math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 38.1s

Alternatives: 8

Speedup: 1.5×

• 49.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (re im)
  :precision binary64
  (* (* 1/2 (sin re)) (+ (exp (- 0 im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (exp((0.0d0 - im)) + exp(im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(N[(1/2 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (re im)
  :precision binary64
  (* (* 1/2 (sin re)) (+ (exp (- 0 im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (exp((0.0d0 - im)) + exp(im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(N[(1/2 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.5× speedup

Math

\[\sin re \cdot \cosh im \]

FPCore

(FPCore (re im)
  :precision binary64
  (* (sin re) (cosh im)))

C

double code(double re, double im) {
	return sin(re) * cosh(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 = sin(re) * cosh(im)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-exp.f64 N/A

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

   8. lift-exp.f64 N/A

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

   9. lift--.f64 N/A

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

   10. sub0-neg N/A

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

   11. cosh-undef N/A

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

   12. associate-*r* N/A

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

   13. metadata-eval N/A

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

   14. cosh-0-rev N/A

       \[\leadsto \sin re \cdot \left(\color{blue}{\cosh 0} \cdot \cosh im\right) \]

   15. cosh-0-rev N/A

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

   16. *-lft-identity N/A

       \[\leadsto \sin re \cdot \color{blue}{\cosh im} \]

   17. lower-*.f64 N/A

       \[\leadsto \color{blue}{\sin re \cdot \cosh im} \]

   18. lower-cosh.f64 100.0%

       \[\leadsto \sin re \cdot \color{blue}{\cosh im} \]

3. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\sin re \cdot \cosh im} \]
4. Add Preprocessing

Alternative 2: 81.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} t_0 := \left|im\right| \cdot \left|im\right|\\ \mathbf{if}\;\left|im\right| \leq 6100000:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;\left|im\right| \leq 399999999999999993113045090218343426990990578856063590215900480871244616433664:\\ \;\;\;\;\left(re \cdot \left(re \cdot \left(\left(\frac{-1}{12} \cdot re\right) \cdot \left(1 - \frac{6}{re \cdot re}\right)\right)\right)\right) \cdot 2\\ \mathbf{elif}\;\left|im\right| \leq 6000000000000000248432939182427692818100624028184179935367669204583630813967171088177433668396288574871737718478838813024662997120339390456701055401984:\\ \;\;\;\;\left(\frac{1}{2} \cdot re\right) \cdot \frac{t\_0 \cdot t\_0 - 2 \cdot 2}{t\_0 - 2}\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 - -2\right) \cdot \left(re \cdot \frac{1}{2}\right)\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0 (* (fabs im) (fabs im))))
  (if (<= (fabs im) 6100000)
    (sin re)
    (if (<=
         (fabs im)
         399999999999999993113045090218343426990990578856063590215900480871244616433664)
      (* (* re (* re (* (* -1/12 re) (- 1 (/ 6 (* re re)))))) 2)
      (if (<=
           (fabs im)
           6000000000000000248432939182427692818100624028184179935367669204583630813967171088177433668396288574871737718478838813024662997120339390456701055401984)
        (* (* 1/2 re) (/ (- (* t_0 t_0) (* 2 2)) (- t_0 2)))
        (* (- t_0 -2) (* re 1/2)))))))

C

double code(double re, double im) {
	double t_0 = fabs(im) * fabs(im);
	double tmp;
	if (fabs(im) <= 6100000.0) {
		tmp = sin(re);
	} else if (fabs(im) <= 4e+77) {
		tmp = (re * (re * ((-0.08333333333333333 * re) * (1.0 - (6.0 / (re * re)))))) * 2.0;
	} else if (fabs(im) <= 6e+150) {
		tmp = (0.5 * re) * (((t_0 * t_0) - (2.0 * 2.0)) / (t_0 - 2.0));
	} else {
		tmp = (t_0 - -2.0) * (re * 0.5);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs(im) * abs(im)
    if (abs(im) <= 6100000.0d0) then
        tmp = sin(re)
    else if (abs(im) <= 4d+77) then
        tmp = (re * (re * (((-0.08333333333333333d0) * re) * (1.0d0 - (6.0d0 / (re * re)))))) * 2.0d0
    else if (abs(im) <= 6d+150) then
        tmp = (0.5d0 * re) * (((t_0 * t_0) - (2.0d0 * 2.0d0)) / (t_0 - 2.0d0))
    else
        tmp = (t_0 - (-2.0d0)) * (re * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.abs(im) * Math.abs(im);
	double tmp;
	if (Math.abs(im) <= 6100000.0) {
		tmp = Math.sin(re);
	} else if (Math.abs(im) <= 4e+77) {
		tmp = (re * (re * ((-0.08333333333333333 * re) * (1.0 - (6.0 / (re * re)))))) * 2.0;
	} else if (Math.abs(im) <= 6e+150) {
		tmp = (0.5 * re) * (((t_0 * t_0) - (2.0 * 2.0)) / (t_0 - 2.0));
	} else {
		tmp = (t_0 - -2.0) * (re * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.fabs(im) * math.fabs(im)
	tmp = 0
	if math.fabs(im) <= 6100000.0:
		tmp = math.sin(re)
	elif math.fabs(im) <= 4e+77:
		tmp = (re * (re * ((-0.08333333333333333 * re) * (1.0 - (6.0 / (re * re)))))) * 2.0
	elif math.fabs(im) <= 6e+150:
		tmp = (0.5 * re) * (((t_0 * t_0) - (2.0 * 2.0)) / (t_0 - 2.0))
	else:
		tmp = (t_0 - -2.0) * (re * 0.5)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(abs(im) * abs(im))
	tmp = 0.0
	if (abs(im) <= 6100000.0)
		tmp = sin(re);
	elseif (abs(im) <= 4e+77)
		tmp = Float64(Float64(re * Float64(re * Float64(Float64(-0.08333333333333333 * re) * Float64(1.0 - Float64(6.0 / Float64(re * re)))))) * 2.0);
	elseif (abs(im) <= 6e+150)
		tmp = Float64(Float64(0.5 * re) * Float64(Float64(Float64(t_0 * t_0) - Float64(2.0 * 2.0)) / Float64(t_0 - 2.0)));
	else
		tmp = Float64(Float64(t_0 - -2.0) * Float64(re * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = abs(im) * abs(im);
	tmp = 0.0;
	if (abs(im) <= 6100000.0)
		tmp = sin(re);
	elseif (abs(im) <= 4e+77)
		tmp = (re * (re * ((-0.08333333333333333 * re) * (1.0 - (6.0 / (re * re)))))) * 2.0;
	elseif (abs(im) <= 6e+150)
		tmp = (0.5 * re) * (((t_0 * t_0) - (2.0 * 2.0)) / (t_0 - 2.0));
	else
		tmp = (t_0 - -2.0) * (re * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Abs[im], $MachinePrecision] * N[Abs[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[im], $MachinePrecision], 6100000], N[Sin[re], $MachinePrecision], If[LessEqual[N[Abs[im], $MachinePrecision], 399999999999999993113045090218343426990990578856063590215900480871244616433664], N[(N[(re * N[(re * N[(N[(-1/12 * re), $MachinePrecision] * N[(1 - N[(6 / N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2), $MachinePrecision], If[LessEqual[N[Abs[im], $MachinePrecision], 6000000000000000248432939182427692818100624028184179935367669204583630813967171088177433668396288574871737718478838813024662997120339390456701055401984], N[(N[(1/2 * re), $MachinePrecision] * N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(2 * 2), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - 2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 - -2), $MachinePrecision] * N[(re * 1/2), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if im < 6.1e6

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 50.3%

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

   5. Taylor expanded in re around 0

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

      1. lower-*.f64 26.7%

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

   7. Applied rewrites 26.7%

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

   8. Taylor expanded in im around 0

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

      1. lower-+.f64 N/A

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

      2. lower-pow.f64 47.5%

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

   10. Applied rewrites 47.5%

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

   11. Taylor expanded in im around 0

       \[\leadsto \color{blue}{\sin re} \]
   12. Step-by-step derivation

       1. lower-sin.f64 50.3%

          \[\leadsto \sin re \]

   13. Applied rewrites 50.3%

       \[\leadsto \color{blue}{\sin re} \]

   if 6.1e6 < im < 3.9999999999999999e77

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 50.3%

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

   5. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 34.7%

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

   7. Applied rewrites 34.7%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 34.7%

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

   9. Applied rewrites 34.7%

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

       1. lift-+.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lift-*.f64 N/A

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

       8. +-commutative N/A

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

       9. metadata-eval N/A

          \[\leadsto \left(re \cdot \left(\left(re \cdot re\right) \cdot \frac{-1}{12} + \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)\right)\right) \cdot 2 \]

       10. sub-flip N/A

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

       11. sub-to-mult-rev N/A

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

       12. lift-/.f64 N/A

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

       13. lift--.f64 N/A

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

       14. *-commutative N/A

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

       15. lift-*.f64 N/A

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

       16. lift-*.f64 N/A

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

       17. associate-*l* N/A

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

       18. *-commutative N/A

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

       19. lift-*.f64 N/A

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

       20. associate-*l* N/A

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

       21. lower-*.f64 N/A

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

       22. lower-*.f64 35.0%

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

       23. lift-/.f64 N/A

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

       24. lift-*.f64 N/A

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

       25. *-commutative N/A

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

   11. Applied rewrites 35.0%

       \[\leadsto \left(re \cdot \left(re \cdot \color{blue}{\left(\left(\frac{-1}{12} \cdot re\right) \cdot \left(1 - \frac{6}{re \cdot re}\right)\right)}\right)\right) \cdot 2 \]

   if 3.9999999999999999e77 < im < 6.0000000000000002e150

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 50.3%

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

   5. Taylor expanded in re around 0

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

      1. lower-*.f64 26.7%

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

   7. Applied rewrites 26.7%

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

   8. Taylor expanded in im around 0

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

      1. lower-+.f64 N/A

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

      2. lower-pow.f64 47.5%

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

   10. Applied rewrites 47.5%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. flip-+ N/A

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

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

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

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

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

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

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

       7. lift-pow.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. lift-pow.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 N/A

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

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

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

       14. lower-unsound--.f64 36.0%

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

       15. lift-pow.f64 N/A

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

       16. unpow2 N/A

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

       17. lower-*.f64 36.0%

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

   12. Applied rewrites 36.0%

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

   if 6.0000000000000002e150 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 50.3%

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

   5. Taylor expanded in re around 0

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

      1. lower-*.f64 26.7%

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

   7. Applied rewrites 26.7%

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

   8. Taylor expanded in im around 0

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

      1. lower-+.f64 N/A

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

      2. lower-pow.f64 47.5%

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

   10. Applied rewrites 47.5%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 47.5%

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

       4. lift-+.f64 N/A

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

       5. +-commutative N/A

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

       6. add-flip N/A

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

       7. lower--.f64 N/A

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

       8. lift-pow.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 N/A

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

       11. metadata-eval 47.5%

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

       12. metadata-eval N/A

           \[\leadsto \left(im \cdot im - -2\right) \cdot \mathsf{Rewrite=>}\left(lift-*.f64, \left(\frac{1}{2} \cdot re\right)\right) \]

       13. metadata-eval N/A

           \[\leadsto \left(im \cdot im - -2\right) \cdot \mathsf{Rewrite<=}\left(*-commutative, \left(re \cdot \frac{1}{2}\right)\right) \]

       14. metadata-eval N/A

           \[\leadsto \left(im \cdot im - -2\right) \cdot \mathsf{Rewrite<=}\left(lift-*.f64, \left(re \cdot \frac{1}{2}\right)\right) \]

   12. Applied rewrites 47.5%

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

Alternative 3: 56.6% accurate, 1.2× speedup

Math

\[\mathsf{copysign}\left(1, re\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{1}{2} \cdot \sin \left(\left|re\right|\right) \leq \frac{5617791046444737}{2808895523222368605827039360607851146278089029597354019897345018089573059460952548948569958162617750330001779372990521213418590137725259726450741103741783193402623334763523207442222181269470220616454421126328215138096104411600982523029892352200425580677351729446660909999175717788745567263052442650378502144}:\\ \;\;\;\;\left(\left|re\right| \cdot \left(\left|re\right| \cdot \left(\left(\frac{-1}{12} \cdot \left|re\right|\right) \cdot \left(1 - \frac{6}{\left|re\right| \cdot \left|re\right|}\right)\right)\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im - -2\right) \cdot \left(\left|re\right| \cdot \frac{1}{2}\right)\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (*
 (copysign 1 re)
 (if (<=
      (* 1/2 (sin (fabs re)))
      5617791046444737/2808895523222368605827039360607851146278089029597354019897345018089573059460952548948569958162617750330001779372990521213418590137725259726450741103741783193402623334763523207442222181269470220616454421126328215138096104411600982523029892352200425580677351729446660909999175717788745567263052442650378502144)
   (*
    (*
     (fabs re)
     (*
      (fabs re)
      (* (* -1/12 (fabs re)) (- 1 (/ 6 (* (fabs re) (fabs re)))))))
    2)
   (* (- (* im im) -2) (* (fabs re) 1/2)))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(fabs(re))) <= 2e-291) {
		tmp = (fabs(re) * (fabs(re) * ((-0.08333333333333333 * fabs(re)) * (1.0 - (6.0 / (fabs(re) * fabs(re))))))) * 2.0;
	} else {
		tmp = ((im * im) - -2.0) * (fabs(re) * 0.5);
	}
	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))) <= 2e-291) {
		tmp = (Math.abs(re) * (Math.abs(re) * ((-0.08333333333333333 * Math.abs(re)) * (1.0 - (6.0 / (Math.abs(re) * Math.abs(re))))))) * 2.0;
	} else {
		tmp = ((im * im) - -2.0) * (Math.abs(re) * 0.5);
	}
	return Math.copySign(1.0, re) * tmp;
}

Python

def code(re, im):
	tmp = 0
	if (0.5 * math.sin(math.fabs(re))) <= 2e-291:
		tmp = (math.fabs(re) * (math.fabs(re) * ((-0.08333333333333333 * math.fabs(re)) * (1.0 - (6.0 / (math.fabs(re) * math.fabs(re))))))) * 2.0
	else:
		tmp = ((im * im) - -2.0) * (math.fabs(re) * 0.5)
	return math.copysign(1.0, re) * tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(abs(re))) <= 2e-291)
		tmp = Float64(Float64(abs(re) * Float64(abs(re) * Float64(Float64(-0.08333333333333333 * abs(re)) * Float64(1.0 - Float64(6.0 / Float64(abs(re) * abs(re))))))) * 2.0);
	else
		tmp = Float64(Float64(Float64(im * im) - -2.0) * Float64(abs(re) * 0.5));
	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))) <= 2e-291)
		tmp = (abs(re) * (abs(re) * ((-0.08333333333333333 * abs(re)) * (1.0 - (6.0 / (abs(re) * abs(re))))))) * 2.0;
	else
		tmp = ((im * im) - -2.0) * (abs(re) * 0.5);
	end
	tmp_2 = (sign(re) * abs(1.0)) * tmp;
end

Wolfram

code[re_, im_] := N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(1/2 * N[Sin[N[Abs[re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 5617791046444737/2808895523222368605827039360607851146278089029597354019897345018089573059460952548948569958162617750330001779372990521213418590137725259726450741103741783193402623334763523207442222181269470220616454421126328215138096104411600982523029892352200425580677351729446660909999175717788745567263052442650378502144], N[(N[(N[Abs[re], $MachinePrecision] * N[(N[Abs[re], $MachinePrecision] * N[(N[(-1/12 * N[Abs[re], $MachinePrecision]), $MachinePrecision] * N[(1 - N[(6 / N[(N[Abs[re], $MachinePrecision] * N[Abs[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2), $MachinePrecision], N[(N[(N[(im * im), $MachinePrecision] - -2), $MachinePrecision] * N[(N[Abs[re], $MachinePrecision] * 1/2), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < 1.9999999999999999e-291

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 50.3%

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

   5. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 34.7%

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

   7. Applied rewrites 34.7%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 34.7%

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

   9. Applied rewrites 34.7%

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

       1. lift-+.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lift-*.f64 N/A

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

       8. +-commutative N/A

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

       9. metadata-eval N/A

          \[\leadsto \left(re \cdot \left(\left(re \cdot re\right) \cdot \frac{-1}{12} + \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)\right)\right) \cdot 2 \]

       10. sub-flip N/A

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

       11. sub-to-mult-rev N/A

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

       12. lift-/.f64 N/A

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

       13. lift--.f64 N/A

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

       14. *-commutative N/A

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

       15. lift-*.f64 N/A

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

       16. lift-*.f64 N/A

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

       17. associate-*l* N/A

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

       18. *-commutative N/A

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

       19. lift-*.f64 N/A

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

       20. associate-*l* N/A

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

       21. lower-*.f64 N/A

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

       22. lower-*.f64 35.0%

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

       23. lift-/.f64 N/A

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

       24. lift-*.f64 N/A

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

       25. *-commutative N/A

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

   11. Applied rewrites 35.0%

       \[\leadsto \left(re \cdot \left(re \cdot \color{blue}{\left(\left(\frac{-1}{12} \cdot re\right) \cdot \left(1 - \frac{6}{re \cdot re}\right)\right)}\right)\right) \cdot 2 \]

   if 1.9999999999999999e-291 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 50.3%

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

   5. Taylor expanded in re around 0

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

      1. lower-*.f64 26.7%

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

   7. Applied rewrites 26.7%

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

   8. Taylor expanded in im around 0

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

      1. lower-+.f64 N/A

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

      2. lower-pow.f64 47.5%

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

   10. Applied rewrites 47.5%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 47.5%

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

       4. lift-+.f64 N/A

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

       5. +-commutative N/A

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

       6. add-flip N/A

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

       7. lower--.f64 N/A

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

       8. lift-pow.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 N/A

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

       11. metadata-eval 47.5%

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

       12. metadata-eval N/A

           \[\leadsto \left(im \cdot im - -2\right) \cdot \mathsf{Rewrite=>}\left(lift-*.f64, \left(\frac{1}{2} \cdot re\right)\right) \]

       13. metadata-eval N/A

           \[\leadsto \left(im \cdot im - -2\right) \cdot \mathsf{Rewrite<=}\left(*-commutative, \left(re \cdot \frac{1}{2}\right)\right) \]

       14. metadata-eval N/A

           \[\leadsto \left(im \cdot im - -2\right) \cdot \mathsf{Rewrite<=}\left(lift-*.f64, \left(re \cdot \frac{1}{2}\right)\right) \]

   12. Applied rewrites 47.5%

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

Alternative 4: 56.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \frac{-1}{12} \cdot \left|re\right|\\ \mathsf{copysign}\left(1, re\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{1}{2} \cdot \sin \left(\left|re\right|\right) \leq \frac{1012011266536553}{101201126653655309176247673359458653524778324882071059178450679013715169783997673445980191850718562247593538932158405955694904368692896738433506699970369254960758712138283180682233453871046608170619883839236372534281003741712346349309051677824579778170405028256179384776166707307615251266093163754323003131653853870546747392}:\\ \;\;\;\;\left(\left|re\right| \cdot \left(\left(\left|re\right| - \frac{\frac{-1}{2}}{t\_0}\right) \cdot t\_0\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im - -2\right) \cdot \left(\left|re\right| \cdot \frac{1}{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0 (* -1/12 (fabs re))))
  (*
   (copysign 1 re)
   (if (<=
        (* 1/2 (sin (fabs re)))
        1012011266536553/101201126653655309176247673359458653524778324882071059178450679013715169783997673445980191850718562247593538932158405955694904368692896738433506699970369254960758712138283180682233453871046608170619883839236372534281003741712346349309051677824579778170405028256179384776166707307615251266093163754323003131653853870546747392)
     (* (* (fabs re) (* (- (fabs re) (/ -1/2 t_0)) t_0)) 2)
     (* (- (* im im) -2) (* (fabs re) 1/2))))))

C

double code(double re, double im) {
	double t_0 = -0.08333333333333333 * fabs(re);
	double tmp;
	if ((0.5 * sin(fabs(re))) <= 1e-308) {
		tmp = (fabs(re) * ((fabs(re) - (-0.5 / t_0)) * t_0)) * 2.0;
	} else {
		tmp = ((im * im) - -2.0) * (fabs(re) * 0.5);
	}
	return copysign(1.0, re) * tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = -0.08333333333333333 * Math.abs(re);
	double tmp;
	if ((0.5 * Math.sin(Math.abs(re))) <= 1e-308) {
		tmp = (Math.abs(re) * ((Math.abs(re) - (-0.5 / t_0)) * t_0)) * 2.0;
	} else {
		tmp = ((im * im) - -2.0) * (Math.abs(re) * 0.5);
	}
	return Math.copySign(1.0, re) * tmp;
}

Python

def code(re, im):
	t_0 = -0.08333333333333333 * math.fabs(re)
	tmp = 0
	if (0.5 * math.sin(math.fabs(re))) <= 1e-308:
		tmp = (math.fabs(re) * ((math.fabs(re) - (-0.5 / t_0)) * t_0)) * 2.0
	else:
		tmp = ((im * im) - -2.0) * (math.fabs(re) * 0.5)
	return math.copysign(1.0, re) * tmp

Julia

function code(re, im)
	t_0 = Float64(-0.08333333333333333 * abs(re))
	tmp = 0.0
	if (Float64(0.5 * sin(abs(re))) <= 1e-308)
		tmp = Float64(Float64(abs(re) * Float64(Float64(abs(re) - Float64(-0.5 / t_0)) * t_0)) * 2.0);
	else
		tmp = Float64(Float64(Float64(im * im) - -2.0) * Float64(abs(re) * 0.5));
	end
	return Float64(copysign(1.0, re) * tmp)
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = -0.08333333333333333 * abs(re);
	tmp = 0.0;
	if ((0.5 * sin(abs(re))) <= 1e-308)
		tmp = (abs(re) * ((abs(re) - (-0.5 / t_0)) * t_0)) * 2.0;
	else
		tmp = ((im * im) - -2.0) * (abs(re) * 0.5);
	end
	tmp_2 = (sign(re) * abs(1.0)) * tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(-1/12 * N[Abs[re], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(1/2 * N[Sin[N[Abs[re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1012011266536553/101201126653655309176247673359458653524778324882071059178450679013715169783997673445980191850718562247593538932158405955694904368692896738433506699970369254960758712138283180682233453871046608170619883839236372534281003741712346349309051677824579778170405028256179384776166707307615251266093163754323003131653853870546747392], N[(N[(N[Abs[re], $MachinePrecision] * N[(N[(N[Abs[re], $MachinePrecision] - N[(-1/2 / t$95$0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] * 2), $MachinePrecision], N[(N[(N[(im * im), $MachinePrecision] - -2), $MachinePrecision] * N[(N[Abs[re], $MachinePrecision] * 1/2), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < 9.9999999999999991e-309

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 50.3%

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

   5. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 34.7%

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

   7. Applied rewrites 34.7%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 34.7%

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

   9. Applied rewrites 34.7%

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

       1. lift-+.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lift-*.f64 N/A

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

       8. +-commutative N/A

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

       9. metadata-eval N/A

          \[\leadsto \left(re \cdot \left(\left(re \cdot re\right) \cdot \frac{-1}{12} + \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)\right)\right) \cdot 2 \]

       10. sub-flip N/A

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

       11. sub-to-mult-rev N/A

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

       12. lift-/.f64 N/A

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

       13. lift--.f64 N/A

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

       14. lift-*.f64 N/A

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

       15. lift-*.f64 N/A

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

       16. associate-*l* N/A

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

       17. *-commutative N/A

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

       18. lift-*.f64 N/A

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

       19. associate-*r* N/A

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

       20. *-commutative N/A

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

       21. lower-*.f64 N/A

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

   11. Applied rewrites 34.6%

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

   if 9.9999999999999991e-309 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 50.3%

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

   5. Taylor expanded in re around 0

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

      1. lower-*.f64 26.7%

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

   7. Applied rewrites 26.7%

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

   8. Taylor expanded in im around 0

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

      1. lower-+.f64 N/A

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

      2. lower-pow.f64 47.5%

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

   10. Applied rewrites 47.5%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 47.5%

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

       4. lift-+.f64 N/A

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

       5. +-commutative N/A

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

       6. add-flip N/A

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

       7. lower--.f64 N/A

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

       8. lift-pow.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 N/A

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

       11. metadata-eval 47.5%

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

       12. metadata-eval N/A

           \[\leadsto \left(im \cdot im - -2\right) \cdot \mathsf{Rewrite=>}\left(lift-*.f64, \left(\frac{1}{2} \cdot re\right)\right) \]

       13. metadata-eval N/A

           \[\leadsto \left(im \cdot im - -2\right) \cdot \mathsf{Rewrite<=}\left(*-commutative, \left(re \cdot \frac{1}{2}\right)\right) \]

       14. metadata-eval N/A

           \[\leadsto \left(im \cdot im - -2\right) \cdot \mathsf{Rewrite<=}\left(lift-*.f64, \left(re \cdot \frac{1}{2}\right)\right) \]

   12. Applied rewrites 47.5%

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

Alternative 5: 56.4% accurate, 1.3× speedup

Math

\[\mathsf{copysign}\left(1, re\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{1}{2} \cdot \sin \left(\left|re\right|\right) \leq \frac{-5764607523034235}{576460752303423488}:\\ \;\;\;\;\left(\left|re\right| \cdot \left(1 \cdot \left(\left(\left|re\right| \cdot \left|re\right|\right) \cdot \frac{-1}{12}\right)\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im - -2\right) \cdot \left(\left|re\right| \cdot \frac{1}{2}\right)\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (*
 (copysign 1 re)
 (if (<= (* 1/2 (sin (fabs re))) -5764607523034235/576460752303423488)
   (* (* (fabs re) (* 1 (* (* (fabs re) (fabs re)) -1/12))) 2)
   (* (- (* im im) -2) (* (fabs re) 1/2)))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(fabs(re))) <= -0.01) {
		tmp = (fabs(re) * (1.0 * ((fabs(re) * fabs(re)) * -0.08333333333333333))) * 2.0;
	} else {
		tmp = ((im * im) - -2.0) * (fabs(re) * 0.5);
	}
	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))) <= -0.01) {
		tmp = (Math.abs(re) * (1.0 * ((Math.abs(re) * Math.abs(re)) * -0.08333333333333333))) * 2.0;
	} else {
		tmp = ((im * im) - -2.0) * (Math.abs(re) * 0.5);
	}
	return Math.copySign(1.0, re) * tmp;
}

Python

def code(re, im):
	tmp = 0
	if (0.5 * math.sin(math.fabs(re))) <= -0.01:
		tmp = (math.fabs(re) * (1.0 * ((math.fabs(re) * math.fabs(re)) * -0.08333333333333333))) * 2.0
	else:
		tmp = ((im * im) - -2.0) * (math.fabs(re) * 0.5)
	return math.copysign(1.0, re) * tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(abs(re))) <= -0.01)
		tmp = Float64(Float64(abs(re) * Float64(1.0 * Float64(Float64(abs(re) * abs(re)) * -0.08333333333333333))) * 2.0);
	else
		tmp = Float64(Float64(Float64(im * im) - -2.0) * Float64(abs(re) * 0.5));
	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))) <= -0.01)
		tmp = (abs(re) * (1.0 * ((abs(re) * abs(re)) * -0.08333333333333333))) * 2.0;
	else
		tmp = ((im * im) - -2.0) * (abs(re) * 0.5);
	end
	tmp_2 = (sign(re) * abs(1.0)) * tmp;
end

Wolfram

code[re_, im_] := N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(1/2 * N[Sin[N[Abs[re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -5764607523034235/576460752303423488], N[(N[(N[Abs[re], $MachinePrecision] * N[(1 * N[(N[(N[Abs[re], $MachinePrecision] * N[Abs[re], $MachinePrecision]), $MachinePrecision] * -1/12), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2), $MachinePrecision], N[(N[(N[(im * im), $MachinePrecision] - -2), $MachinePrecision] * N[(N[Abs[re], $MachinePrecision] * 1/2), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < -0.01

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 50.3%

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

   5. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 34.7%

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

   7. Applied rewrites 34.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. sub-to-mult N/A

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

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

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

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

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

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

         \[\leadsto \left(re \cdot \left(\left(1 - \frac{\mathsf{neg}\left(\frac{1}{2}\right)}{\frac{-1}{12} \cdot {re}^{2}}\right) \cdot \left(\frac{-1}{12} \cdot {re}^{2}\right)\right)\right) \cdot 2 \]

      8. metadata-eval 22.7%

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 22.7%

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 22.7%

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 22.7%

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

      18. lift-pow.f64 N/A

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

      19. unpow2 N/A

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

      20. lower-*.f64 22.7%

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

   9. Applied rewrites 22.7%

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

   10. Taylor expanded in re around inf

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

   12. Applied rewrites 10.9%

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

   if -0.01 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 50.3%

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

   5. Taylor expanded in re around 0

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

      1. lower-*.f64 26.7%

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

   7. Applied rewrites 26.7%

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

   8. Taylor expanded in im around 0

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

      1. lower-+.f64 N/A

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

      2. lower-pow.f64 47.5%

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

   10. Applied rewrites 47.5%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 47.5%

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

       4. lift-+.f64 N/A

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

       5. +-commutative N/A

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

       6. add-flip N/A

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

       7. lower--.f64 N/A

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

       8. lift-pow.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 N/A

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

       11. metadata-eval 47.5%

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

       12. metadata-eval N/A

           \[\leadsto \left(im \cdot im - -2\right) \cdot \mathsf{Rewrite=>}\left(lift-*.f64, \left(\frac{1}{2} \cdot re\right)\right) \]

       13. metadata-eval N/A

           \[\leadsto \left(im \cdot im - -2\right) \cdot \mathsf{Rewrite<=}\left(*-commutative, \left(re \cdot \frac{1}{2}\right)\right) \]

       14. metadata-eval N/A

           \[\leadsto \left(im \cdot im - -2\right) \cdot \mathsf{Rewrite<=}\left(lift-*.f64, \left(re \cdot \frac{1}{2}\right)\right) \]

   12. Applied rewrites 47.5%

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

Alternative 6: 55.8% accurate, 0.7× speedup

Math

\[\mathsf{copysign}\left(1, re\right) \cdot \begin{array}{l} \mathbf{if}\;\left(\frac{1}{2} \cdot \sin \left(\left|re\right|\right)\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq \frac{-5764607523034235}{288230376151711744}:\\ \;\;\;\;\left(\left|re\right| \cdot \left(\frac{1}{2} + \left(\frac{-1}{12} \cdot \left|re\right|\right) \cdot \left|re\right|\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im - -2\right) \cdot \left(\left|re\right| \cdot \frac{1}{2}\right)\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (*
 (copysign 1 re)
 (if (<=
      (* (* 1/2 (sin (fabs re))) (+ (exp (- 0 im)) (exp im)))
      -5764607523034235/288230376151711744)
   (* (* (fabs re) (+ 1/2 (* (* -1/12 (fabs re)) (fabs re)))) 2)
   (* (- (* im im) -2) (* (fabs re) 1/2)))))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(fabs(re))) * (exp((0.0 - im)) + exp(im))) <= -0.02) {
		tmp = (fabs(re) * (0.5 + ((-0.08333333333333333 * fabs(re)) * fabs(re)))) * 2.0;
	} else {
		tmp = ((im * im) - -2.0) * (fabs(re) * 0.5);
	}
	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))) * (Math.exp((0.0 - im)) + Math.exp(im))) <= -0.02) {
		tmp = (Math.abs(re) * (0.5 + ((-0.08333333333333333 * Math.abs(re)) * Math.abs(re)))) * 2.0;
	} else {
		tmp = ((im * im) - -2.0) * (Math.abs(re) * 0.5);
	}
	return Math.copySign(1.0, re) * tmp;
}

Python

def code(re, im):
	tmp = 0
	if ((0.5 * math.sin(math.fabs(re))) * (math.exp((0.0 - im)) + math.exp(im))) <= -0.02:
		tmp = (math.fabs(re) * (0.5 + ((-0.08333333333333333 * math.fabs(re)) * math.fabs(re)))) * 2.0
	else:
		tmp = ((im * im) - -2.0) * (math.fabs(re) * 0.5)
	return math.copysign(1.0, re) * tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(abs(re))) * Float64(exp(Float64(0.0 - im)) + exp(im))) <= -0.02)
		tmp = Float64(Float64(abs(re) * Float64(0.5 + Float64(Float64(-0.08333333333333333 * abs(re)) * abs(re)))) * 2.0);
	else
		tmp = Float64(Float64(Float64(im * im) - -2.0) * Float64(abs(re) * 0.5));
	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))) * (exp((0.0 - im)) + exp(im))) <= -0.02)
		tmp = (abs(re) * (0.5 + ((-0.08333333333333333 * abs(re)) * abs(re)))) * 2.0;
	else
		tmp = ((im * im) - -2.0) * (abs(re) * 0.5);
	end
	tmp_2 = (sign(re) * abs(1.0)) * tmp;
end

Wolfram

code[re_, im_] := N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[(1/2 * N[Sin[N[Abs[re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5764607523034235/288230376151711744], N[(N[(N[Abs[re], $MachinePrecision] * N[(1/2 + N[(N[(-1/12 * N[Abs[re], $MachinePrecision]), $MachinePrecision] * N[Abs[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2), $MachinePrecision], N[(N[(N[(im * im), $MachinePrecision] - -2), $MachinePrecision] * N[(N[Abs[re], $MachinePrecision] * 1/2), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -0.02

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 50.3%

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

   5. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 34.7%

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

   7. Applied rewrites 34.7%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 34.7%

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

   9. Applied rewrites 34.7%

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

   if -0.02 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 50.3%

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

   5. Taylor expanded in re around 0

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

      1. lower-*.f64 26.7%

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

   7. Applied rewrites 26.7%

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

   8. Taylor expanded in im around 0

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

      1. lower-+.f64 N/A

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

      2. lower-pow.f64 47.5%

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

   10. Applied rewrites 47.5%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 47.5%

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

       4. lift-+.f64 N/A

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

       5. +-commutative N/A

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

       6. add-flip N/A

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

       7. lower--.f64 N/A

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

       8. lift-pow.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 N/A

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

       11. metadata-eval 47.5%

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

       12. metadata-eval N/A

           \[\leadsto \left(im \cdot im - -2\right) \cdot \mathsf{Rewrite=>}\left(lift-*.f64, \left(\frac{1}{2} \cdot re\right)\right) \]

       13. metadata-eval N/A

           \[\leadsto \left(im \cdot im - -2\right) \cdot \mathsf{Rewrite<=}\left(*-commutative, \left(re \cdot \frac{1}{2}\right)\right) \]

       14. metadata-eval N/A

           \[\leadsto \left(im \cdot im - -2\right) \cdot \mathsf{Rewrite<=}\left(lift-*.f64, \left(re \cdot \frac{1}{2}\right)\right) \]

   12. Applied rewrites 47.5%

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

Alternative 7: 47.5% accurate, 16.7× speedup

Math

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

FPCore

(FPCore (re im)
  :precision binary64
  (* (- (* im im) -2) (* re 1/2)))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(re, im):
	return ((im * im) - -2.0) * (re * 0.5)

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(N[(N[(im * im), $MachinePrecision] - -2), $MachinePrecision] * N[(re * 1/2), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in im around 0

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

4. Applied rewrites 50.3%

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

5. Taylor expanded in re around 0

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

   1. lower-*.f64 26.7%

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

7. Applied rewrites 26.7%

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

8. Taylor expanded in im around 0

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

   1. lower-+.f64 N/A

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

   2. lower-pow.f64 47.5%

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

10. Applied rewrites 47.5%

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

    1. lift-*.f64 N/A

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

    2. *-commutative N/A

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

    3. lower-*.f64 47.5%

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

    4. lift-+.f64 N/A

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

    5. +-commutative N/A

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

    6. add-flip N/A

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

    7. lower--.f64 N/A

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

    8. lift-pow.f64 N/A

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

    9. unpow2 N/A

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

    10. lower-*.f64 N/A

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

    11. metadata-eval 47.5%

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

    12. metadata-eval N/A

        \[\leadsto \left(im \cdot im - -2\right) \cdot \mathsf{Rewrite=>}\left(lift-*.f64, \left(\frac{1}{2} \cdot re\right)\right) \]

    13. metadata-eval N/A

        \[\leadsto \left(im \cdot im - -2\right) \cdot \mathsf{Rewrite<=}\left(*-commutative, \left(re \cdot \frac{1}{2}\right)\right) \]

    14. metadata-eval N/A

        \[\leadsto \left(im \cdot im - -2\right) \cdot \mathsf{Rewrite<=}\left(lift-*.f64, \left(re \cdot \frac{1}{2}\right)\right) \]

12. Applied rewrites 47.5%

    \[\leadsto \color{blue}{\left(im \cdot im - -2\right) \cdot \left(re \cdot \frac{1}{2}\right)} \]
13. Add Preprocessing

Alternative 8: 26.7% accurate, 28.8× speedup

Math

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

FPCore

(FPCore (re im)
  :precision binary64
  (* (* 1/2 re) 2))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

public static double code(double re, double im) {
	return (0.5 * re) * 2.0;
}

Python

def code(re, im):
	return (0.5 * re) * 2.0

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(N[(1/2 * re), $MachinePrecision] * 2), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in im around 0

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

4. Applied rewrites 50.3%

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

5. Taylor expanded in re around 0

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

   1. lower-*.f64 26.7%

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

7. Applied rewrites 26.7%

   \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot re\right)} \cdot 2 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025271 -o generate:evaluate
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 1/2 (sin re)) (+ (exp (- 0 im)) (exp im))))