math.cos on complex, imaginary part

Percentage Accurate: 66.3% → 99.8%

Time: 6.2s

Alternatives: 25

Speedup: 2.3×

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

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ \mathbf{if}\;im \leq -0.065:\\ \;\;\;\;t\_0 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;im \leq 3.7:\\ \;\;\;\;t\_0 \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(1 - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re))))
   (if (<= im -0.065)
     (* t_0 (- (exp (- im)) (exp im)))
     (if (<= im 3.7)
       (*
        t_0
        (*
         (-
          (*
           (-
            (*
             (*
              (- (* -0.0003968253968253968 (* im im)) 0.016666666666666666)
              im)
             im)
            0.3333333333333333)
           (* im im))
          2.0)
         im))
       (* t_0 (- 1.0 (exp im)))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sin(re);
	double tmp;
	if (im <= -0.065) {
		tmp = t_0 * (exp(-im) - exp(im));
	} else if (im <= 3.7) {
		tmp = t_0 * ((((((((-0.0003968253968253968 * (im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	} else {
		tmp = t_0 * (1.0 - exp(im));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * sin(re)
    if (im <= (-0.065d0)) then
        tmp = t_0 * (exp(-im) - exp(im))
    else if (im <= 3.7d0) then
        tmp = t_0 * (((((((((-0.0003968253968253968d0) * (im * im)) - 0.016666666666666666d0) * im) * im) - 0.3333333333333333d0) * (im * im)) - 2.0d0) * im)
    else
        tmp = t_0 * (1.0d0 - exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.sin(re);
	double tmp;
	if (im <= -0.065) {
		tmp = t_0 * (Math.exp(-im) - Math.exp(im));
	} else if (im <= 3.7) {
		tmp = t_0 * ((((((((-0.0003968253968253968 * (im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	} else {
		tmp = t_0 * (1.0 - Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sin(re)
	tmp = 0
	if im <= -0.065:
		tmp = t_0 * (math.exp(-im) - math.exp(im))
	elif im <= 3.7:
		tmp = t_0 * ((((((((-0.0003968253968253968 * (im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im)
	else:
		tmp = t_0 * (1.0 - math.exp(im))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sin(re))
	tmp = 0.0
	if (im <= -0.065)
		tmp = Float64(t_0 * Float64(exp(Float64(-im)) - exp(im)));
	elseif (im <= 3.7)
		tmp = Float64(t_0 * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * Float64(im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * Float64(im * im)) - 2.0) * im));
	else
		tmp = Float64(t_0 * Float64(1.0 - exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * sin(re);
	tmp = 0.0;
	if (im <= -0.065)
		tmp = t_0 * (exp(-im) - exp(im));
	elseif (im <= 3.7)
		tmp = t_0 * ((((((((-0.0003968253968253968 * (im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	else
		tmp = t_0 * (1.0 - exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -0.065], N[(t$95$0 * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 3.7], N[(t$95$0 * N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.016666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(1.0 - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < -0.065000000000000002

   1. Initial program 100.0%

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

   if -0.065000000000000002 < im < 3.7000000000000002

   1. Initial program 31.4%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.6%

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

   if 3.7000000000000002 < im

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 99.9%

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

Alternative 2: 73.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(\mathsf{fma}\left(-1, im, 1\right) - \mathsf{fma}\left(\mathsf{fma}\left(im, 0.5, 1\right), im, 1\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(-\sin re\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.004166666666666667 \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (- (exp (- im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (* (* 0.5 re) (- (fma -1.0 im 1.0) (fma (fma im 0.5 1.0) im 1.0)))
     (if (<= t_0 0.0)
       (* (- (sin re)) im)
       (*
        (*
         (fma
          (- (* 0.004166666666666667 (* re re)) 0.08333333333333333)
          (* re re)
          0.5)
         re)
        (*
         (-
          (*
           (-
            (*
             (*
              (- (* -0.0003968253968253968 (* im im)) 0.016666666666666666)
              im)
             im)
            0.3333333333333333)
           (* im im))
          2.0)
         im))))))

C

double code(double re, double im) {
	double t_0 = (0.5 * sin(re)) * (exp(-im) - exp(im));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (0.5 * re) * (fma(-1.0, im, 1.0) - fma(fma(im, 0.5, 1.0), im, 1.0));
	} else if (t_0 <= 0.0) {
		tmp = -sin(re) * im;
	} else {
		tmp = (fma(((0.004166666666666667 * (re * re)) - 0.08333333333333333), (re * re), 0.5) * re) * ((((((((-0.0003968253968253968 * (im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - exp(im)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(0.5 * re) * Float64(fma(-1.0, im, 1.0) - fma(fma(im, 0.5, 1.0), im, 1.0)));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(-sin(re)) * im);
	else
		tmp = Float64(Float64(fma(Float64(Float64(0.004166666666666667 * Float64(re * re)) - 0.08333333333333333), Float64(re * re), 0.5) * re) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * Float64(im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * Float64(im * im)) - 2.0) * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(0.5 * re), $MachinePrecision] * N[(N[(-1.0 * im + 1.0), $MachinePrecision] - N[(N[(im * 0.5 + 1.0), $MachinePrecision] * im + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[((-N[Sin[re], $MachinePrecision]) * im), $MachinePrecision], N[(N[(N[(N[(N[(0.004166666666666667 * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.08333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.016666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 49.8%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 38.5%

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

   9. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 32.7

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

   11. Applied rewrites 32.7%

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

   12. Taylor expanded in im around 0

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

       1. mul-1-neg N/A

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

       2. +-commutative N/A

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

       3. mul-1-neg N/A

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

       4. lower-fma.f64 32.7

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

   14. Applied rewrites 32.7%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.0

   1. Initial program 30.8%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lift-sin.f64 99.0

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

   5. Applied rewrites 99.0%

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

   if 0.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 98.2%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 87.6%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 66.0

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

   8. Applied rewrites 66.0%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.004166666666666667 \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right)} \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 39.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \leq -5 \cdot 10^{-8}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(1 - \mathsf{fma}\left(im \cdot 0.5, im, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666 \cdot im, re \cdot re, -im\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))) -5e-8)
   (* (* 0.5 re) (- 1.0 (fma (* im 0.5) im 1.0)))
   (* (fma (* 0.16666666666666666 im) (* re re) (- im)) re)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(-im) - exp(im))) <= -5e-8) {
		tmp = (0.5 * re) * (1.0 - fma((im * 0.5), im, 1.0));
	} else {
		tmp = fma((0.16666666666666666 * im), (re * re), -im) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - exp(im))) <= -5e-8)
		tmp = Float64(Float64(0.5 * re) * Float64(1.0 - fma(Float64(im * 0.5), im, 1.0)));
	else
		tmp = Float64(fma(Float64(0.16666666666666666 * im), Float64(re * re), Float64(-im)) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-8], N[(N[(0.5 * re), $MachinePrecision] * N[(1.0 - N[(N[(im * 0.5), $MachinePrecision] * im + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.16666666666666666 * im), $MachinePrecision] * N[(re * re), $MachinePrecision] + (-im)), $MachinePrecision] * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -4.9999999999999998e-8

   1. Initial program 99.8%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 49.5%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 38.0%

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

   9. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 32.3

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

   11. Applied rewrites 32.3%

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

   12. Taylor expanded in im around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 32.3

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

   14. Applied rewrites 32.3%

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

   if -4.9999999999999998e-8 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 54.9%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lift-sin.f64 66.3

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

   5. Applied rewrites 66.3%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lift-neg.f64 41.6

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

   8. Applied rewrites 41.6%

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

   9. Taylor expanded in re around 0

      \[\leadsto \mathsf{fma}\left(\frac{1}{6} \cdot im, re \cdot re, -im\right) \cdot re \]
   10. Step-by-step derivation

       1. lift-*.f64 42.1

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

   11. Applied rewrites 42.1%

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

Alternative 4: 39.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \leq -2000000:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(1 - \mathsf{fma}\left(im \cdot im, 0.5, im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666 \cdot im, re \cdot re, -im\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))) -2000000.0)
   (* (* 0.5 re) (- 1.0 (fma (* im im) 0.5 im)))
   (* (fma (* 0.16666666666666666 im) (* re re) (- im)) re)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(-im) - exp(im))) <= -2000000.0) {
		tmp = (0.5 * re) * (1.0 - fma((im * im), 0.5, im));
	} else {
		tmp = fma((0.16666666666666666 * im), (re * re), -im) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - exp(im))) <= -2000000.0)
		tmp = Float64(Float64(0.5 * re) * Float64(1.0 - fma(Float64(im * im), 0.5, im)));
	else
		tmp = Float64(fma(Float64(0.16666666666666666 * im), Float64(re * re), Float64(-im)) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2000000.0], N[(N[(0.5 * re), $MachinePrecision] * N[(1.0 - N[(N[(im * im), $MachinePrecision] * 0.5 + im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.16666666666666666 * im), $MachinePrecision] * N[(re * re), $MachinePrecision] + (-im)), $MachinePrecision] * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -2e6

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 49.8%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 38.4%

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

   9. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 32.6

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

   11. Applied rewrites 32.6%

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

   12. Taylor expanded in im around inf

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

       1. distribute-lft-in N/A

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

       2. inv-pow N/A

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

       3. pow-prod-up N/A

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

       4. metadata-eval N/A

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

       5. unpow1 N/A

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

       6. lower-fma.f64 N/A

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

       7. pow2 N/A

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

       8. lift-*.f64 32.6

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

   14. Applied rewrites 32.6%

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

   if -2e6 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 55.1%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lift-sin.f64 66.3

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

   5. Applied rewrites 66.3%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lift-neg.f64 41.5

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

   8. Applied rewrites 41.5%

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

   9. Taylor expanded in re around 0

      \[\leadsto \mathsf{fma}\left(\frac{1}{6} \cdot im, re \cdot re, -im\right) \cdot re \]
   10. Step-by-step derivation

       1. lift-*.f64 41.9

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

   11. Applied rewrites 41.9%

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

Alternative 5: 39.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \leq -2000000:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(1 - \left(im \cdot im\right) \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666 \cdot im, re \cdot re, -im\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))) -2000000.0)
   (* (* 0.5 re) (- 1.0 (* (* im im) 0.5)))
   (* (fma (* 0.16666666666666666 im) (* re re) (- im)) re)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(-im) - exp(im))) <= -2000000.0) {
		tmp = (0.5 * re) * (1.0 - ((im * im) * 0.5));
	} else {
		tmp = fma((0.16666666666666666 * im), (re * re), -im) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - exp(im))) <= -2000000.0)
		tmp = Float64(Float64(0.5 * re) * Float64(1.0 - Float64(Float64(im * im) * 0.5)));
	else
		tmp = Float64(fma(Float64(0.16666666666666666 * im), Float64(re * re), Float64(-im)) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2000000.0], N[(N[(0.5 * re), $MachinePrecision] * N[(1.0 - N[(N[(im * im), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.16666666666666666 * im), $MachinePrecision] * N[(re * re), $MachinePrecision] + (-im)), $MachinePrecision] * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -2e6

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 49.8%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 38.4%

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

   9. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 32.6

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

   11. Applied rewrites 32.6%

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

   12. Taylor expanded in im around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 32.6

          \[\leadsto \left(0.5 \cdot re\right) \cdot \left(1 - \left(im \cdot im\right) \cdot 0.5\right) \]

   14. Applied rewrites 32.6%

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

   if -2e6 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 55.1%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lift-sin.f64 66.3

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

   5. Applied rewrites 66.3%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lift-neg.f64 41.5

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

   8. Applied rewrites 41.5%

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

   9. Taylor expanded in re around 0

      \[\leadsto \mathsf{fma}\left(\frac{1}{6} \cdot im, re \cdot re, -im\right) \cdot re \]
   10. Step-by-step derivation

       1. lift-*.f64 41.9

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

   11. Applied rewrites 41.9%

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

Alternative 6: 53.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ \mathbf{if}\;t\_0 \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666 \cdot im, re \cdot re, -im\right) \cdot re\\ \mathbf{elif}\;t\_0 \leq 0.3:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(\left(-0.3333333333333333 \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(re \cdot re\right) \cdot im\right) \cdot -0.008333333333333333, re \cdot re, -im\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re))))
   (if (<= t_0 -0.005)
     (* (fma (* 0.16666666666666666 im) (* re re) (- im)) re)
     (if (<= t_0 0.3)
       (* (* 0.5 re) (* (- (* -0.3333333333333333 (* im im)) 2.0) im))
       (*
        (fma (* (* (* re re) im) -0.008333333333333333) (* re re) (- im))
        re)))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sin(re);
	double tmp;
	if (t_0 <= -0.005) {
		tmp = fma((0.16666666666666666 * im), (re * re), -im) * re;
	} else if (t_0 <= 0.3) {
		tmp = (0.5 * re) * (((-0.3333333333333333 * (im * im)) - 2.0) * im);
	} else {
		tmp = fma((((re * re) * im) * -0.008333333333333333), (re * re), -im) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * sin(re))
	tmp = 0.0
	if (t_0 <= -0.005)
		tmp = Float64(fma(Float64(0.16666666666666666 * im), Float64(re * re), Float64(-im)) * re);
	elseif (t_0 <= 0.3)
		tmp = Float64(Float64(0.5 * re) * Float64(Float64(Float64(-0.3333333333333333 * Float64(im * im)) - 2.0) * im));
	else
		tmp = Float64(fma(Float64(Float64(Float64(re * re) * im) * -0.008333333333333333), Float64(re * re), Float64(-im)) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.005], N[(N[(N[(0.16666666666666666 * im), $MachinePrecision] * N[(re * re), $MachinePrecision] + (-im)), $MachinePrecision] * re), $MachinePrecision], If[LessEqual[t$95$0, 0.3], N[(N[(0.5 * re), $MachinePrecision] * N[(N[(N[(-0.3333333333333333 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(re * re), $MachinePrecision] * im), $MachinePrecision] * -0.008333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + (-im)), $MachinePrecision] * re), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 55.7%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lift-sin.f64 50.4

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

   5. Applied rewrites 50.4%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lift-neg.f64 23.8

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

   8. Applied rewrites 23.8%

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

   9. Taylor expanded in re around 0

      \[\leadsto \mathsf{fma}\left(\frac{1}{6} \cdot im, re \cdot re, -im\right) \cdot re \]
   10. Step-by-step derivation

       1. lift-*.f64 23.1

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

   11. Applied rewrites 23.1%

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

   if -0.0050000000000000001 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < 0.299999999999999989

   1. Initial program 73.1%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 84.0

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

   5. Applied rewrites 84.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 73.2%

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

   if 0.299999999999999989 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 56.4%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lift-sin.f64 49.9

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

   5. Applied rewrites 49.9%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lift-neg.f64 23.4

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

   8. Applied rewrites 23.4%

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

   9. Taylor expanded in re around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 23.4

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

   11. Applied rewrites 23.4%

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

Alternative 7: 99.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ \mathbf{if}\;im \leq -3.3 \cdot 10^{+44}:\\ \;\;\;\;t\_0 \cdot \left(\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \mathbf{elif}\;im \leq -15.6:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{-im} - 1\right)\\ \mathbf{elif}\;im \leq 3.7:\\ \;\;\;\;t\_0 \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(1 - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re))))
   (if (<= im -3.3e+44)
     (*
      t_0
      (*
       (-
        (*
         (-
          (* (* (* (* -0.0003968253968253968 im) im) im) im)
          0.3333333333333333)
         (* im im))
        2.0)
       im))
     (if (<= im -15.6)
       (* (* 0.5 re) (- (exp (- im)) 1.0))
       (if (<= im 3.7)
         (*
          t_0
          (*
           (-
            (*
             (-
              (*
               (*
                (- (* -0.0003968253968253968 (* im im)) 0.016666666666666666)
                im)
               im)
              0.3333333333333333)
             (* im im))
            2.0)
           im))
         (* t_0 (- 1.0 (exp im))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sin(re);
	double tmp;
	if (im <= -3.3e+44) {
		tmp = t_0 * ((((((((-0.0003968253968253968 * im) * im) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	} else if (im <= -15.6) {
		tmp = (0.5 * re) * (exp(-im) - 1.0);
	} else if (im <= 3.7) {
		tmp = t_0 * ((((((((-0.0003968253968253968 * (im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	} else {
		tmp = t_0 * (1.0 - exp(im));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * sin(re)
    if (im <= (-3.3d+44)) then
        tmp = t_0 * (((((((((-0.0003968253968253968d0) * im) * im) * im) * im) - 0.3333333333333333d0) * (im * im)) - 2.0d0) * im)
    else if (im <= (-15.6d0)) then
        tmp = (0.5d0 * re) * (exp(-im) - 1.0d0)
    else if (im <= 3.7d0) then
        tmp = t_0 * (((((((((-0.0003968253968253968d0) * (im * im)) - 0.016666666666666666d0) * im) * im) - 0.3333333333333333d0) * (im * im)) - 2.0d0) * im)
    else
        tmp = t_0 * (1.0d0 - exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.sin(re);
	double tmp;
	if (im <= -3.3e+44) {
		tmp = t_0 * ((((((((-0.0003968253968253968 * im) * im) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	} else if (im <= -15.6) {
		tmp = (0.5 * re) * (Math.exp(-im) - 1.0);
	} else if (im <= 3.7) {
		tmp = t_0 * ((((((((-0.0003968253968253968 * (im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	} else {
		tmp = t_0 * (1.0 - Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sin(re)
	tmp = 0
	if im <= -3.3e+44:
		tmp = t_0 * ((((((((-0.0003968253968253968 * im) * im) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im)
	elif im <= -15.6:
		tmp = (0.5 * re) * (math.exp(-im) - 1.0)
	elif im <= 3.7:
		tmp = t_0 * ((((((((-0.0003968253968253968 * (im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im)
	else:
		tmp = t_0 * (1.0 - math.exp(im))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sin(re))
	tmp = 0.0
	if (im <= -3.3e+44)
		tmp = Float64(t_0 * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * im) * im) * im) * im) - 0.3333333333333333) * Float64(im * im)) - 2.0) * im));
	elseif (im <= -15.6)
		tmp = Float64(Float64(0.5 * re) * Float64(exp(Float64(-im)) - 1.0));
	elseif (im <= 3.7)
		tmp = Float64(t_0 * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * Float64(im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * Float64(im * im)) - 2.0) * im));
	else
		tmp = Float64(t_0 * Float64(1.0 - exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * sin(re);
	tmp = 0.0;
	if (im <= -3.3e+44)
		tmp = t_0 * ((((((((-0.0003968253968253968 * im) * im) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	elseif (im <= -15.6)
		tmp = (0.5 * re) * (exp(-im) - 1.0);
	elseif (im <= 3.7)
		tmp = t_0 * ((((((((-0.0003968253968253968 * (im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	else
		tmp = t_0 * (1.0 - exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -3.3e+44], N[(t$95$0 * N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, -15.6], N[(N[(0.5 * re), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 3.7], N[(t$95$0 * N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.016666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(1.0 - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -3.30000000000000013e44

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in im around inf

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

      1. pow2 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 100.0

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   8. Applied rewrites 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   if -3.30000000000000013e44 < im < -15.5999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 77.8%

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

   if -15.5999999999999996 < im < 3.7000000000000002

   1. Initial program 31.6%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.5%

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

   if 3.7000000000000002 < im

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 99.9%

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

Alternative 8: 99.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ \mathbf{if}\;im \leq -3.8:\\ \;\;\;\;t\_0 \cdot \left(e^{-im} - 1\right)\\ \mathbf{elif}\;im \leq 3.7:\\ \;\;\;\;t\_0 \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(1 - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re))))
   (if (<= im -3.8)
     (* t_0 (- (exp (- im)) 1.0))
     (if (<= im 3.7)
       (*
        t_0
        (*
         (-
          (*
           (-
            (*
             (*
              (- (* -0.0003968253968253968 (* im im)) 0.016666666666666666)
              im)
             im)
            0.3333333333333333)
           (* im im))
          2.0)
         im))
       (* t_0 (- 1.0 (exp im)))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sin(re);
	double tmp;
	if (im <= -3.8) {
		tmp = t_0 * (exp(-im) - 1.0);
	} else if (im <= 3.7) {
		tmp = t_0 * ((((((((-0.0003968253968253968 * (im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	} else {
		tmp = t_0 * (1.0 - exp(im));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * sin(re)
    if (im <= (-3.8d0)) then
        tmp = t_0 * (exp(-im) - 1.0d0)
    else if (im <= 3.7d0) then
        tmp = t_0 * (((((((((-0.0003968253968253968d0) * (im * im)) - 0.016666666666666666d0) * im) * im) - 0.3333333333333333d0) * (im * im)) - 2.0d0) * im)
    else
        tmp = t_0 * (1.0d0 - exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.sin(re);
	double tmp;
	if (im <= -3.8) {
		tmp = t_0 * (Math.exp(-im) - 1.0);
	} else if (im <= 3.7) {
		tmp = t_0 * ((((((((-0.0003968253968253968 * (im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	} else {
		tmp = t_0 * (1.0 - Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sin(re)
	tmp = 0
	if im <= -3.8:
		tmp = t_0 * (math.exp(-im) - 1.0)
	elif im <= 3.7:
		tmp = t_0 * ((((((((-0.0003968253968253968 * (im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im)
	else:
		tmp = t_0 * (1.0 - math.exp(im))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sin(re))
	tmp = 0.0
	if (im <= -3.8)
		tmp = Float64(t_0 * Float64(exp(Float64(-im)) - 1.0));
	elseif (im <= 3.7)
		tmp = Float64(t_0 * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * Float64(im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * Float64(im * im)) - 2.0) * im));
	else
		tmp = Float64(t_0 * Float64(1.0 - exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * sin(re);
	tmp = 0.0;
	if (im <= -3.8)
		tmp = t_0 * (exp(-im) - 1.0);
	elseif (im <= 3.7)
		tmp = t_0 * ((((((((-0.0003968253968253968 * (im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	else
		tmp = t_0 * (1.0 - exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -3.8], N[(t$95$0 * N[(N[Exp[(-im)], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 3.7], N[(t$95$0 * N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.016666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(1.0 - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < -3.7999999999999998

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 99.9%

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

   if -3.7999999999999998 < im < 3.7000000000000002

   1. Initial program 31.6%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.5%

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

   if 3.7000000000000002 < im

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 99.9%

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

Alternative 9: 59.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.005:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(-0.3333333333333333 \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.004166666666666667 \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.005)
   (*
    (* (fma (* re re) -0.08333333333333333 0.5) re)
    (* (- (* -0.3333333333333333 (* im im)) 2.0) im))
   (*
    (*
     (fma
      (- (* 0.004166666666666667 (* re re)) 0.08333333333333333)
      (* re re)
      0.5)
     re)
    (*
     (-
      (*
       (-
        (*
         (* (- (* -0.0003968253968253968 (* im im)) 0.016666666666666666) im)
         im)
        0.3333333333333333)
       (* im im))
      2.0)
     im))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.005) {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * re) * (((-0.3333333333333333 * (im * im)) - 2.0) * im);
	} else {
		tmp = (fma(((0.004166666666666667 * (re * re)) - 0.08333333333333333), (re * re), 0.5) * re) * ((((((((-0.0003968253968253968 * (im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.005)
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re) * Float64(Float64(Float64(-0.3333333333333333 * Float64(im * im)) - 2.0) * im));
	else
		tmp = Float64(Float64(fma(Float64(Float64(0.004166666666666667 * Float64(re * re)) - 0.08333333333333333), Float64(re * re), 0.5) * re) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * Float64(im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * Float64(im * im)) - 2.0) * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.005], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(-0.3333333333333333 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.004166666666666667 * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.08333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.016666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 55.7%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 83.8

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

   5. Applied rewrites 83.8%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 26.2

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

   8. Applied rewrites 26.2%

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

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

   1. Initial program 69.9%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 93.0%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 70.9

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

   8. Applied rewrites 70.9%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.004166666666666667 \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right)} \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 58.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq 2 \cdot 10^{-49}:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.004166666666666667 \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) 2e-49)
   (*
    (* (fma (* re re) -0.08333333333333333 0.5) re)
    (*
     (-
      (* (- (* (* -0.016666666666666666 im) im) 0.3333333333333333) (* im im))
      2.0)
     im))
   (*
    (*
     (fma
      (- (* 0.004166666666666667 (* re re)) 0.08333333333333333)
      (* re re)
      0.5)
     re)
    (*
     (-
      (*
       (-
        (* (* (* (* -0.0003968253968253968 im) im) im) im)
        0.3333333333333333)
       (* im im))
      2.0)
     im))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= 2e-49) {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * re) * ((((((-0.016666666666666666 * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	} else {
		tmp = (fma(((0.004166666666666667 * (re * re)) - 0.08333333333333333), (re * re), 0.5) * re) * ((((((((-0.0003968253968253968 * im) * im) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= 2e-49)
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re) * Float64(Float64(Float64(Float64(Float64(Float64(-0.016666666666666666 * im) * im) - 0.3333333333333333) * Float64(im * im)) - 2.0) * im));
	else
		tmp = Float64(Float64(fma(Float64(Float64(0.004166666666666667 * Float64(re * re)) - 0.08333333333333333), Float64(re * re), 0.5) * re) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * im) * im) * im) * im) - 0.3333333333333333) * Float64(im * im)) - 2.0) * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], 2e-49], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(N[(N[(N[(-0.016666666666666666 * im), $MachinePrecision] * im), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.004166666666666667 * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.08333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < 1.99999999999999987e-49

   1. Initial program 70.6%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 92.7%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 90.1%

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

   9. Taylor expanded in re around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 67.7

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

   11. Applied rewrites 67.7%

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

   if 1.99999999999999987e-49 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 55.4%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 93.4%

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

   6. Taylor expanded in im around inf

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

      1. pow2 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 93.2

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   8. Applied rewrites 93.2%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   9. Taylor expanded in re around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. lower--.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 N/A

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

       10. pow2 N/A

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

       11. lift-*.f64 35.0

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

   11. Applied rewrites 35.0%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.004166666666666667 \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right)} \cdot \left(\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 95.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ \mathbf{if}\;im \leq -3.3 \cdot 10^{+44}:\\ \;\;\;\;t\_0 \cdot \left(\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \mathbf{elif}\;im \leq -15.6:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{-im} - 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re))))
   (if (<= im -3.3e+44)
     (*
      t_0
      (*
       (-
        (*
         (-
          (* (* (* (* -0.0003968253968253968 im) im) im) im)
          0.3333333333333333)
         (* im im))
        2.0)
       im))
     (if (<= im -15.6)
       (* (* 0.5 re) (- (exp (- im)) 1.0))
       (*
        t_0
        (*
         (-
          (*
           (-
            (*
             (*
              (- (* -0.0003968253968253968 (* im im)) 0.016666666666666666)
              im)
             im)
            0.3333333333333333)
           (* im im))
          2.0)
         im))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sin(re);
	double tmp;
	if (im <= -3.3e+44) {
		tmp = t_0 * ((((((((-0.0003968253968253968 * im) * im) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	} else if (im <= -15.6) {
		tmp = (0.5 * re) * (exp(-im) - 1.0);
	} else {
		tmp = t_0 * ((((((((-0.0003968253968253968 * (im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * sin(re)
    if (im <= (-3.3d+44)) then
        tmp = t_0 * (((((((((-0.0003968253968253968d0) * im) * im) * im) * im) - 0.3333333333333333d0) * (im * im)) - 2.0d0) * im)
    else if (im <= (-15.6d0)) then
        tmp = (0.5d0 * re) * (exp(-im) - 1.0d0)
    else
        tmp = t_0 * (((((((((-0.0003968253968253968d0) * (im * im)) - 0.016666666666666666d0) * im) * im) - 0.3333333333333333d0) * (im * im)) - 2.0d0) * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.sin(re);
	double tmp;
	if (im <= -3.3e+44) {
		tmp = t_0 * ((((((((-0.0003968253968253968 * im) * im) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	} else if (im <= -15.6) {
		tmp = (0.5 * re) * (Math.exp(-im) - 1.0);
	} else {
		tmp = t_0 * ((((((((-0.0003968253968253968 * (im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sin(re)
	tmp = 0
	if im <= -3.3e+44:
		tmp = t_0 * ((((((((-0.0003968253968253968 * im) * im) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im)
	elif im <= -15.6:
		tmp = (0.5 * re) * (math.exp(-im) - 1.0)
	else:
		tmp = t_0 * ((((((((-0.0003968253968253968 * (im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sin(re))
	tmp = 0.0
	if (im <= -3.3e+44)
		tmp = Float64(t_0 * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * im) * im) * im) * im) - 0.3333333333333333) * Float64(im * im)) - 2.0) * im));
	elseif (im <= -15.6)
		tmp = Float64(Float64(0.5 * re) * Float64(exp(Float64(-im)) - 1.0));
	else
		tmp = Float64(t_0 * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * Float64(im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * Float64(im * im)) - 2.0) * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * sin(re);
	tmp = 0.0;
	if (im <= -3.3e+44)
		tmp = t_0 * ((((((((-0.0003968253968253968 * im) * im) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	elseif (im <= -15.6)
		tmp = (0.5 * re) * (exp(-im) - 1.0);
	else
		tmp = t_0 * ((((((((-0.0003968253968253968 * (im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -3.3e+44], N[(t$95$0 * N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, -15.6], N[(N[(0.5 * re), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.016666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < -3.30000000000000013e44

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in im around inf

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

      1. pow2 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 100.0

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   8. Applied rewrites 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   if -3.30000000000000013e44 < im < -15.5999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 77.8%

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

   if -15.5999999999999996 < im

   1. Initial program 54.8%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 94.9%

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

Alternative 12: 58.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq 0.004:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) 0.004)
   (*
    (* (fma (* re re) -0.08333333333333333 0.5) re)
    (*
     (-
      (* (- (* (* -0.016666666666666666 im) im) 0.3333333333333333) (* im im))
      2.0)
     im))
   (*
    (* 0.5 re)
    (*
     (-
      (*
       (-
        (*
         (* (- (* -0.0003968253968253968 (* im im)) 0.016666666666666666) im)
         im)
        0.3333333333333333)
       (* im im))
      2.0)
     im))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= 0.004) {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * re) * ((((((-0.016666666666666666 * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	} else {
		tmp = (0.5 * re) * ((((((((-0.0003968253968253968 * (im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= 0.004)
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re) * Float64(Float64(Float64(Float64(Float64(Float64(-0.016666666666666666 * im) * im) - 0.3333333333333333) * Float64(im * im)) - 2.0) * im));
	else
		tmp = Float64(Float64(0.5 * re) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * Float64(im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * Float64(im * im)) - 2.0) * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], 0.004], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(N[(N[(N[(-0.016666666666666666 * im), $MachinePrecision] * im), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.016666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < 0.0040000000000000001

   1. Initial program 69.9%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 92.8%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 90.2%

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

   9. Taylor expanded in re around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 68.8

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

   11. Applied rewrites 68.8%

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

   if 0.0040000000000000001 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 55.4%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 93.2%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 25.8%

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

Alternative 13: 95.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \mathbf{if}\;im \leq -3.3 \cdot 10^{+44}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im \leq -15.6:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{-im} - 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (*
          (* 0.5 (sin re))
          (*
           (-
            (*
             (-
              (* (* (* (* -0.0003968253968253968 im) im) im) im)
              0.3333333333333333)
             (* im im))
            2.0)
           im))))
   (if (<= im -3.3e+44)
     t_0
     (if (<= im -15.6) (* (* 0.5 re) (- (exp (- im)) 1.0)) t_0))))

C

double code(double re, double im) {
	double t_0 = (0.5 * sin(re)) * ((((((((-0.0003968253968253968 * im) * im) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	double tmp;
	if (im <= -3.3e+44) {
		tmp = t_0;
	} else if (im <= -15.6) {
		tmp = (0.5 * re) * (exp(-im) - 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (0.5d0 * sin(re)) * (((((((((-0.0003968253968253968d0) * im) * im) * im) * im) - 0.3333333333333333d0) * (im * im)) - 2.0d0) * im)
    if (im <= (-3.3d+44)) then
        tmp = t_0
    else if (im <= (-15.6d0)) then
        tmp = (0.5d0 * re) * (exp(-im) - 1.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (0.5 * Math.sin(re)) * ((((((((-0.0003968253968253968 * im) * im) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	double tmp;
	if (im <= -3.3e+44) {
		tmp = t_0;
	} else if (im <= -15.6) {
		tmp = (0.5 * re) * (Math.exp(-im) - 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (0.5 * math.sin(re)) * ((((((((-0.0003968253968253968 * im) * im) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im)
	tmp = 0
	if im <= -3.3e+44:
		tmp = t_0
	elif im <= -15.6:
		tmp = (0.5 * re) * (math.exp(-im) - 1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * im) * im) * im) * im) - 0.3333333333333333) * Float64(im * im)) - 2.0) * im))
	tmp = 0.0
	if (im <= -3.3e+44)
		tmp = t_0;
	elseif (im <= -15.6)
		tmp = Float64(Float64(0.5 * re) * Float64(exp(Float64(-im)) - 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (0.5 * sin(re)) * ((((((((-0.0003968253968253968 * im) * im) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	tmp = 0.0;
	if (im <= -3.3e+44)
		tmp = t_0;
	elseif (im <= -15.6)
		tmp = (0.5 * re) * (exp(-im) - 1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -3.3e+44], t$95$0, If[LessEqual[im, -15.6], N[(N[(0.5 * re), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if im < -3.30000000000000013e44 or -15.5999999999999996 < im

   1. Initial program 65.1%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 96.0%

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

   6. Taylor expanded in im around inf

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

      1. pow2 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 95.9

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   8. Applied rewrites 95.9%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   if -3.30000000000000013e44 < im < -15.5999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 77.8%

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

Alternative 14: 58.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq 0.004:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) 0.004)
   (*
    (* (fma (* re re) -0.08333333333333333 0.5) re)
    (*
     (-
      (* (- (* (* -0.016666666666666666 im) im) 0.3333333333333333) (* im im))
      2.0)
     im))
   (*
    (* re 0.5)
    (*
     (-
      (*
       (-
        (* (* (* (* -0.0003968253968253968 im) im) im) im)
        0.3333333333333333)
       (* im im))
      2.0)
     im))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= 0.004) {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * re) * ((((((-0.016666666666666666 * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	} else {
		tmp = (re * 0.5) * ((((((((-0.0003968253968253968 * im) * im) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= 0.004)
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re) * Float64(Float64(Float64(Float64(Float64(Float64(-0.016666666666666666 * im) * im) - 0.3333333333333333) * Float64(im * im)) - 2.0) * im));
	else
		tmp = Float64(Float64(re * 0.5) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * im) * im) * im) * im) - 0.3333333333333333) * Float64(im * im)) - 2.0) * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], 0.004], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(N[(N[(N[(-0.016666666666666666 * im), $MachinePrecision] * im), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[(re * 0.5), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < 0.0040000000000000001

   1. Initial program 69.9%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 92.8%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 90.2%

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

   9. Taylor expanded in re around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 68.8

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

   11. Applied rewrites 68.8%

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

   if 0.0040000000000000001 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 55.4%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 93.2%

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

   6. Taylor expanded in im around inf

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

      1. pow2 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 93.1

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   8. Applied rewrites 93.1%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   9. Taylor expanded in re around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 25.8

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

   11. Applied rewrites 25.8%

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

Alternative 15: 59.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 55.7%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 83.8

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

   5. Applied rewrites 83.8%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 26.2

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

   8. Applied rewrites 26.2%

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

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

   1. Initial program 69.9%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 93.0%

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

   6. Taylor expanded in im around inf

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

      1. pow2 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 92.8

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   8. Applied rewrites 92.8%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   9. Taylor expanded in re around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 70.5

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

   11. Applied rewrites 70.5%

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

Alternative 16: 92.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ \mathbf{if}\;im \leq -8 \cdot 10^{+102}:\\ \;\;\;\;t\_0 \cdot \left(\left(-0.3333333333333333 \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \mathbf{elif}\;im \leq -15.6:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{-im} - 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re))))
   (if (<= im -8e+102)
     (* t_0 (* (- (* -0.3333333333333333 (* im im)) 2.0) im))
     (if (<= im -15.6)
       (* (* 0.5 re) (- (exp (- im)) 1.0))
       (*
        t_0
        (*
         (-
          (*
           (* (- (* -0.016666666666666666 (* im im)) 0.3333333333333333) im)
           im)
          2.0)
         im))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sin(re);
	double tmp;
	if (im <= -8e+102) {
		tmp = t_0 * (((-0.3333333333333333 * (im * im)) - 2.0) * im);
	} else if (im <= -15.6) {
		tmp = (0.5 * re) * (exp(-im) - 1.0);
	} else {
		tmp = t_0 * ((((((-0.016666666666666666 * (im * im)) - 0.3333333333333333) * im) * im) - 2.0) * im);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * sin(re)
    if (im <= (-8d+102)) then
        tmp = t_0 * ((((-0.3333333333333333d0) * (im * im)) - 2.0d0) * im)
    else if (im <= (-15.6d0)) then
        tmp = (0.5d0 * re) * (exp(-im) - 1.0d0)
    else
        tmp = t_0 * (((((((-0.016666666666666666d0) * (im * im)) - 0.3333333333333333d0) * im) * im) - 2.0d0) * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.sin(re);
	double tmp;
	if (im <= -8e+102) {
		tmp = t_0 * (((-0.3333333333333333 * (im * im)) - 2.0) * im);
	} else if (im <= -15.6) {
		tmp = (0.5 * re) * (Math.exp(-im) - 1.0);
	} else {
		tmp = t_0 * ((((((-0.016666666666666666 * (im * im)) - 0.3333333333333333) * im) * im) - 2.0) * im);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sin(re)
	tmp = 0
	if im <= -8e+102:
		tmp = t_0 * (((-0.3333333333333333 * (im * im)) - 2.0) * im)
	elif im <= -15.6:
		tmp = (0.5 * re) * (math.exp(-im) - 1.0)
	else:
		tmp = t_0 * ((((((-0.016666666666666666 * (im * im)) - 0.3333333333333333) * im) * im) - 2.0) * im)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sin(re))
	tmp = 0.0
	if (im <= -8e+102)
		tmp = Float64(t_0 * Float64(Float64(Float64(-0.3333333333333333 * Float64(im * im)) - 2.0) * im));
	elseif (im <= -15.6)
		tmp = Float64(Float64(0.5 * re) * Float64(exp(Float64(-im)) - 1.0));
	else
		tmp = Float64(t_0 * Float64(Float64(Float64(Float64(Float64(Float64(-0.016666666666666666 * Float64(im * im)) - 0.3333333333333333) * im) * im) - 2.0) * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * sin(re);
	tmp = 0.0;
	if (im <= -8e+102)
		tmp = t_0 * (((-0.3333333333333333 * (im * im)) - 2.0) * im);
	elseif (im <= -15.6)
		tmp = (0.5 * re) * (exp(-im) - 1.0);
	else
		tmp = t_0 * ((((((-0.016666666666666666 * (im * im)) - 0.3333333333333333) * im) * im) - 2.0) * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -8e+102], N[(t$95$0 * N[(N[(N[(-0.3333333333333333 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, -15.6], N[(N[(0.5 * re), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(N[(N[(N[(N[(-0.016666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < -7.99999999999999982e102

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -7.99999999999999982e102 < im < -15.5999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 74.9%

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

   if -15.5999999999999996 < im

   1. Initial program 54.8%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 93.0

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

   5. Applied rewrites 93.0%

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

Alternative 17: 95.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(\left(-0.3333333333333333 \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \mathbf{if}\;im \leq -8 \cdot 10^{+102}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im \leq -15.6:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{-im} - 1\right)\\ \mathbf{elif}\;im \leq 9.2:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im \leq 8.2 \cdot 10^{+102}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(1 - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (*
          (* 0.5 (sin re))
          (* (- (* -0.3333333333333333 (* im im)) 2.0) im))))
   (if (<= im -8e+102)
     t_0
     (if (<= im -15.6)
       (* (* 0.5 re) (- (exp (- im)) 1.0))
       (if (<= im 9.2)
         t_0
         (if (<= im 8.2e+102) (* (* 0.5 re) (- 1.0 (exp im))) t_0))))))

C

double code(double re, double im) {
	double t_0 = (0.5 * sin(re)) * (((-0.3333333333333333 * (im * im)) - 2.0) * im);
	double tmp;
	if (im <= -8e+102) {
		tmp = t_0;
	} else if (im <= -15.6) {
		tmp = (0.5 * re) * (exp(-im) - 1.0);
	} else if (im <= 9.2) {
		tmp = t_0;
	} else if (im <= 8.2e+102) {
		tmp = (0.5 * re) * (1.0 - exp(im));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (0.5d0 * sin(re)) * ((((-0.3333333333333333d0) * (im * im)) - 2.0d0) * im)
    if (im <= (-8d+102)) then
        tmp = t_0
    else if (im <= (-15.6d0)) then
        tmp = (0.5d0 * re) * (exp(-im) - 1.0d0)
    else if (im <= 9.2d0) then
        tmp = t_0
    else if (im <= 8.2d+102) then
        tmp = (0.5d0 * re) * (1.0d0 - exp(im))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (0.5 * Math.sin(re)) * (((-0.3333333333333333 * (im * im)) - 2.0) * im);
	double tmp;
	if (im <= -8e+102) {
		tmp = t_0;
	} else if (im <= -15.6) {
		tmp = (0.5 * re) * (Math.exp(-im) - 1.0);
	} else if (im <= 9.2) {
		tmp = t_0;
	} else if (im <= 8.2e+102) {
		tmp = (0.5 * re) * (1.0 - Math.exp(im));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (0.5 * math.sin(re)) * (((-0.3333333333333333 * (im * im)) - 2.0) * im)
	tmp = 0
	if im <= -8e+102:
		tmp = t_0
	elif im <= -15.6:
		tmp = (0.5 * re) * (math.exp(-im) - 1.0)
	elif im <= 9.2:
		tmp = t_0
	elif im <= 8.2e+102:
		tmp = (0.5 * re) * (1.0 - math.exp(im))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(Float64(Float64(-0.3333333333333333 * Float64(im * im)) - 2.0) * im))
	tmp = 0.0
	if (im <= -8e+102)
		tmp = t_0;
	elseif (im <= -15.6)
		tmp = Float64(Float64(0.5 * re) * Float64(exp(Float64(-im)) - 1.0));
	elseif (im <= 9.2)
		tmp = t_0;
	elseif (im <= 8.2e+102)
		tmp = Float64(Float64(0.5 * re) * Float64(1.0 - exp(im)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (0.5 * sin(re)) * (((-0.3333333333333333 * (im * im)) - 2.0) * im);
	tmp = 0.0;
	if (im <= -8e+102)
		tmp = t_0;
	elseif (im <= -15.6)
		tmp = (0.5 * re) * (exp(-im) - 1.0);
	elseif (im <= 9.2)
		tmp = t_0;
	elseif (im <= 8.2e+102)
		tmp = (0.5 * re) * (1.0 - exp(im));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(-0.3333333333333333 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -8e+102], t$95$0, If[LessEqual[im, -15.6], N[(N[(0.5 * re), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 9.2], t$95$0, If[LessEqual[im, 8.2e+102], N[(N[(0.5 * re), $MachinePrecision] * N[(1.0 - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -7.99999999999999982e102 or -15.5999999999999996 < im < 9.1999999999999993 or 8.1999999999999999e102 < im

   1. Initial program 59.7%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 99.5

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

   5. Applied rewrites 99.5%

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

   if -7.99999999999999982e102 < im < -15.5999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 74.9%

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

   if 9.1999999999999993 < im < 8.1999999999999999e102

   1. Initial program 99.7%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 75.3%

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

Alternative 18: 54.3% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 70.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 83.7

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

   5. Applied rewrites 83.7%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 64.3

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

   8. Applied rewrites 64.3%

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

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

   1. Initial program 55.3%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lift-sin.f64 51.1

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

   5. Applied rewrites 51.1%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lift-neg.f64 24.1

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

   8. Applied rewrites 24.1%

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

   10. Applied rewrites 24.1%

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

Alternative 19: 93.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\sin re \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\right) \cdot im\\ \mathbf{if}\;im \leq -8.5 \cdot 10^{+146}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im \leq -15.6:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{-im} - 1\right)\\ \mathbf{elif}\;im \leq 9.2:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im \leq 5.6 \cdot 10^{+104}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(1 - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* (sin re) (fma (* -0.16666666666666666 im) im -1.0)) im)))
   (if (<= im -8.5e+146)
     t_0
     (if (<= im -15.6)
       (* (* 0.5 re) (- (exp (- im)) 1.0))
       (if (<= im 9.2)
         t_0
         (if (<= im 5.6e+104) (* (* 0.5 re) (- 1.0 (exp im))) t_0))))))

C

double code(double re, double im) {
	double t_0 = (sin(re) * fma((-0.16666666666666666 * im), im, -1.0)) * im;
	double tmp;
	if (im <= -8.5e+146) {
		tmp = t_0;
	} else if (im <= -15.6) {
		tmp = (0.5 * re) * (exp(-im) - 1.0);
	} else if (im <= 9.2) {
		tmp = t_0;
	} else if (im <= 5.6e+104) {
		tmp = (0.5 * re) * (1.0 - exp(im));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(sin(re) * fma(Float64(-0.16666666666666666 * im), im, -1.0)) * im)
	tmp = 0.0
	if (im <= -8.5e+146)
		tmp = t_0;
	elseif (im <= -15.6)
		tmp = Float64(Float64(0.5 * re) * Float64(exp(Float64(-im)) - 1.0));
	elseif (im <= 9.2)
		tmp = t_0;
	elseif (im <= 5.6e+104)
		tmp = Float64(Float64(0.5 * re) * Float64(1.0 - exp(im)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Sin[re], $MachinePrecision] * N[(N[(-0.16666666666666666 * im), $MachinePrecision] * im + -1.0), $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[im, -8.5e+146], t$95$0, If[LessEqual[im, -15.6], N[(N[(0.5 * re), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 9.2], t$95$0, If[LessEqual[im, 5.6e+104], N[(N[(0.5 * re), $MachinePrecision] * N[(1.0 - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -8.5e146 or -15.5999999999999996 < im < 9.1999999999999993 or 5.6e104 < im

   1. Initial program 57.7%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 97.6

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

   5. Applied rewrites 97.6%

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

   if -8.5e146 < im < -15.5999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 75.0%

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

   if 9.1999999999999993 < im < 5.6e104

   1. Initial program 99.7%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 75.3%

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

Alternative 20: 53.3% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 55.7%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lift-sin.f64 50.4

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

   5. Applied rewrites 50.4%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lift-neg.f64 23.8

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

   8. Applied rewrites 23.8%

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

   9. Taylor expanded in re around 0

      \[\leadsto \mathsf{fma}\left(\frac{1}{6} \cdot im, re \cdot re, -im\right) \cdot re \]
   10. Step-by-step derivation

       1. lift-*.f64 23.1

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

   11. Applied rewrites 23.1%

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

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

   1. Initial program 69.9%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 83.7

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

   5. Applied rewrites 83.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 63.5%

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

Alternative 21: 86.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\\ \mathbf{if}\;im \leq -2.15 \cdot 10^{+176}:\\ \;\;\;\;t\_0 \cdot \left(\left(-0.3333333333333333 \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \mathbf{elif}\;im \leq -15.6:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{-im} - 1\right)\\ \mathbf{elif}\;im \leq 6.2:\\ \;\;\;\;\left(-\sin re\right) \cdot im\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+64}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(1 - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (fma (* re re) -0.08333333333333333 0.5) re)))
   (if (<= im -2.15e+176)
     (* t_0 (* (- (* -0.3333333333333333 (* im im)) 2.0) im))
     (if (<= im -15.6)
       (* (* 0.5 re) (- (exp (- im)) 1.0))
       (if (<= im 6.2)
         (* (- (sin re)) im)
         (if (<= im 2e+64)
           (* (* 0.5 re) (- 1.0 (exp im)))
           (*
            t_0
            (*
             (-
              (*
               (- (* (* -0.016666666666666666 im) im) 0.3333333333333333)
               (* im im))
              2.0)
             im))))))))

C

double code(double re, double im) {
	double t_0 = fma((re * re), -0.08333333333333333, 0.5) * re;
	double tmp;
	if (im <= -2.15e+176) {
		tmp = t_0 * (((-0.3333333333333333 * (im * im)) - 2.0) * im);
	} else if (im <= -15.6) {
		tmp = (0.5 * re) * (exp(-im) - 1.0);
	} else if (im <= 6.2) {
		tmp = -sin(re) * im;
	} else if (im <= 2e+64) {
		tmp = (0.5 * re) * (1.0 - exp(im));
	} else {
		tmp = t_0 * ((((((-0.016666666666666666 * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re)
	tmp = 0.0
	if (im <= -2.15e+176)
		tmp = Float64(t_0 * Float64(Float64(Float64(-0.3333333333333333 * Float64(im * im)) - 2.0) * im));
	elseif (im <= -15.6)
		tmp = Float64(Float64(0.5 * re) * Float64(exp(Float64(-im)) - 1.0));
	elseif (im <= 6.2)
		tmp = Float64(Float64(-sin(re)) * im);
	elseif (im <= 2e+64)
		tmp = Float64(Float64(0.5 * re) * Float64(1.0 - exp(im)));
	else
		tmp = Float64(t_0 * Float64(Float64(Float64(Float64(Float64(Float64(-0.016666666666666666 * im) * im) - 0.3333333333333333) * Float64(im * im)) - 2.0) * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision]}, If[LessEqual[im, -2.15e+176], N[(t$95$0 * N[(N[(N[(-0.3333333333333333 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, -15.6], N[(N[(0.5 * re), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 6.2], N[((-N[Sin[re], $MachinePrecision]) * im), $MachinePrecision], If[LessEqual[im, 2e+64], N[(N[(0.5 * re), $MachinePrecision] * N[(1.0 - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(N[(N[(N[(N[(-0.016666666666666666 * im), $MachinePrecision] * im), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if im < -2.15000000000000013e176

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 77.0

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

   8. Applied rewrites 77.0%

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

   if -2.15000000000000013e176 < im < -15.5999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 74.1%

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

   if -15.5999999999999996 < im < 6.20000000000000018

   1. Initial program 31.6%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lift-sin.f64 98.6

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

   5. Applied rewrites 98.6%

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

   if 6.20000000000000018 < im < 2.00000000000000004e64

   1. Initial program 99.5%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 75.4%

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

   if 2.00000000000000004e64 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in re around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 74.3

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

   11. Applied rewrites 74.3%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)} \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 22: 84.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\\ \mathbf{if}\;im \leq -2.15 \cdot 10^{+176}:\\ \;\;\;\;t\_0 \cdot \left(\left(-0.3333333333333333 \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \mathbf{elif}\;im \leq -0.0068:\\ \;\;\;\;\left(\mathsf{fma}\left(0.004166666666666667 \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \mathbf{elif}\;im \leq 6.2:\\ \;\;\;\;\left(-\sin re\right) \cdot im\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+64}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(1 - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (fma (* re re) -0.08333333333333333 0.5) re)))
   (if (<= im -2.15e+176)
     (* t_0 (* (- (* -0.3333333333333333 (* im im)) 2.0) im))
     (if (<= im -0.0068)
       (*
        (*
         (fma
          (- (* 0.004166666666666667 (* re re)) 0.08333333333333333)
          (* re re)
          0.5)
         re)
        (*
         (-
          (*
           (-
            (*
             (*
              (- (* -0.0003968253968253968 (* im im)) 0.016666666666666666)
              im)
             im)
            0.3333333333333333)
           (* im im))
          2.0)
         im))
       (if (<= im 6.2)
         (* (- (sin re)) im)
         (if (<= im 2e+64)
           (* (* 0.5 re) (- 1.0 (exp im)))
           (*
            t_0
            (*
             (-
              (*
               (- (* (* -0.016666666666666666 im) im) 0.3333333333333333)
               (* im im))
              2.0)
             im))))))))

C

double code(double re, double im) {
	double t_0 = fma((re * re), -0.08333333333333333, 0.5) * re;
	double tmp;
	if (im <= -2.15e+176) {
		tmp = t_0 * (((-0.3333333333333333 * (im * im)) - 2.0) * im);
	} else if (im <= -0.0068) {
		tmp = (fma(((0.004166666666666667 * (re * re)) - 0.08333333333333333), (re * re), 0.5) * re) * ((((((((-0.0003968253968253968 * (im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	} else if (im <= 6.2) {
		tmp = -sin(re) * im;
	} else if (im <= 2e+64) {
		tmp = (0.5 * re) * (1.0 - exp(im));
	} else {
		tmp = t_0 * ((((((-0.016666666666666666 * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re)
	tmp = 0.0
	if (im <= -2.15e+176)
		tmp = Float64(t_0 * Float64(Float64(Float64(-0.3333333333333333 * Float64(im * im)) - 2.0) * im));
	elseif (im <= -0.0068)
		tmp = Float64(Float64(fma(Float64(Float64(0.004166666666666667 * Float64(re * re)) - 0.08333333333333333), Float64(re * re), 0.5) * re) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * Float64(im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * Float64(im * im)) - 2.0) * im));
	elseif (im <= 6.2)
		tmp = Float64(Float64(-sin(re)) * im);
	elseif (im <= 2e+64)
		tmp = Float64(Float64(0.5 * re) * Float64(1.0 - exp(im)));
	else
		tmp = Float64(t_0 * Float64(Float64(Float64(Float64(Float64(Float64(-0.016666666666666666 * im) * im) - 0.3333333333333333) * Float64(im * im)) - 2.0) * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision]}, If[LessEqual[im, -2.15e+176], N[(t$95$0 * N[(N[(N[(-0.3333333333333333 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, -0.0068], N[(N[(N[(N[(N[(0.004166666666666667 * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.08333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.016666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 6.2], N[((-N[Sin[re], $MachinePrecision]) * im), $MachinePrecision], If[LessEqual[im, 2e+64], N[(N[(0.5 * re), $MachinePrecision] * N[(1.0 - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(N[(N[(N[(N[(-0.016666666666666666 * im), $MachinePrecision] * im), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if im < -2.15000000000000013e176

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 77.0

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

   8. Applied rewrites 77.0%

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

   if -2.15000000000000013e176 < im < -0.00679999999999999962

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 77.0%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 61.0

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

   8. Applied rewrites 61.0%

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

   if -0.00679999999999999962 < im < 6.20000000000000018

   1. Initial program 31.3%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lift-sin.f64 98.9

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

   5. Applied rewrites 98.9%

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

   if 6.20000000000000018 < im < 2.00000000000000004e64

   1. Initial program 99.5%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 75.4%

      \[\leadsto \left(0.5 \cdot \color{blue}{re}\right) \cdot \left(1 - e^{im}\right) \]

   if 2.00000000000000004e64 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

   5. Applied rewrites 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]

   6. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot \color{blue}{re}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot \color{blue}{re}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       3. +-commutative N/A

          \[\leadsto \left(\left(\frac{-1}{12} \cdot {re}^{2} + \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       4. *-commutative N/A

          \[\leadsto \left(\left({re}^{2} \cdot \frac{-1}{12} + \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       5. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left({re}^{2}, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       6. unpow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       7. lower-*.f64 74.3

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   11. Applied rewrites 74.3%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)} \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 23: 35.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq 0.004:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666 \cdot im, re \cdot re, -im\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(-re\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) 0.004)
   (* (fma (* 0.16666666666666666 im) (* re re) (- im)) re)
   (* (- re) im)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= 0.004) {
		tmp = fma((0.16666666666666666 * im), (re * re), -im) * re;
	} else {
		tmp = -re * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= 0.004)
		tmp = Float64(fma(Float64(0.16666666666666666 * im), Float64(re * re), Float64(-im)) * re);
	else
		tmp = Float64(Float64(-re) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], 0.004], N[(N[(N[(0.16666666666666666 * im), $MachinePrecision] * N[(re * re), $MachinePrecision] + (-im)), $MachinePrecision] * re), $MachinePrecision], N[((-re) * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < 0.0040000000000000001

   1. Initial program 69.9%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto -1 \cdot \left(\sin re \cdot \color{blue}{im}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \sin re\right) \cdot \color{blue}{im} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot \sin re\right) \cdot \color{blue}{im} \]

      4. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\sin re\right)\right) \cdot im \]

      5. lower-neg.f64 N/A

         \[\leadsto \left(-\sin re\right) \cdot im \]

      6. lift-sin.f64 50.7

         \[\leadsto \left(-\sin re\right) \cdot im \]

   5. Applied rewrites 50.7%

      \[\leadsto \color{blue}{\left(-\sin re\right) \cdot im} \]

   6. Taylor expanded in re around 0

      \[\leadsto re \cdot \color{blue}{\left(-1 \cdot im + {re}^{2} \cdot \left(\frac{-1}{120} \cdot \left(im \cdot {re}^{2}\right) + \frac{1}{6} \cdot im\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-1 \cdot im + {re}^{2} \cdot \left(\frac{-1}{120} \cdot \left(im \cdot {re}^{2}\right) + \frac{1}{6} \cdot im\right)\right) \cdot re \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot im + {re}^{2} \cdot \left(\frac{-1}{120} \cdot \left(im \cdot {re}^{2}\right) + \frac{1}{6} \cdot im\right)\right) \cdot re \]

      3. +-commutative N/A

         \[\leadsto \left({re}^{2} \cdot \left(\frac{-1}{120} \cdot \left(im \cdot {re}^{2}\right) + \frac{1}{6} \cdot im\right) + -1 \cdot im\right) \cdot re \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\frac{-1}{120} \cdot \left(im \cdot {re}^{2}\right) + \frac{1}{6} \cdot im\right) \cdot {re}^{2} + -1 \cdot im\right) \cdot re \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{120} \cdot \left(im \cdot {re}^{2}\right) + \frac{1}{6} \cdot im, {re}^{2}, -1 \cdot im\right) \cdot re \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{120}, im \cdot {re}^{2}, \frac{1}{6} \cdot im\right), {re}^{2}, -1 \cdot im\right) \cdot re \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{120}, {re}^{2} \cdot im, \frac{1}{6} \cdot im\right), {re}^{2}, -1 \cdot im\right) \cdot re \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{120}, {re}^{2} \cdot im, \frac{1}{6} \cdot im\right), {re}^{2}, -1 \cdot im\right) \cdot re \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{120}, \left(re \cdot re\right) \cdot im, \frac{1}{6} \cdot im\right), {re}^{2}, -1 \cdot im\right) \cdot re \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{120}, \left(re \cdot re\right) \cdot im, \frac{1}{6} \cdot im\right), {re}^{2}, -1 \cdot im\right) \cdot re \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{120}, \left(re \cdot re\right) \cdot im, \frac{1}{6} \cdot im\right), {re}^{2}, -1 \cdot im\right) \cdot re \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{120}, \left(re \cdot re\right) \cdot im, \frac{1}{6} \cdot im\right), re \cdot re, -1 \cdot im\right) \cdot re \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{120}, \left(re \cdot re\right) \cdot im, \frac{1}{6} \cdot im\right), re \cdot re, -1 \cdot im\right) \cdot re \]

      14. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{120}, \left(re \cdot re\right) \cdot im, \frac{1}{6} \cdot im\right), re \cdot re, \mathsf{neg}\left(im\right)\right) \cdot re \]

      15. lift-neg.f64 41.6

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333, \left(re \cdot re\right) \cdot im, 0.16666666666666666 \cdot im\right), re \cdot re, -im\right) \cdot re \]

   8. Applied rewrites 41.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333, \left(re \cdot re\right) \cdot im, 0.16666666666666666 \cdot im\right), re \cdot re, -im\right) \cdot \color{blue}{re} \]

   9. Taylor expanded in re around 0

      \[\leadsto \mathsf{fma}\left(\frac{1}{6} \cdot im, re \cdot re, -im\right) \cdot re \]
   10. Step-by-step derivation

       1. lift-*.f64 41.4

          \[\leadsto \mathsf{fma}\left(0.16666666666666666 \cdot im, re \cdot re, -im\right) \cdot re \]

   11. Applied rewrites 41.4%

       \[\leadsto \mathsf{fma}\left(0.16666666666666666 \cdot im, re \cdot re, -im\right) \cdot re \]

   if 0.0040000000000000001 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 55.4%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto -1 \cdot \left(\sin re \cdot \color{blue}{im}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \sin re\right) \cdot \color{blue}{im} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot \sin re\right) \cdot \color{blue}{im} \]

      4. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\sin re\right)\right) \cdot im \]

      5. lower-neg.f64 N/A

         \[\leadsto \left(-\sin re\right) \cdot im \]

      6. lift-sin.f64 51.0

         \[\leadsto \left(-\sin re\right) \cdot im \]

   5. Applied rewrites 51.0%

      \[\leadsto \color{blue}{\left(-\sin re\right) \cdot im} \]

   6. Taylor expanded in re around 0

      \[\leadsto \left(-re\right) \cdot im \]
   7. Step-by-step derivation

   8. Applied rewrites 15.3%

      \[\leadsto \left(-re\right) \cdot im \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 32.9% accurate, 39.5× speedup

Math

\[\begin{array}{l} \\ \left(-re\right) \cdot im \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (- re) im))

C

double code(double re, double im) {
	return -re * im;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = -re * im
end function

Java

public static double code(double re, double im) {
	return -re * im;
}

Python

def code(re, im):
	return -re * im

Julia

function code(re, im)
	return Float64(Float64(-re) * im)
end

MATLAB

function tmp = code(re, im)
	tmp = -re * im;
end

Wolfram

code[re_, im_] := N[((-re) * im), $MachinePrecision]

Derivation

1. Initial program 66.3%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in im around 0

   \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto -1 \cdot \left(\sin re \cdot \color{blue}{im}\right) \]

   2. associate-*r* N/A

      \[\leadsto \left(-1 \cdot \sin re\right) \cdot \color{blue}{im} \]

   3. lower-*.f64 N/A

      \[\leadsto \left(-1 \cdot \sin re\right) \cdot \color{blue}{im} \]

   4. mul-1-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(\sin re\right)\right) \cdot im \]

   5. lower-neg.f64 N/A

      \[\leadsto \left(-\sin re\right) \cdot im \]

   6. lift-sin.f64 50.7

      \[\leadsto \left(-\sin re\right) \cdot im \]

5. Applied rewrites 50.7%

   \[\leadsto \color{blue}{\left(-\sin re\right) \cdot im} \]

6. Taylor expanded in re around 0

   \[\leadsto \left(-re\right) \cdot im \]
7. Step-by-step derivation

8. Applied rewrites 32.9%

   \[\leadsto \left(-re\right) \cdot im \]
9. Add Preprocessing

Alternative 25: 14.7% accurate, 52.7× speedup

Math

\[\begin{array}{l} \\ 0 \cdot im \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* 0.0 im))

C

double code(double re, double im) {
	return 0.0 * im;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.0d0 * im
end function

Java

public static double code(double re, double im) {
	return 0.0 * im;
}

Python

def code(re, im):
	return 0.0 * im

Julia

function code(re, im)
	return Float64(0.0 * im)
end

MATLAB

function tmp = code(re, im)
	tmp = 0.0 * im;
end

Wolfram

code[re_, im_] := N[(0.0 * im), $MachinePrecision]

Derivation

1. Initial program 66.3%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in im around 0

   \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto -1 \cdot \left(\sin re \cdot \color{blue}{im}\right) \]

   2. associate-*r* N/A

      \[\leadsto \left(-1 \cdot \sin re\right) \cdot \color{blue}{im} \]

   3. lower-*.f64 N/A

      \[\leadsto \left(-1 \cdot \sin re\right) \cdot \color{blue}{im} \]

   4. mul-1-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(\sin re\right)\right) \cdot im \]

   5. lower-neg.f64 N/A

      \[\leadsto \left(-\sin re\right) \cdot im \]

   6. lift-sin.f64 50.7

      \[\leadsto \left(-\sin re\right) \cdot im \]

5. Applied rewrites 50.7%

   \[\leadsto \color{blue}{\left(-\sin re\right) \cdot im} \]
6. Step-by-step derivation

   1. lift-neg.f64 N/A

      \[\leadsto \left(\mathsf{neg}\left(\sin re\right)\right) \cdot im \]

   2. lift-sin.f64 N/A

      \[\leadsto \left(\mathsf{neg}\left(\sin re\right)\right) \cdot im \]

   3. sin-+PI-rev N/A

      \[\leadsto \sin \left(re + \mathsf{PI}\left(\right)\right) \cdot im \]

   4. sin-sum N/A

      \[\leadsto \left(\sin re \cdot \cos \mathsf{PI}\left(\right) + \cos re \cdot \sin \mathsf{PI}\left(\right)\right) \cdot im \]

   5. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sin re, \cos \mathsf{PI}\left(\right), \cos re \cdot \sin \mathsf{PI}\left(\right)\right) \cdot im \]

   6. lift-sin.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sin re, \cos \mathsf{PI}\left(\right), \cos re \cdot \sin \mathsf{PI}\left(\right)\right) \cdot im \]

   7. lower-cos.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sin re, \cos \mathsf{PI}\left(\right), \cos re \cdot \sin \mathsf{PI}\left(\right)\right) \cdot im \]

   8. lower-PI.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sin re, \cos \pi, \cos re \cdot \sin \mathsf{PI}\left(\right)\right) \cdot im \]

   9. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sin re, \cos \pi, \cos re \cdot \sin \mathsf{PI}\left(\right)\right) \cdot im \]

   10. lower-cos.f64 N/A

       \[\leadsto \mathsf{fma}\left(\sin re, \cos \pi, \cos re \cdot \sin \mathsf{PI}\left(\right)\right) \cdot im \]

   11. lower-sin.f64 N/A

       \[\leadsto \mathsf{fma}\left(\sin re, \cos \pi, \cos re \cdot \sin \mathsf{PI}\left(\right)\right) \cdot im \]

   12. lower-PI.f64 28.2

       \[\leadsto \mathsf{fma}\left(\sin re, \cos \pi, \cos re \cdot \sin \pi\right) \cdot im \]

7. Applied rewrites 28.2%

   \[\leadsto \mathsf{fma}\left(\sin re, \cos \pi, \cos re \cdot \sin \pi\right) \cdot im \]

8. Taylor expanded in re around 0

   \[\leadsto \sin \mathsf{PI}\left(\right) \cdot im \]
9. Step-by-step derivation

   1. sin-PI 14.7

      \[\leadsto 0 \cdot im \]

10. Applied rewrites 14.7%

    \[\leadsto 0 \cdot im \]
11. Add Preprocessing

Developer Target 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|im\right| < 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(0.16666666666666666 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (< (fabs im) 1.0)
   (-
    (*
     (sin re)
     (+
      (+ im (* (* (* 0.16666666666666666 im) im) im))
      (* (* (* (* (* 0.008333333333333333 im) im) im) im) im))))
   (* (* 0.5 (sin re)) (- (exp (- im)) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if (fabs(im) < 1.0) {
		tmp = -(sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * sin(re)) * (exp(-im) - exp(im));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (abs(im) < 1.0d0) then
        tmp = -(sin(re) * ((im + (((0.16666666666666666d0 * im) * im) * im)) + (((((0.008333333333333333d0 * im) * im) * im) * im) * im)))
    else
        tmp = (0.5d0 * sin(re)) * (exp(-im) - exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.abs(im) < 1.0) {
		tmp = -(Math.sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * Math.sin(re)) * (Math.exp(-im) - Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.fabs(im) < 1.0:
		tmp = -(math.sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)))
	else:
		tmp = (0.5 * math.sin(re)) * (math.exp(-im) - math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (abs(im) < 1.0)
		tmp = Float64(-Float64(sin(re) * Float64(Float64(im + Float64(Float64(Float64(0.16666666666666666 * im) * im) * im)) + Float64(Float64(Float64(Float64(Float64(0.008333333333333333 * im) * im) * im) * im) * im))));
	else
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (abs(im) < 1.0)
		tmp = -(sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	else
		tmp = (0.5 * sin(re)) * (exp(-im) - exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Less[N[Abs[im], $MachinePrecision], 1.0], (-N[(N[Sin[re], $MachinePrecision] * N[(N[(im + N[(N[(N[(0.16666666666666666 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[(0.008333333333333333 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2025089 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (fabs im) 1) (- (* (sin re) (+ im (* 1/6 im im im) (* 1/120 im im im im im)))) (* (* 1/2 (sin re)) (- (exp (- im)) (exp im)))))

  (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))