%

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

   4. Applied rewrites 90.8%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(-0.016666666666666666 \cdot {im}^{2} - 0.3333333333333333\right) - 2\right)\right)} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. metadata-eval N/A

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

      3. associate--r+ N/A

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

      4. sub-to-mult N/A

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

      5. lower-unsound-*.f64 N/A

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

   6. Applied rewrites 90.8%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left(\left(1 - \frac{1}{\left(\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 1}\right) \cdot \color{blue}{\left(\left(\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 1\right)}\right)\right) \]

   if 2.3e10 < im < 1.02e62

   1. Initial program 65.6%

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

   2. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 52.0%

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

   4. Applied rewrites 52.0%

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

   5. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 35.5%

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

   7. Applied rewrites 35.5%

      \[\leadsto -1 \cdot \left(im \cdot \left(re \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot {re}^{2}\right)}\right)\right) \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. metadata-eval N/A

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

      5. sub-to-mult N/A

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

      6. lower-unsound-*.f64 N/A

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

      7. lower-unsound--.f64 N/A

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

      8. lower-unsound-/.f64 23.5%

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 23.5%

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 23.5%

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 23.5%

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

      18. lift-pow.f64 N/A

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

      19. unpow2 N/A

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

      20. lower-*.f64 23.5%

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

   9. Applied rewrites 23.5%

      \[\leadsto -1 \cdot \left(im \cdot \left(re \cdot \left(\left(1 - \frac{-1}{\left(re \cdot re\right) \cdot -0.16666666666666666}\right) \cdot \left(\left(re \cdot re\right) \cdot \color{blue}{-0.16666666666666666}\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*r* N/A

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

       5. associate-*r* N/A

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

       6. lower-*.f64 N/A

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

   11. Applied rewrites 35.6%

       \[\leadsto -1 \cdot \left(im \cdot \left(\left(re \cdot \left(\left(\left(1 - \frac{6}{re \cdot re}\right) \cdot re\right) \cdot re\right)\right) \cdot -0.16666666666666666\right)\right) \]

   if 1.02e62 < im

   1. Initial program 65.6%

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

   2. Taylor expanded in im around 0

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 90.8%

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

   4. Applied rewrites 90.8%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(-0.016666666666666666 \cdot {im}^{2} - 0.3333333333333333\right) - 2\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 90.8%

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 90.8%

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

   6. Applied rewrites 90.8%

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

Alternative 4: 93.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \left|im\right| \cdot \left|im\right|\\ t_1 := 0.016666666666666666 \cdot t\_0\\ \mathsf{copysign}\left(1, im\right) \cdot \begin{array}{l} \mathbf{if}\;\left|im\right| \leq 23000000000:\\ \;\;\;\;\left(\sin re \cdot 0.5\right) \cdot \left(\left(-2 - \frac{t\_1 \cdot t\_1 - 0.3333333333333333 \cdot 0.3333333333333333}{t\_1 - 0.3333333333333333} \cdot t\_0\right) \cdot \left|im\right|\right)\\ \mathbf{elif}\;\left|im\right| \leq 1.02 \cdot 10^{+62}:\\ \;\;\;\;-1 \cdot \left(\left|im\right| \cdot \left(\left(re \cdot \left(\left(\left(1 - \frac{6}{re \cdot re}\right) \cdot re\right) \cdot re\right)\right) \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(\left|im\right| \cdot \left(\left(\left(-0.016666666666666666 \cdot t\_0 - 0.3333333333333333\right) \cdot \left|im\right|\right) \cdot \left|im\right| - 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0 (* (fabs im) (fabs im)))
       (t_1 (* 0.016666666666666666 t_0)))
  (*
   (copysign 1.0 im)
   (if (<= (fabs im) 23000000000.0)
     (*
      (* (sin re) 0.5)
      (*
       (-
        -2.0
        (*
         (/
          (- (* t_1 t_1) (* 0.3333333333333333 0.3333333333333333))
          (- t_1 0.3333333333333333))
         t_0))
       (fabs im)))
     (if (<= (fabs im) 1.02e+62)
       (*
        -1.0
        (*
         (fabs im)
         (*
          (* re (* (* (- 1.0 (/ 6.0 (* re re))) re) re))
          -0.16666666666666666)))
       (*
        (* 0.5 (sin re))
        (*
         (fabs im)
         (-
          (*
           (*
            (- (* -0.016666666666666666 t_0) 0.3333333333333333)
            (fabs im))
           (fabs im))
          2.0))))))))

C

double code(double re, double im) {
	double t_0 = fabs(im) * fabs(im);
	double t_1 = 0.016666666666666666 * t_0;
	double tmp;
	if (fabs(im) <= 23000000000.0) {
		tmp = (sin(re) * 0.5) * ((-2.0 - ((((t_1 * t_1) - (0.3333333333333333 * 0.3333333333333333)) / (t_1 - 0.3333333333333333)) * t_0)) * fabs(im));
	} else if (fabs(im) <= 1.02e+62) {
		tmp = -1.0 * (fabs(im) * ((re * (((1.0 - (6.0 / (re * re))) * re) * re)) * -0.16666666666666666));
	} else {
		tmp = (0.5 * sin(re)) * (fabs(im) * (((((-0.016666666666666666 * t_0) - 0.3333333333333333) * fabs(im)) * fabs(im)) - 2.0));
	}
	return copysign(1.0, im) * tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = Math.abs(im) * Math.abs(im);
	double t_1 = 0.016666666666666666 * t_0;
	double tmp;
	if (Math.abs(im) <= 23000000000.0) {
		tmp = (Math.sin(re) * 0.5) * ((-2.0 - ((((t_1 * t_1) - (0.3333333333333333 * 0.3333333333333333)) / (t_1 - 0.3333333333333333)) * t_0)) * Math.abs(im));
	} else if (Math.abs(im) <= 1.02e+62) {
		tmp = -1.0 * (Math.abs(im) * ((re * (((1.0 - (6.0 / (re * re))) * re) * re)) * -0.16666666666666666));
	} else {
		tmp = (0.5 * Math.sin(re)) * (Math.abs(im) * (((((-0.016666666666666666 * t_0) - 0.3333333333333333) * Math.abs(im)) * Math.abs(im)) - 2.0));
	}
	return Math.copySign(1.0, im) * tmp;
}

Python

def code(re, im):
	t_0 = math.fabs(im) * math.fabs(im)
	t_1 = 0.016666666666666666 * t_0
	tmp = 0
	if math.fabs(im) <= 23000000000.0:
		tmp = (math.sin(re) * 0.5) * ((-2.0 - ((((t_1 * t_1) - (0.3333333333333333 * 0.3333333333333333)) / (t_1 - 0.3333333333333333)) * t_0)) * math.fabs(im))
	elif math.fabs(im) <= 1.02e+62:
		tmp = -1.0 * (math.fabs(im) * ((re * (((1.0 - (6.0 / (re * re))) * re) * re)) * -0.16666666666666666))
	else:
		tmp = (0.5 * math.sin(re)) * (math.fabs(im) * (((((-0.016666666666666666 * t_0) - 0.3333333333333333) * math.fabs(im)) * math.fabs(im)) - 2.0))
	return math.copysign(1.0, im) * tmp

Julia

function code(re, im)
	t_0 = Float64(abs(im) * abs(im))
	t_1 = Float64(0.016666666666666666 * t_0)
	tmp = 0.0
	if (abs(im) <= 23000000000.0)
		tmp = Float64(Float64(sin(re) * 0.5) * Float64(Float64(-2.0 - Float64(Float64(Float64(Float64(t_1 * t_1) - Float64(0.3333333333333333 * 0.3333333333333333)) / Float64(t_1 - 0.3333333333333333)) * t_0)) * abs(im)));
	elseif (abs(im) <= 1.02e+62)
		tmp = Float64(-1.0 * Float64(abs(im) * Float64(Float64(re * Float64(Float64(Float64(1.0 - Float64(6.0 / Float64(re * re))) * re) * re)) * -0.16666666666666666)));
	else
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(abs(im) * Float64(Float64(Float64(Float64(Float64(-0.016666666666666666 * t_0) - 0.3333333333333333) * abs(im)) * abs(im)) - 2.0)));
	end
	return Float64(copysign(1.0, im) * tmp)
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = abs(im) * abs(im);
	t_1 = 0.016666666666666666 * t_0;
	tmp = 0.0;
	if (abs(im) <= 23000000000.0)
		tmp = (sin(re) * 0.5) * ((-2.0 - ((((t_1 * t_1) - (0.3333333333333333 * 0.3333333333333333)) / (t_1 - 0.3333333333333333)) * t_0)) * abs(im));
	elseif (abs(im) <= 1.02e+62)
		tmp = -1.0 * (abs(im) * ((re * (((1.0 - (6.0 / (re * re))) * re) * re)) * -0.16666666666666666));
	else
		tmp = (0.5 * sin(re)) * (abs(im) * (((((-0.016666666666666666 * t_0) - 0.3333333333333333) * abs(im)) * abs(im)) - 2.0));
	end
	tmp_2 = (sign(im) * abs(1.0)) * tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Abs[im], $MachinePrecision] * N[Abs[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.016666666666666666 * t$95$0), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[im], $MachinePrecision], 23000000000.0], N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[(-2.0 - N[(N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(0.3333333333333333 * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 - 0.3333333333333333), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] * N[Abs[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[im], $MachinePrecision], 1.02e+62], N[(-1.0 * N[(N[Abs[im], $MachinePrecision] * N[(N[(re * N[(N[(N[(1.0 - N[(6.0 / N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[im], $MachinePrecision] * N[(N[(N[(N[(N[(-0.016666666666666666 * t$95$0), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[Abs[im], $MachinePrecision]), $MachinePrecision] * N[Abs[im], $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 2.3e10

   1. Initial program 65.6%

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

   2. Taylor expanded in im around 0

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 90.8%

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

   4. Applied rewrites 90.8%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(-0.016666666666666666 \cdot {im}^{2} - 0.3333333333333333\right) - 2\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 90.8%

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 90.8%

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

   6. Applied rewrites 90.8%

      \[\leadsto \color{blue}{\left(\sin re \cdot 0.5\right) \cdot \left(\left(-2 - \left(0.3333333333333333 - -0.016666666666666666 \cdot \left(im \cdot im\right)\right) \cdot \left(im \cdot im\right)\right) \cdot im\right)} \]
   7. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. flip-+ N/A

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

      5. lower-unsound-/.f64 N/A

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

      6. lower-unsound--.f64 N/A

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

      7. lower-unsound-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-lft-neg-out N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-lft-neg-out N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower-unsound-*.f64 N/A

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

      17. lower-unsound--.f64 N/A

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

   8. Applied rewrites 65.4%

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

   if 2.3e10 < im < 1.02e62

   1. Initial program 65.6%

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

   2. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 52.0%

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

   4. Applied rewrites 52.0%

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

   5. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 35.5%

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

   7. Applied rewrites 35.5%

      \[\leadsto -1 \cdot \left(im \cdot \left(re \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot {re}^{2}\right)}\right)\right) \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. metadata-eval N/A

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

      5. sub-to-mult N/A

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

      6. lower-unsound-*.f64 N/A

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

      7. lower-unsound--.f64 N/A

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

      8. lower-unsound-/.f64 23.5%

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 23.5%

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 23.5%

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 23.5%

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

      18. lift-pow.f64 N/A

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

      19. unpow2 N/A

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

      20. lower-*.f64 23.5%

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

   9. Applied rewrites 23.5%

      \[\leadsto -1 \cdot \left(im \cdot \left(re \cdot \left(\left(1 - \frac{-1}{\left(re \cdot re\right) \cdot -0.16666666666666666}\right) \cdot \left(\left(re \cdot re\right) \cdot \color{blue}{-0.16666666666666666}\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*r* N/A

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

       5. associate-*r* N/A

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

       6. lower-*.f64 N/A

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

   11. Applied rewrites 35.6%

       \[\leadsto -1 \cdot \left(im \cdot \left(\left(re \cdot \left(\left(\left(1 - \frac{6}{re \cdot re}\right) \cdot re\right) \cdot re\right)\right) \cdot -0.16666666666666666\right)\right) \]

   if 1.02e62 < im

   1. Initial program 65.6%

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

   2. Taylor expanded in im around 0

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 90.8%

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

   4. Applied rewrites 90.8%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(-0.016666666666666666 \cdot {im}^{2} - 0.3333333333333333\right) - 2\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 90.8%

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 90.8%

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

   6. Applied rewrites 90.8%

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

Alternative 5: 93.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(\left|im\right| \cdot \left(\left(\left(-0.016666666666666666 \cdot \left(\left|im\right| \cdot \left|im\right|\right) - 0.3333333333333333\right) \cdot \left|im\right|\right) \cdot \left|im\right| - 2\right)\right)\\ \mathsf{copysign}\left(1, im\right) \cdot \begin{array}{l} \mathbf{if}\;\left|im\right| \leq 23000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\left|im\right| \leq 1.02 \cdot 10^{+62}:\\ \;\;\;\;-1 \cdot \left(\left|im\right| \cdot \left(\left(re \cdot \left(\left(\left(1 - \frac{6}{re \cdot re}\right) \cdot re\right) \cdot re\right)\right) \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0
        (*
         (* 0.5 (sin re))
         (*
          (fabs im)
          (-
           (*
            (*
             (-
              (* -0.016666666666666666 (* (fabs im) (fabs im)))
              0.3333333333333333)
             (fabs im))
            (fabs im))
           2.0)))))
  (*
   (copysign 1.0 im)
   (if (<= (fabs im) 23000000000.0)
     t_0
     (if (<= (fabs im) 1.02e+62)
       (*
        -1.0
        (*
         (fabs im)
         (*
          (* re (* (* (- 1.0 (/ 6.0 (* re re))) re) re))
          -0.16666666666666666)))
       t_0)))))

C

double code(double re, double im) {
	double t_0 = (0.5 * sin(re)) * (fabs(im) * (((((-0.016666666666666666 * (fabs(im) * fabs(im))) - 0.3333333333333333) * fabs(im)) * fabs(im)) - 2.0));
	double tmp;
	if (fabs(im) <= 23000000000.0) {
		tmp = t_0;
	} else if (fabs(im) <= 1.02e+62) {
		tmp = -1.0 * (fabs(im) * ((re * (((1.0 - (6.0 / (re * re))) * re) * re)) * -0.16666666666666666));
	} else {
		tmp = t_0;
	}
	return copysign(1.0, im) * tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = (0.5 * Math.sin(re)) * (Math.abs(im) * (((((-0.016666666666666666 * (Math.abs(im) * Math.abs(im))) - 0.3333333333333333) * Math.abs(im)) * Math.abs(im)) - 2.0));
	double tmp;
	if (Math.abs(im) <= 23000000000.0) {
		tmp = t_0;
	} else if (Math.abs(im) <= 1.02e+62) {
		tmp = -1.0 * (Math.abs(im) * ((re * (((1.0 - (6.0 / (re * re))) * re) * re)) * -0.16666666666666666));
	} else {
		tmp = t_0;
	}
	return Math.copySign(1.0, im) * tmp;
}

Python

def code(re, im):
	t_0 = (0.5 * math.sin(re)) * (math.fabs(im) * (((((-0.016666666666666666 * (math.fabs(im) * math.fabs(im))) - 0.3333333333333333) * math.fabs(im)) * math.fabs(im)) - 2.0))
	tmp = 0
	if math.fabs(im) <= 23000000000.0:
		tmp = t_0
	elif math.fabs(im) <= 1.02e+62:
		tmp = -1.0 * (math.fabs(im) * ((re * (((1.0 - (6.0 / (re * re))) * re) * re)) * -0.16666666666666666))
	else:
		tmp = t_0
	return math.copysign(1.0, im) * tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(abs(im) * Float64(Float64(Float64(Float64(Float64(-0.016666666666666666 * Float64(abs(im) * abs(im))) - 0.3333333333333333) * abs(im)) * abs(im)) - 2.0)))
	tmp = 0.0
	if (abs(im) <= 23000000000.0)
		tmp = t_0;
	elseif (abs(im) <= 1.02e+62)
		tmp = Float64(-1.0 * Float64(abs(im) * Float64(Float64(re * Float64(Float64(Float64(1.0 - Float64(6.0 / Float64(re * re))) * re) * re)) * -0.16666666666666666)));
	else
		tmp = t_0;
	end
	return Float64(copysign(1.0, im) * tmp)
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (0.5 * sin(re)) * (abs(im) * (((((-0.016666666666666666 * (abs(im) * abs(im))) - 0.3333333333333333) * abs(im)) * abs(im)) - 2.0));
	tmp = 0.0;
	if (abs(im) <= 23000000000.0)
		tmp = t_0;
	elseif (abs(im) <= 1.02e+62)
		tmp = -1.0 * (abs(im) * ((re * (((1.0 - (6.0 / (re * re))) * re) * re)) * -0.16666666666666666));
	else
		tmp = t_0;
	end
	tmp_2 = (sign(im) * abs(1.0)) * tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[im], $MachinePrecision] * N[(N[(N[(N[(N[(-0.016666666666666666 * N[(N[Abs[im], $MachinePrecision] * N[Abs[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[Abs[im], $MachinePrecision]), $MachinePrecision] * N[Abs[im], $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[im], $MachinePrecision], 23000000000.0], t$95$0, If[LessEqual[N[Abs[im], $MachinePrecision], 1.02e+62], N[(-1.0 * N[(N[Abs[im], $MachinePrecision] * N[(N[(re * N[(N[(N[(1.0 - N[(6.0 / N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2.3e10 or 1.02e62 < im

   1. Initial program 65.6%

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

   2. Taylor expanded in im around 0

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 90.8%

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

   4. Applied rewrites 90.8%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(-0.016666666666666666 \cdot {im}^{2} - 0.3333333333333333\right) - 2\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 90.8%

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 90.8%

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

   6. Applied rewrites 90.8%

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

   if 2.3e10 < im < 1.02e62

   1. Initial program 65.6%

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

   2. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 52.0%

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

   4. Applied rewrites 52.0%

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

   5. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 35.5%

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

   7. Applied rewrites 35.5%

      \[\leadsto -1 \cdot \left(im \cdot \left(re \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot {re}^{2}\right)}\right)\right) \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. metadata-eval N/A

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

      5. sub-to-mult N/A

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

      6. lower-unsound-*.f64 N/A

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

      7. lower-unsound--.f64 N/A

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

      8. lower-unsound-/.f64 23.5%

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 23.5%

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 23.5%

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 23.5%

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

      18. lift-pow.f64 N/A

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

      19. unpow2 N/A

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

      20. lower-*.f64 23.5%

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

   9. Applied rewrites 23.5%

      \[\leadsto -1 \cdot \left(im \cdot \left(re \cdot \left(\left(1 - \frac{-1}{\left(re \cdot re\right) \cdot -0.16666666666666666}\right) \cdot \left(\left(re \cdot re\right) \cdot \color{blue}{-0.16666666666666666}\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*r* N/A

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

       5. associate-*r* N/A

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

       6. lower-*.f64 N/A

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

   11. Applied rewrites 35.6%

       \[\leadsto -1 \cdot \left(im \cdot \left(\left(re \cdot \left(\left(\left(1 - \frac{6}{re \cdot re}\right) \cdot re\right) \cdot re\right)\right) \cdot -0.16666666666666666\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 91.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \left(\sin re \cdot 0.5\right) \cdot \left(\left(-2 - 0.3333333333333333 \cdot \left(\left|im\right| \cdot \left|im\right|\right)\right) \cdot \left|im\right|\right)\\ \mathsf{copysign}\left(1, im\right) \cdot \begin{array}{l} \mathbf{if}\;\left|im\right| \leq 23000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\left|im\right| \leq 8.2 \cdot 10^{+102}:\\ \;\;\;\;-1 \cdot \left(\left|im\right| \cdot \left(\left(re \cdot \left(\left(\left(1 - \frac{6}{re \cdot re}\right) \cdot re\right) \cdot re\right)\right) \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0
        (*
         (* (sin re) 0.5)
         (*
          (- -2.0 (* 0.3333333333333333 (* (fabs im) (fabs im))))
          (fabs im)))))
  (*
   (copysign 1.0 im)
   (if (<= (fabs im) 23000000000.0)
     t_0
     (if (<= (fabs im) 8.2e+102)
       (*
        -1.0
        (*
         (fabs im)
         (*
          (* re (* (* (- 1.0 (/ 6.0 (* re re))) re) re))
          -0.16666666666666666)))
       t_0)))))

C

double code(double re, double im) {
	double t_0 = (sin(re) * 0.5) * ((-2.0 - (0.3333333333333333 * (fabs(im) * fabs(im)))) * fabs(im));
	double tmp;
	if (fabs(im) <= 23000000000.0) {
		tmp = t_0;
	} else if (fabs(im) <= 8.2e+102) {
		tmp = -1.0 * (fabs(im) * ((re * (((1.0 - (6.0 / (re * re))) * re) * re)) * -0.16666666666666666));
	} else {
		tmp = t_0;
	}
	return copysign(1.0, im) * tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = (Math.sin(re) * 0.5) * ((-2.0 - (0.3333333333333333 * (Math.abs(im) * Math.abs(im)))) * Math.abs(im));
	double tmp;
	if (Math.abs(im) <= 23000000000.0) {
		tmp = t_0;
	} else if (Math.abs(im) <= 8.2e+102) {
		tmp = -1.0 * (Math.abs(im) * ((re * (((1.0 - (6.0 / (re * re))) * re) * re)) * -0.16666666666666666));
	} else {
		tmp = t_0;
	}
	return Math.copySign(1.0, im) * tmp;
}

Python

def code(re, im):
	t_0 = (math.sin(re) * 0.5) * ((-2.0 - (0.3333333333333333 * (math.fabs(im) * math.fabs(im)))) * math.fabs(im))
	tmp = 0
	if math.fabs(im) <= 23000000000.0:
		tmp = t_0
	elif math.fabs(im) <= 8.2e+102:
		tmp = -1.0 * (math.fabs(im) * ((re * (((1.0 - (6.0 / (re * re))) * re) * re)) * -0.16666666666666666))
	else:
		tmp = t_0
	return math.copysign(1.0, im) * tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(sin(re) * 0.5) * Float64(Float64(-2.0 - Float64(0.3333333333333333 * Float64(abs(im) * abs(im)))) * abs(im)))
	tmp = 0.0
	if (abs(im) <= 23000000000.0)
		tmp = t_0;
	elseif (abs(im) <= 8.2e+102)
		tmp = Float64(-1.0 * Float64(abs(im) * Float64(Float64(re * Float64(Float64(Float64(1.0 - Float64(6.0 / Float64(re * re))) * re) * re)) * -0.16666666666666666)));
	else
		tmp = t_0;
	end
	return Float64(copysign(1.0, im) * tmp)
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (sin(re) * 0.5) * ((-2.0 - (0.3333333333333333 * (abs(im) * abs(im)))) * abs(im));
	tmp = 0.0;
	if (abs(im) <= 23000000000.0)
		tmp = t_0;
	elseif (abs(im) <= 8.2e+102)
		tmp = -1.0 * (abs(im) * ((re * (((1.0 - (6.0 / (re * re))) * re) * re)) * -0.16666666666666666));
	else
		tmp = t_0;
	end
	tmp_2 = (sign(im) * abs(1.0)) * tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[(-2.0 - N[(0.3333333333333333 * N[(N[Abs[im], $MachinePrecision] * N[Abs[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[im], $MachinePrecision], 23000000000.0], t$95$0, If[LessEqual[N[Abs[im], $MachinePrecision], 8.2e+102], N[(-1.0 * N[(N[Abs[im], $MachinePrecision] * N[(N[(re * N[(N[(N[(1.0 - N[(6.0 / N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2.3e10 or 8.1999999999999999e102 < im

   1. Initial program 65.6%

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

   2. Taylor expanded in im around 0

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 90.8%

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

   4. Applied rewrites 90.8%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(-0.016666666666666666 \cdot {im}^{2} - 0.3333333333333333\right) - 2\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 90.8%

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 90.8%

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

   6. Applied rewrites 90.8%

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

   7. Taylor expanded in im around 0

      \[\leadsto \left(\sin re \cdot 0.5\right) \cdot \left(\left(-2 - \frac{1}{3} \cdot \left(im \cdot im\right)\right) \cdot im\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 84.4%

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

   if 2.3e10 < im < 8.1999999999999999e102

   1. Initial program 65.6%

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

   2. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 52.0%

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

   4. Applied rewrites 52.0%

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

   5. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 35.5%

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

   7. Applied rewrites 35.5%

      \[\leadsto -1 \cdot \left(im \cdot \left(re \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot {re}^{2}\right)}\right)\right) \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. metadata-eval N/A

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

      5. sub-to-mult N/A

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

      6. lower-unsound-*.f64 N/A

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

      7. lower-unsound--.f64 N/A

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

      8. lower-unsound-/.f64 23.5%

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 23.5%

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 23.5%

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 23.5%

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

      18. lift-pow.f64 N/A

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

      19. unpow2 N/A

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

      20. lower-*.f64 23.5%

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

   9. Applied rewrites 23.5%

      \[\leadsto -1 \cdot \left(im \cdot \left(re \cdot \left(\left(1 - \frac{-1}{\left(re \cdot re\right) \cdot -0.16666666666666666}\right) \cdot \left(\left(re \cdot re\right) \cdot \color{blue}{-0.16666666666666666}\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*r* N/A

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

       5. associate-*r* N/A

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

       6. lower-*.f64 N/A

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

   11. Applied rewrites 35.6%

       \[\leadsto -1 \cdot \left(im \cdot \left(\left(re \cdot \left(\left(\left(1 - \frac{6}{re \cdot re}\right) \cdot re\right) \cdot re\right)\right) \cdot -0.16666666666666666\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 90.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left|im\right| \cdot \left|im\right|\\ t_1 := -\left|im\right|\\ t_2 := \sin \left(\left|re\right|\right)\\ t_3 := \left(0.5 \cdot t\_2\right) \cdot \left(e^{t\_1} - e^{\left|im\right|}\right)\\ \mathsf{copysign}\left(1, re\right) \cdot \left(\mathsf{copysign}\left(1, im\right) \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\left(0.5 \cdot \left|re\right|\right) \cdot \left(\left(-2 - \left(0.3333333333333333 - -0.016666666666666666 \cdot t\_0\right) \cdot t\_0\right) \cdot \left|im\right|\right)\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{-7}:\\ \;\;\;\;t\_2 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(-\left(\left(\left|re\right| \cdot \left|re\right|\right) \cdot -0.16666666666666666 - -1\right) \cdot \left|re\right|\right) \cdot \left|im\right|\\ \end{array}\right) \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0 (* (fabs im) (fabs im)))
       (t_1 (- (fabs im)))
       (t_2 (sin (fabs re)))
       (t_3 (* (* 0.5 t_2) (- (exp t_1) (exp (fabs im))))))
  (*
   (copysign 1.0 re)
   (*
    (copysign 1.0 im)
    (if (<= t_3 (- INFINITY))
      (*
       (* 0.5 (fabs re))
       (*
        (-
         -2.0
         (* (- 0.3333333333333333 (* -0.016666666666666666 t_0)) t_0))
        (fabs im)))
      (if (<= t_3 2e-7)
        (* t_2 t_1)
        (*
         (-
          (*
           (- (* (* (fabs re) (fabs re)) -0.16666666666666666) -1.0)
           (fabs re)))
         (fabs im))))))))

C

double code(double re, double im) {
	double t_0 = fabs(im) * fabs(im);
	double t_1 = -fabs(im);
	double t_2 = sin(fabs(re));
	double t_3 = (0.5 * t_2) * (exp(t_1) - exp(fabs(im)));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = (0.5 * fabs(re)) * ((-2.0 - ((0.3333333333333333 - (-0.016666666666666666 * t_0)) * t_0)) * fabs(im));
	} else if (t_3 <= 2e-7) {
		tmp = t_2 * t_1;
	} else {
		tmp = -((((fabs(re) * fabs(re)) * -0.16666666666666666) - -1.0) * fabs(re)) * fabs(im);
	}
	return copysign(1.0, re) * (copysign(1.0, im) * tmp);
}

Java

public static double code(double re, double im) {
	double t_0 = Math.abs(im) * Math.abs(im);
	double t_1 = -Math.abs(im);
	double t_2 = Math.sin(Math.abs(re));
	double t_3 = (0.5 * t_2) * (Math.exp(t_1) - Math.exp(Math.abs(im)));
	double tmp;
	if (t_3 <= -Double.POSITIVE_INFINITY) {
		tmp = (0.5 * Math.abs(re)) * ((-2.0 - ((0.3333333333333333 - (-0.016666666666666666 * t_0)) * t_0)) * Math.abs(im));
	} else if (t_3 <= 2e-7) {
		tmp = t_2 * t_1;
	} else {
		tmp = -((((Math.abs(re) * Math.abs(re)) * -0.16666666666666666) - -1.0) * Math.abs(re)) * Math.abs(im);
	}
	return Math.copySign(1.0, re) * (Math.copySign(1.0, im) * tmp);
}

Python

def code(re, im):
	t_0 = math.fabs(im) * math.fabs(im)
	t_1 = -math.fabs(im)
	t_2 = math.sin(math.fabs(re))
	t_3 = (0.5 * t_2) * (math.exp(t_1) - math.exp(math.fabs(im)))
	tmp = 0
	if t_3 <= -math.inf:
		tmp = (0.5 * math.fabs(re)) * ((-2.0 - ((0.3333333333333333 - (-0.016666666666666666 * t_0)) * t_0)) * math.fabs(im))
	elif t_3 <= 2e-7:
		tmp = t_2 * t_1
	else:
		tmp = -((((math.fabs(re) * math.fabs(re)) * -0.16666666666666666) - -1.0) * math.fabs(re)) * math.fabs(im)
	return math.copysign(1.0, re) * (math.copysign(1.0, im) * tmp)

Julia

function code(re, im)
	t_0 = Float64(abs(im) * abs(im))
	t_1 = Float64(-abs(im))
	t_2 = sin(abs(re))
	t_3 = Float64(Float64(0.5 * t_2) * Float64(exp(t_1) - exp(abs(im))))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(Float64(0.5 * abs(re)) * Float64(Float64(-2.0 - Float64(Float64(0.3333333333333333 - Float64(-0.016666666666666666 * t_0)) * t_0)) * abs(im)));
	elseif (t_3 <= 2e-7)
		tmp = Float64(t_2 * t_1);
	else
		tmp = Float64(Float64(-Float64(Float64(Float64(Float64(abs(re) * abs(re)) * -0.16666666666666666) - -1.0) * abs(re))) * abs(im));
	end
	return Float64(copysign(1.0, re) * Float64(copysign(1.0, im) * tmp))
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = abs(im) * abs(im);
	t_1 = -abs(im);
	t_2 = sin(abs(re));
	t_3 = (0.5 * t_2) * (exp(t_1) - exp(abs(im)));
	tmp = 0.0;
	if (t_3 <= -Inf)
		tmp = (0.5 * abs(re)) * ((-2.0 - ((0.3333333333333333 - (-0.016666666666666666 * t_0)) * t_0)) * abs(im));
	elseif (t_3 <= 2e-7)
		tmp = t_2 * t_1;
	else
		tmp = -((((abs(re) * abs(re)) * -0.16666666666666666) - -1.0) * abs(re)) * abs(im);
	end
	tmp_2 = (sign(re) * abs(1.0)) * ((sign(im) * abs(1.0)) * tmp);
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Abs[im], $MachinePrecision] * N[Abs[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[Abs[im], $MachinePrecision])}, Block[{t$95$2 = N[Sin[N[Abs[re], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(0.5 * t$95$2), $MachinePrecision] * N[(N[Exp[t$95$1], $MachinePrecision] - N[Exp[N[Abs[im], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$3, (-Infinity)], N[(N[(0.5 * N[Abs[re], $MachinePrecision]), $MachinePrecision] * N[(N[(-2.0 - N[(N[(0.3333333333333333 - N[(-0.016666666666666666 * t$95$0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] * N[Abs[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e-7], N[(t$95$2 * t$95$1), $MachinePrecision], N[((-N[(N[(N[(N[(N[Abs[re], $MachinePrecision] * N[Abs[re], $MachinePrecision]), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - -1.0), $MachinePrecision] * N[Abs[re], $MachinePrecision]), $MachinePrecision]) * N[Abs[im], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 65.6%

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

   2. Taylor expanded in im around 0

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 90.8%

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

   4. Applied rewrites 90.8%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(-0.016666666666666666 \cdot {im}^{2} - 0.3333333333333333\right) - 2\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 90.8%

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 90.8%

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

   6. Applied rewrites 90.8%

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

   7. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot re\right)} \cdot \left(\left(-2 - \left(0.3333333333333333 - -0.016666666666666666 \cdot \left(im \cdot im\right)\right) \cdot \left(im \cdot im\right)\right) \cdot im\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 57.8%

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

   9. Applied rewrites 57.8%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 1.9999999999999999e-7

   1. Initial program 65.6%

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

   2. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 52.0%

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

   4. Applied rewrites 52.0%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. lift-neg.f64 N/A

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

      7. lower-*.f64 52.0%

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

   6. Applied rewrites 52.0%

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

   if 1.9999999999999999e-7 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 65.6%

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

   2. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 52.0%

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

   4. Applied rewrites 52.0%

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

   5. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 35.5%

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

   7. Applied rewrites 35.5%

      \[\leadsto -1 \cdot \left(im \cdot \left(re \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot {re}^{2}\right)}\right)\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-neg-in N/A

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

      6. lower-*.f64 N/A

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

   9. Applied rewrites 35.5%

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

Alternative 8: 67.5% accurate, 1.1× speedup

Math

\[\mathsf{copysign}\left(1, re\right) \cdot \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin \left(\left|re\right|\right) \leq 10^{-305}:\\ \;\;\;\;-1 \cdot \left(im \cdot \left(\left(\left|re\right| \cdot \left(\left(\left(1 - \frac{6}{\left|re\right| \cdot \left|re\right|}\right) \cdot \left|re\right|\right) \cdot \left|re\right|\right)\right) \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \left|re\right|\right) \cdot \left(\left(-2 - \left(0.3333333333333333 - -0.016666666666666666 \cdot \left(im \cdot im\right)\right) \cdot \left(im \cdot im\right)\right) \cdot im\right)\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (*
 (copysign 1.0 re)
 (if (<= (* 0.5 (sin (fabs re))) 1e-305)
   (*
    -1.0
    (*
     im
     (*
      (*
       (fabs re)
       (*
        (* (- 1.0 (/ 6.0 (* (fabs re) (fabs re)))) (fabs re))
        (fabs re)))
      -0.16666666666666666)))
   (*
    (* 0.5 (fabs re))
    (*
     (-
      -2.0
      (*
       (- 0.3333333333333333 (* -0.016666666666666666 (* im im)))
       (* im im)))
     im)))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(fabs(re))) <= 1e-305) {
		tmp = -1.0 * (im * ((fabs(re) * (((1.0 - (6.0 / (fabs(re) * fabs(re)))) * fabs(re)) * fabs(re))) * -0.16666666666666666));
	} else {
		tmp = (0.5 * fabs(re)) * ((-2.0 - ((0.3333333333333333 - (-0.016666666666666666 * (im * im))) * (im * im))) * im);
	}
	return copysign(1.0, re) * tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if ((0.5 * Math.sin(Math.abs(re))) <= 1e-305) {
		tmp = -1.0 * (im * ((Math.abs(re) * (((1.0 - (6.0 / (Math.abs(re) * Math.abs(re)))) * Math.abs(re)) * Math.abs(re))) * -0.16666666666666666));
	} else {
		tmp = (0.5 * Math.abs(re)) * ((-2.0 - ((0.3333333333333333 - (-0.016666666666666666 * (im * im))) * (im * im))) * im);
	}
	return Math.copySign(1.0, re) * tmp;
}

Python

def code(re, im):
	tmp = 0
	if (0.5 * math.sin(math.fabs(re))) <= 1e-305:
		tmp = -1.0 * (im * ((math.fabs(re) * (((1.0 - (6.0 / (math.fabs(re) * math.fabs(re)))) * math.fabs(re)) * math.fabs(re))) * -0.16666666666666666))
	else:
		tmp = (0.5 * math.fabs(re)) * ((-2.0 - ((0.3333333333333333 - (-0.016666666666666666 * (im * im))) * (im * im))) * im)
	return math.copysign(1.0, re) * tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(abs(re))) <= 1e-305)
		tmp = Float64(-1.0 * Float64(im * Float64(Float64(abs(re) * Float64(Float64(Float64(1.0 - Float64(6.0 / Float64(abs(re) * abs(re)))) * abs(re)) * abs(re))) * -0.16666666666666666)));
	else
		tmp = Float64(Float64(0.5 * abs(re)) * Float64(Float64(-2.0 - Float64(Float64(0.3333333333333333 - Float64(-0.016666666666666666 * Float64(im * im))) * Float64(im * im))) * im));
	end
	return Float64(copysign(1.0, re) * tmp)
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((0.5 * sin(abs(re))) <= 1e-305)
		tmp = -1.0 * (im * ((abs(re) * (((1.0 - (6.0 / (abs(re) * abs(re)))) * abs(re)) * abs(re))) * -0.16666666666666666));
	else
		tmp = (0.5 * abs(re)) * ((-2.0 - ((0.3333333333333333 - (-0.016666666666666666 * (im * im))) * (im * im))) * im);
	end
	tmp_2 = (sign(re) * abs(1.0)) * tmp;
end

Wolfram

code[re_, im_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(0.5 * N[Sin[N[Abs[re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e-305], N[(-1.0 * N[(im * N[(N[(N[Abs[re], $MachinePrecision] * N[(N[(N[(1.0 - N[(6.0 / N[(N[Abs[re], $MachinePrecision] * N[Abs[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[re], $MachinePrecision]), $MachinePrecision] * N[Abs[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Abs[re], $MachinePrecision]), $MachinePrecision] * N[(N[(-2.0 - N[(N[(0.3333333333333333 - N[(-0.016666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < 1e-305

   1. Initial program 65.6%

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

   2. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 52.0%

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

   4. Applied rewrites 52.0%

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

   5. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 35.5%

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

   7. Applied rewrites 35.5%

      \[\leadsto -1 \cdot \left(im \cdot \left(re \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot {re}^{2}\right)}\right)\right) \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. metadata-eval N/A

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

      5. sub-to-mult N/A

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

      6. lower-unsound-*.f64 N/A

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

      7. lower-unsound--.f64 N/A

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

      8. lower-unsound-/.f64 23.5%

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 23.5%

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 23.5%

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 23.5%

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

      18. lift-pow.f64 N/A

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

      19. unpow2 N/A

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

      20. lower-*.f64 23.5%

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

   9. Applied rewrites 23.5%

      \[\leadsto -1 \cdot \left(im \cdot \left(re \cdot \left(\left(1 - \frac{-1}{\left(re \cdot re\right) \cdot -0.16666666666666666}\right) \cdot \left(\left(re \cdot re\right) \cdot \color{blue}{-0.16666666666666666}\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*r* N/A

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

       5. associate-*r* N/A

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

       6. lower-*.f64 N/A

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

   11. Applied rewrites 35.6%

       \[\leadsto -1 \cdot \left(im \cdot \left(\left(re \cdot \left(\left(\left(1 - \frac{6}{re \cdot re}\right) \cdot re\right) \cdot re\right)\right) \cdot -0.16666666666666666\right)\right) \]

   if 1e-305 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 65.6%

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

   2. Taylor expanded in im around 0

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 90.8%

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

   4. Applied rewrites 90.8%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(-0.016666666666666666 \cdot {im}^{2} - 0.3333333333333333\right) - 2\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 90.8%

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 90.8%

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

   6. Applied rewrites 90.8%

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

   7. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot re\right)} \cdot \left(\left(-2 - \left(0.3333333333333333 - -0.016666666666666666 \cdot \left(im \cdot im\right)\right) \cdot \left(im \cdot im\right)\right) \cdot im\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 57.8%

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

   9. Applied rewrites 57.8%

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

Alternative 9: 67.4% accurate, 1.2× speedup

Math

\[\mathsf{copysign}\left(1, re\right) \cdot \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin \left(\left|re\right|\right) \leq -0.005:\\ \;\;\;\;\left(-\left(\left(\left|re\right| \cdot \left|re\right|\right) \cdot -0.16666666666666666 - -1\right) \cdot \left|re\right|\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \left|re\right|\right) \cdot \left(\left(-2 - \left(0.3333333333333333 - -0.016666666666666666 \cdot \left(im \cdot im\right)\right) \cdot \left(im \cdot im\right)\right) \cdot im\right)\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (*
 (copysign 1.0 re)
 (if (<= (* 0.5 (sin (fabs re))) -0.005)
   (*
    (-
     (*
      (- (* (* (fabs re) (fabs re)) -0.16666666666666666) -1.0)
      (fabs re)))
    im)
   (*
    (* 0.5 (fabs re))
    (*
     (-
      -2.0
      (*
       (- 0.3333333333333333 (* -0.016666666666666666 (* im im)))
       (* im im)))
     im)))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(fabs(re))) <= -0.005) {
		tmp = -((((fabs(re) * fabs(re)) * -0.16666666666666666) - -1.0) * fabs(re)) * im;
	} else {
		tmp = (0.5 * fabs(re)) * ((-2.0 - ((0.3333333333333333 - (-0.016666666666666666 * (im * im))) * (im * im))) * im);
	}
	return copysign(1.0, re) * tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if ((0.5 * Math.sin(Math.abs(re))) <= -0.005) {
		tmp = -((((Math.abs(re) * Math.abs(re)) * -0.16666666666666666) - -1.0) * Math.abs(re)) * im;
	} else {
		tmp = (0.5 * Math.abs(re)) * ((-2.0 - ((0.3333333333333333 - (-0.016666666666666666 * (im * im))) * (im * im))) * im);
	}
	return Math.copySign(1.0, re) * tmp;
}

Python

def code(re, im):
	tmp = 0
	if (0.5 * math.sin(math.fabs(re))) <= -0.005:
		tmp = -((((math.fabs(re) * math.fabs(re)) * -0.16666666666666666) - -1.0) * math.fabs(re)) * im
	else:
		tmp = (0.5 * math.fabs(re)) * ((-2.0 - ((0.3333333333333333 - (-0.016666666666666666 * (im * im))) * (im * im))) * im)
	return math.copysign(1.0, re) * tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(abs(re))) <= -0.005)
		tmp = Float64(Float64(-Float64(Float64(Float64(Float64(abs(re) * abs(re)) * -0.16666666666666666) - -1.0) * abs(re))) * im);
	else
		tmp = Float64(Float64(0.5 * abs(re)) * Float64(Float64(-2.0 - Float64(Float64(0.3333333333333333 - Float64(-0.016666666666666666 * Float64(im * im))) * Float64(im * im))) * im));
	end
	return Float64(copysign(1.0, re) * tmp)
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((0.5 * sin(abs(re))) <= -0.005)
		tmp = -((((abs(re) * abs(re)) * -0.16666666666666666) - -1.0) * abs(re)) * im;
	else
		tmp = (0.5 * abs(re)) * ((-2.0 - ((0.3333333333333333 - (-0.016666666666666666 * (im * im))) * (im * im))) * im);
	end
	tmp_2 = (sign(re) * abs(1.0)) * tmp;
end

Wolfram

code[re_, im_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(0.5 * N[Sin[N[Abs[re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.005], N[((-N[(N[(N[(N[(N[Abs[re], $MachinePrecision] * N[Abs[re], $MachinePrecision]), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - -1.0), $MachinePrecision] * N[Abs[re], $MachinePrecision]), $MachinePrecision]) * im), $MachinePrecision], N[(N[(0.5 * N[Abs[re], $MachinePrecision]), $MachinePrecision] * N[(N[(-2.0 - N[(N[(0.3333333333333333 - N[(-0.016666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision]), $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 65.6%

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

   2. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 52.0%

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

   4. Applied rewrites 52.0%

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

   5. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 35.5%

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

   7. Applied rewrites 35.5%

      \[\leadsto -1 \cdot \left(im \cdot \left(re \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot {re}^{2}\right)}\right)\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-neg-in N/A

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

      6. lower-*.f64 N/A

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

   9. Applied rewrites 35.5%

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

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

   1. Initial program 65.6%

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

   2. Taylor expanded in im around 0

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 90.8%

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

   4. Applied rewrites 90.8%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(-0.016666666666666666 \cdot {im}^{2} - 0.3333333333333333\right) - 2\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 90.8%

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 90.8%

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

   6. Applied rewrites 90.8%

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

   7. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot re\right)} \cdot \left(\left(-2 - \left(0.3333333333333333 - -0.016666666666666666 \cdot \left(im \cdot im\right)\right) \cdot \left(im \cdot im\right)\right) \cdot im\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 57.8%

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

   9. Applied rewrites 57.8%

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

Alternative 10: 63.2% accurate, 1.3× speedup

Math

\[\mathsf{copysign}\left(1, re\right) \cdot \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin \left(\left|re\right|\right) \leq -0.005:\\ \;\;\;\;\left(-\left(\left(\left|re\right| \cdot \left|re\right|\right) \cdot -0.16666666666666666 - -1\right) \cdot \left|re\right|\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \left|re\right|\right) \cdot \left(\left(-2 - 0.3333333333333333 \cdot \left(im \cdot im\right)\right) \cdot im\right)\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (*
 (copysign 1.0 re)
 (if (<= (* 0.5 (sin (fabs re))) -0.005)
   (*
    (-
     (*
      (- (* (* (fabs re) (fabs re)) -0.16666666666666666) -1.0)
      (fabs re)))
    im)
   (*
    (* 0.5 (fabs re))
    (* (- -2.0 (* 0.3333333333333333 (* im im))) im)))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(fabs(re))) <= -0.005) {
		tmp = -((((fabs(re) * fabs(re)) * -0.16666666666666666) - -1.0) * fabs(re)) * im;
	} else {
		tmp = (0.5 * fabs(re)) * ((-2.0 - (0.3333333333333333 * (im * im))) * im);
	}
	return copysign(1.0, re) * tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if ((0.5 * Math.sin(Math.abs(re))) <= -0.005) {
		tmp = -((((Math.abs(re) * Math.abs(re)) * -0.16666666666666666) - -1.0) * Math.abs(re)) * im;
	} else {
		tmp = (0.5 * Math.abs(re)) * ((-2.0 - (0.3333333333333333 * (im * im))) * im);
	}
	return Math.copySign(1.0, re) * tmp;
}

Python

def code(re, im):
	tmp = 0
	if (0.5 * math.sin(math.fabs(re))) <= -0.005:
		tmp = -((((math.fabs(re) * math.fabs(re)) * -0.16666666666666666) - -1.0) * math.fabs(re)) * im
	else:
		tmp = (0.5 * math.fabs(re)) * ((-2.0 - (0.3333333333333333 * (im * im))) * im)
	return math.copysign(1.0, re) * tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(abs(re))) <= -0.005)
		tmp = Float64(Float64(-Float64(Float64(Float64(Float64(abs(re) * abs(re)) * -0.16666666666666666) - -1.0) * abs(re))) * im);
	else
		tmp = Float64(Float64(0.5 * abs(re)) * Float64(Float64(-2.0 - Float64(0.3333333333333333 * Float64(im * im))) * im));
	end
	return Float64(copysign(1.0, re) * tmp)
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((0.5 * sin(abs(re))) <= -0.005)
		tmp = -((((abs(re) * abs(re)) * -0.16666666666666666) - -1.0) * abs(re)) * im;
	else
		tmp = (0.5 * abs(re)) * ((-2.0 - (0.3333333333333333 * (im * im))) * im);
	end
	tmp_2 = (sign(re) * abs(1.0)) * tmp;
end

Wolfram

code[re_, im_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(0.5 * N[Sin[N[Abs[re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.005], N[((-N[(N[(N[(N[(N[Abs[re], $MachinePrecision] * N[Abs[re], $MachinePrecision]), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - -1.0), $MachinePrecision] * N[Abs[re], $MachinePrecision]), $MachinePrecision]) * im), $MachinePrecision], N[(N[(0.5 * N[Abs[re], $MachinePrecision]), $MachinePrecision] * N[(N[(-2.0 - N[(0.3333333333333333 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision]), $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 65.6%

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

   2. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 52.0%

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

   4. Applied rewrites 52.0%

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

   5. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 35.5%

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

   7. Applied rewrites 35.5%

      \[\leadsto -1 \cdot \left(im \cdot \left(re \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot {re}^{2}\right)}\right)\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-neg-in N/A

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

      6. lower-*.f64 N/A

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

   9. Applied rewrites 35.5%

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

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

   1. Initial program 65.6%

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

   2. Taylor expanded in im around 0

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 90.8%

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

   4. Applied rewrites 90.8%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(-0.016666666666666666 \cdot {im}^{2} - 0.3333333333333333\right) - 2\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 90.8%

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 90.8%

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

   6. Applied rewrites 90.8%

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

   7. Taylor expanded in im around 0

      \[\leadsto \left(\sin re \cdot 0.5\right) \cdot \left(\left(-2 - \frac{1}{3} \cdot \left(im \cdot im\right)\right) \cdot im\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 84.4%

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

   10. Taylor expanded in re around 0

       \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot re\right)} \cdot \left(\left(-2 - 0.3333333333333333 \cdot \left(im \cdot im\right)\right) \cdot im\right) \]
   11. Step-by-step derivation

       1. lower-*.f64 53.5%

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

   12. Applied rewrites 53.5%

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

Alternative 11: 42.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double re, double im) {
	double tmp;
	if ((0.5 * Math.sin(Math.abs(re))) <= -0.005) {
		tmp = -((((Math.abs(re) * Math.abs(re)) * -0.16666666666666666) - -1.0) * Math.abs(re)) * im;
	} else {
		tmp = -(im * Math.abs(re));
	}
	return Math.copySign(1.0, re) * tmp;
}

Python

def code(re, im):
	tmp = 0
	if (0.5 * math.sin(math.fabs(re))) <= -0.005:
		tmp = -((((math.fabs(re) * math.fabs(re)) * -0.16666666666666666) - -1.0) * math.fabs(re)) * im
	else:
		tmp = -(im * math.fabs(re))
	return math.copysign(1.0, re) * tmp

Julia

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

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((0.5 * sin(abs(re))) <= -0.005)
		tmp = -((((abs(re) * abs(re)) * -0.16666666666666666) - -1.0) * abs(re)) * im;
	else
		tmp = -(im * abs(re));
	end
	tmp_2 = (sign(re) * abs(1.0)) * tmp;
end

Wolfram

code[re_, im_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(0.5 * N[Sin[N[Abs[re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.005], N[((-N[(N[(N[(N[(N[Abs[re], $MachinePrecision] * N[Abs[re], $MachinePrecision]), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - -1.0), $MachinePrecision] * N[Abs[re], $MachinePrecision]), $MachinePrecision]) * im), $MachinePrecision], (-N[(im * N[Abs[re], $MachinePrecision]), $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 65.6%

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

   2. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 52.0%

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

   4. Applied rewrites 52.0%

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

   5. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 35.5%

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

   7. Applied rewrites 35.5%

      \[\leadsto -1 \cdot \left(im \cdot \left(re \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot {re}^{2}\right)}\right)\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-neg-in N/A

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

      6. lower-*.f64 N/A

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

   9. Applied rewrites 35.5%

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

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

   1. Initial program 65.6%

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

   2. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 52.0%

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

   4. Applied rewrites 52.0%

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

   5. Taylor expanded in re around 0

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

      1. lower-*.f64 32.8%

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

   7. Applied rewrites 32.8%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 32.8%

         \[\leadsto -im \cdot re \]

   9. Applied rewrites 32.8%

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

Alternative 12: 32.8% accurate, 39.5× speedup

Math

\[-im \cdot re \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.6%

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

2. Taylor expanded in im around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sin.f64 52.0%

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

4. Applied rewrites 52.0%

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

5. Taylor expanded in re around 0

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

   1. lower-*.f64 32.8%

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

7. Applied rewrites 32.8%

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

   1. lift-*.f64 N/A

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

   2. mul-1-neg N/A

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

   3. lower-neg.f64 32.8%

      \[\leadsto -im \cdot re \]

9. Applied rewrites 32.8%

   \[\leadsto -im \cdot re \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025258 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"
  :precision binary64
  (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))