math.cos on complex, imaginary part

Percentage Accurate: 65.6% → 99.7%

Time: 7.0s

Alternatives: 19

Speedup: 1.4×

• 49.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 65.6% 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: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ t_1 := e^{-im}\\ \mathbf{if}\;im \leq -3:\\ \;\;\;\;t\_0 \cdot \left(t\_1 - 1\right)\\ \mathbf{elif}\;im \leq 0.014:\\ \;\;\;\;\mathsf{fma}\left(\left(\sin re \cdot \mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right)\right) \cdot im, im, -\sin re\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(t\_1 - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re))) (t_1 (exp (- im))))
   (if (<= im -3.0)
     (* t_0 (- t_1 1.0))
     (if (<= im 0.014)
       (*
        (fma
         (*
          (*
           (sin re)
           (fma -0.008333333333333333 (* im im) -0.16666666666666666))
          im)
         im
         (- (sin re)))
        im)
       (* t_0 (- t_1 (exp im)))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sin(re);
	double t_1 = exp(-im);
	double tmp;
	if (im <= -3.0) {
		tmp = t_0 * (t_1 - 1.0);
	} else if (im <= 0.014) {
		tmp = fma(((sin(re) * fma(-0.008333333333333333, (im * im), -0.16666666666666666)) * im), im, -sin(re)) * im;
	} else {
		tmp = t_0 * (t_1 - exp(im));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * sin(re))
	t_1 = exp(Float64(-im))
	tmp = 0.0
	if (im <= -3.0)
		tmp = Float64(t_0 * Float64(t_1 - 1.0));
	elseif (im <= 0.014)
		tmp = Float64(fma(Float64(Float64(sin(re) * fma(-0.008333333333333333, Float64(im * im), -0.16666666666666666)) * im), im, Float64(-sin(re))) * im);
	else
		tmp = Float64(t_0 * Float64(t_1 - exp(im)));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Exp[(-im)], $MachinePrecision]}, If[LessEqual[im, -3.0], N[(t$95$0 * N[(t$95$1 - 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 0.014], N[(N[(N[(N[(N[Sin[re], $MachinePrecision] * N[(-0.008333333333333333 * N[(im * im), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision] * im + (-N[Sin[re], $MachinePrecision])), $MachinePrecision] * im), $MachinePrecision], N[(t$95$0 * N[(t$95$1 - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -3

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 99.7%

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

   if -3 < im < 0.0140000000000000003

   1. Initial program 30.9%

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

   2. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 99.6%

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

   if 0.0140000000000000003 < im

   1. Initial program 99.9%

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

Alternative 2: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ t_1 := e^{-im}\\ \mathbf{if}\;im \leq -3:\\ \;\;\;\;t\_0 \cdot \left(t\_1 - 1\right)\\ \mathbf{elif}\;im \leq 0.014:\\ \;\;\;\;\left(\sin re \cdot 0.5\right) \cdot \left(\left(\left(\left(\left(im \cdot im\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(t\_1 - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re))) (t_1 (exp (- im))))
   (if (<= im -3.0)
     (* t_0 (- t_1 1.0))
     (if (<= im 0.014)
       (*
        (* (sin re) 0.5)
        (*
         (-
          (*
           (* (- (* (* im im) -0.016666666666666666) 0.3333333333333333) im)
           im)
          2.0)
         im))
       (* t_0 (- t_1 (exp im)))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sin(re);
	double t_1 = exp(-im);
	double tmp;
	if (im <= -3.0) {
		tmp = t_0 * (t_1 - 1.0);
	} else if (im <= 0.014) {
		tmp = (sin(re) * 0.5) * (((((((im * im) * -0.016666666666666666) - 0.3333333333333333) * im) * im) - 2.0) * im);
	} else {
		tmp = t_0 * (t_1 - 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) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * sin(re)
    t_1 = exp(-im)
    if (im <= (-3.0d0)) then
        tmp = t_0 * (t_1 - 1.0d0)
    else if (im <= 0.014d0) then
        tmp = (sin(re) * 0.5d0) * (((((((im * im) * (-0.016666666666666666d0)) - 0.3333333333333333d0) * im) * im) - 2.0d0) * im)
    else
        tmp = t_0 * (t_1 - 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 t_1 = Math.exp(-im);
	double tmp;
	if (im <= -3.0) {
		tmp = t_0 * (t_1 - 1.0);
	} else if (im <= 0.014) {
		tmp = (Math.sin(re) * 0.5) * (((((((im * im) * -0.016666666666666666) - 0.3333333333333333) * im) * im) - 2.0) * im);
	} else {
		tmp = t_0 * (t_1 - Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sin(re)
	t_1 = math.exp(-im)
	tmp = 0
	if im <= -3.0:
		tmp = t_0 * (t_1 - 1.0)
	elif im <= 0.014:
		tmp = (math.sin(re) * 0.5) * (((((((im * im) * -0.016666666666666666) - 0.3333333333333333) * im) * im) - 2.0) * im)
	else:
		tmp = t_0 * (t_1 - math.exp(im))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sin(re))
	t_1 = exp(Float64(-im))
	tmp = 0.0
	if (im <= -3.0)
		tmp = Float64(t_0 * Float64(t_1 - 1.0));
	elseif (im <= 0.014)
		tmp = Float64(Float64(sin(re) * 0.5) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(im * im) * -0.016666666666666666) - 0.3333333333333333) * im) * im) - 2.0) * im));
	else
		tmp = Float64(t_0 * Float64(t_1 - exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * sin(re);
	t_1 = exp(-im);
	tmp = 0.0;
	if (im <= -3.0)
		tmp = t_0 * (t_1 - 1.0);
	elseif (im <= 0.014)
		tmp = (sin(re) * 0.5) * (((((((im * im) * -0.016666666666666666) - 0.3333333333333333) * im) * im) - 2.0) * im);
	else
		tmp = t_0 * (t_1 - exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Exp[(-im)], $MachinePrecision]}, If[LessEqual[im, -3.0], N[(t$95$0 * N[(t$95$1 - 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 0.014], N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(im * im), $MachinePrecision] * -0.016666666666666666), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(t$95$1 - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -3

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 99.7%

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

   if -3 < im < 0.0140000000000000003

   1. Initial program 30.9%

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

   2. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sin.f64 99.6

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 99.6

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

   6. Applied rewrites 99.6%

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

   if 0.0140000000000000003 < im

   1. Initial program 99.9%

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

Alternative 3: 98.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\sin re \cdot 0.5\right) \cdot \left(\left(\left(\left(\left(im \cdot im\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right)\\ \mathbf{if}\;im \leq -3:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - 1\right)\\ \mathbf{elif}\;im \leq 52000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im \leq 1.02 \cdot 10^{+62}:\\ \;\;\;\;\left(\left(1 - e^{im}\right) \cdot 0.5\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (*
          (* (sin re) 0.5)
          (*
           (-
            (*
             (* (- (* (* im im) -0.016666666666666666) 0.3333333333333333) im)
             im)
            2.0)
           im))))
   (if (<= im -3.0)
     (* (* 0.5 (sin re)) (- (exp (- im)) 1.0))
     (if (<= im 52000.0)
       t_0
       (if (<= im 1.02e+62) (* (* (- 1.0 (exp im)) 0.5) re) t_0)))))

C

double code(double re, double im) {
	double t_0 = (sin(re) * 0.5) * (((((((im * im) * -0.016666666666666666) - 0.3333333333333333) * im) * im) - 2.0) * im);
	double tmp;
	if (im <= -3.0) {
		tmp = (0.5 * sin(re)) * (exp(-im) - 1.0);
	} else if (im <= 52000.0) {
		tmp = t_0;
	} else if (im <= 1.02e+62) {
		tmp = ((1.0 - exp(im)) * 0.5) * re;
	} 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 = (sin(re) * 0.5d0) * (((((((im * im) * (-0.016666666666666666d0)) - 0.3333333333333333d0) * im) * im) - 2.0d0) * im)
    if (im <= (-3.0d0)) then
        tmp = (0.5d0 * sin(re)) * (exp(-im) - 1.0d0)
    else if (im <= 52000.0d0) then
        tmp = t_0
    else if (im <= 1.02d+62) then
        tmp = ((1.0d0 - exp(im)) * 0.5d0) * re
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (Math.sin(re) * 0.5) * (((((((im * im) * -0.016666666666666666) - 0.3333333333333333) * im) * im) - 2.0) * im);
	double tmp;
	if (im <= -3.0) {
		tmp = (0.5 * Math.sin(re)) * (Math.exp(-im) - 1.0);
	} else if (im <= 52000.0) {
		tmp = t_0;
	} else if (im <= 1.02e+62) {
		tmp = ((1.0 - Math.exp(im)) * 0.5) * re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (math.sin(re) * 0.5) * (((((((im * im) * -0.016666666666666666) - 0.3333333333333333) * im) * im) - 2.0) * im)
	tmp = 0
	if im <= -3.0:
		tmp = (0.5 * math.sin(re)) * (math.exp(-im) - 1.0)
	elif im <= 52000.0:
		tmp = t_0
	elif im <= 1.02e+62:
		tmp = ((1.0 - math.exp(im)) * 0.5) * re
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(sin(re) * 0.5) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(im * im) * -0.016666666666666666) - 0.3333333333333333) * im) * im) - 2.0) * im))
	tmp = 0.0
	if (im <= -3.0)
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - 1.0));
	elseif (im <= 52000.0)
		tmp = t_0;
	elseif (im <= 1.02e+62)
		tmp = Float64(Float64(Float64(1.0 - exp(im)) * 0.5) * re);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (sin(re) * 0.5) * (((((((im * im) * -0.016666666666666666) - 0.3333333333333333) * im) * im) - 2.0) * im);
	tmp = 0.0;
	if (im <= -3.0)
		tmp = (0.5 * sin(re)) * (exp(-im) - 1.0);
	elseif (im <= 52000.0)
		tmp = t_0;
	elseif (im <= 1.02e+62)
		tmp = ((1.0 - exp(im)) * 0.5) * re;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(im * im), $MachinePrecision] * -0.016666666666666666), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -3.0], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 52000.0], t$95$0, If[LessEqual[im, 1.02e+62], N[(N[(N[(1.0 - N[Exp[im], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * re), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -3

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 99.7%

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

   if -3 < im < 52000 or 1.02000000000000002e62 < im

   1. Initial program 51.1%

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

   2. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 99.2

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

   4. Applied rewrites 99.2%

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

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sin.f64 99.2

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 99.2

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

   6. Applied rewrites 99.2%

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

   if 52000 < im < 1.02000000000000002e62

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift--.f64 76.3

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

   4. Applied rewrites 76.3%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 76.3%

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

Alternative 4: 96.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\sin re \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\right) \cdot im\\ \mathbf{if}\;im \leq -2.1:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - 1\right)\\ \mathbf{elif}\;im \leq 52000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im \leq 3 \cdot 10^{+107}:\\ \;\;\;\;\left(\left(1 - e^{im}\right) \cdot 0.5\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* (sin re) (fma (* -0.16666666666666666 im) im -1.0)) im)))
   (if (<= im -2.1)
     (* (* 0.5 (sin re)) (- (exp (- im)) 1.0))
     (if (<= im 52000.0)
       t_0
       (if (<= im 3e+107) (* (* (- 1.0 (exp im)) 0.5) re) 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 <= -2.1) {
		tmp = (0.5 * sin(re)) * (exp(-im) - 1.0);
	} else if (im <= 52000.0) {
		tmp = t_0;
	} else if (im <= 3e+107) {
		tmp = ((1.0 - exp(im)) * 0.5) * re;
	} 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 <= -2.1)
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - 1.0));
	elseif (im <= 52000.0)
		tmp = t_0;
	elseif (im <= 3e+107)
		tmp = Float64(Float64(Float64(1.0 - exp(im)) * 0.5) * re);
	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, -2.1], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 52000.0], t$95$0, If[LessEqual[im, 3e+107], N[(N[(N[(1.0 - N[Exp[im], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * re), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -2.10000000000000009

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 99.7%

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

   if -2.10000000000000009 < im < 52000 or 3.00000000000000023e107 < im

   1. Initial program 48.5%

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

   2. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right)\right)} \]
   3. 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 97.1

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

   4. Applied rewrites 97.1%

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

   if 52000 < im < 3.00000000000000023e107

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift--.f64 75.5

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

   4. Applied rewrites 75.5%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 75.5%

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

Alternative 5: 92.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\sin re \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\right) \cdot im\\ \mathbf{if}\;im \leq -3.6 \cdot 10^{+96}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im \leq -6.6:\\ \;\;\;\;\left(\left(e^{-im} - 1\right) \cdot 0.5\right) \cdot re\\ \mathbf{elif}\;im \leq 52000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im \leq 3 \cdot 10^{+107}:\\ \;\;\;\;\left(\left(1 - e^{im}\right) \cdot 0.5\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* (sin re) (fma (* -0.16666666666666666 im) im -1.0)) im)))
   (if (<= im -3.6e+96)
     t_0
     (if (<= im -6.6)
       (* (* (- (exp (- im)) 1.0) 0.5) re)
       (if (<= im 52000.0)
         t_0
         (if (<= im 3e+107) (* (* (- 1.0 (exp im)) 0.5) re) 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 <= -3.6e+96) {
		tmp = t_0;
	} else if (im <= -6.6) {
		tmp = ((exp(-im) - 1.0) * 0.5) * re;
	} else if (im <= 52000.0) {
		tmp = t_0;
	} else if (im <= 3e+107) {
		tmp = ((1.0 - exp(im)) * 0.5) * re;
	} 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 <= -3.6e+96)
		tmp = t_0;
	elseif (im <= -6.6)
		tmp = Float64(Float64(Float64(exp(Float64(-im)) - 1.0) * 0.5) * re);
	elseif (im <= 52000.0)
		tmp = t_0;
	elseif (im <= 3e+107)
		tmp = Float64(Float64(Float64(1.0 - exp(im)) * 0.5) * re);
	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, -3.6e+96], t$95$0, If[LessEqual[im, -6.6], N[(N[(N[(N[Exp[(-im)], $MachinePrecision] - 1.0), $MachinePrecision] * 0.5), $MachinePrecision] * re), $MachinePrecision], If[LessEqual[im, 52000.0], t$95$0, If[LessEqual[im, 3e+107], N[(N[(N[(1.0 - N[Exp[im], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * re), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -3.60000000000000013e96 or -6.5999999999999996 < im < 52000 or 3.00000000000000023e107 < im

   1. Initial program 59.1%

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

   2. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right)\right)} \]
   3. 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 95.4

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

   4. Applied rewrites 95.4%

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

   if -3.60000000000000013e96 < im < -6.5999999999999996

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift--.f64 72.2

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

   4. Applied rewrites 72.2%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 71.9%

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

   if 52000 < im < 3.00000000000000023e107

   1. Initial program 59.1%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift--.f64 47.6

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

   4. Applied rewrites 47.6%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 32.4%

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

Alternative 6: 85.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -5:\\ \;\;\;\;\left(\left(e^{-im} - 1\right) \cdot 0.5\right) \cdot re\\ \mathbf{elif}\;im \leq 5.5:\\ \;\;\;\;\left(-\sin re\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 - e^{im}\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -5.0)
   (* (* (- (exp (- im)) 1.0) 0.5) re)
   (if (<= im 5.5)
     (* (- (sin re)) im)
     (* (* (- 1.0 (exp im)) (fma (* re re) -0.08333333333333333 0.5)) re))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -5.0) {
		tmp = ((exp(-im) - 1.0) * 0.5) * re;
	} else if (im <= 5.5) {
		tmp = -sin(re) * im;
	} else {
		tmp = ((1.0 - exp(im)) * fma((re * re), -0.08333333333333333, 0.5)) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -5.0)
		tmp = Float64(Float64(Float64(exp(Float64(-im)) - 1.0) * 0.5) * re);
	elseif (im <= 5.5)
		tmp = Float64(Float64(-sin(re)) * im);
	else
		tmp = Float64(Float64(Float64(1.0 - exp(im)) * fma(Float64(re * re), -0.08333333333333333, 0.5)) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, -5.0], N[(N[(N[(N[Exp[(-im)], $MachinePrecision] - 1.0), $MachinePrecision] * 0.5), $MachinePrecision] * re), $MachinePrecision], If[LessEqual[im, 5.5], N[((-N[Sin[re], $MachinePrecision]) * im), $MachinePrecision], N[(N[(N[(1.0 - N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < -5

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift--.f64 72.5

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

   4. Applied rewrites 72.5%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 72.4%

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

   if -5 < im < 5.5

   1. Initial program 31.2%

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

   2. Taylor expanded in im around 0

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

      1. *-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 98.7

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

   4. Applied rewrites 98.7%

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

   if 5.5 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 73.7%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 73.7%

      \[\leadsto \left(\left(1 - e^{im}\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 54.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\\ \mathbf{if}\;0.5 \cdot \sin re \leq 4 \cdot 10^{-157}:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.16666666666666666, 1\right) \cdot re\right) \cdot \left(t\_0 \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(re \cdot re\right) - 0.16666666666666666, re \cdot re, 1\right) \cdot re\right) \cdot t\_0\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (fma (* -0.16666666666666666 im) im -1.0)))
   (if (<= (* 0.5 (sin re)) 4e-157)
     (* (* (fma (* re re) -0.16666666666666666 1.0) re) (* t_0 im))
     (*
      (*
       (*
        (fma
         (- (* 0.008333333333333333 (* re re)) 0.16666666666666666)
         (* re re)
         1.0)
        re)
       t_0)
      im))))

C

double code(double re, double im) {
	double t_0 = fma((-0.16666666666666666 * im), im, -1.0);
	double tmp;
	if ((0.5 * sin(re)) <= 4e-157) {
		tmp = (fma((re * re), -0.16666666666666666, 1.0) * re) * (t_0 * im);
	} else {
		tmp = ((fma(((0.008333333333333333 * (re * re)) - 0.16666666666666666), (re * re), 1.0) * re) * t_0) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(Float64(-0.16666666666666666 * im), im, -1.0)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= 4e-157)
		tmp = Float64(Float64(fma(Float64(re * re), -0.16666666666666666, 1.0) * re) * Float64(t_0 * im));
	else
		tmp = Float64(Float64(Float64(fma(Float64(Float64(0.008333333333333333 * Float64(re * re)) - 0.16666666666666666), Float64(re * re), 1.0) * re) * t_0) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(-0.16666666666666666 * im), $MachinePrecision] * im + -1.0), $MachinePrecision]}, If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], 4e-157], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * re), $MachinePrecision] * N[(t$95$0 * im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.008333333333333333 * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * re), $MachinePrecision] * t$95$0), $MachinePrecision] * im), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < 3.99999999999999977e-157

   1. Initial program 69.6%

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

   2. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right)\right)} \]
   3. 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 80.0

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

   4. Applied rewrites 80.0%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 55.5

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

   7. Applied rewrites 55.5%

      \[\leadsto \left(\left(\mathsf{fma}\left(re \cdot re, -0.16666666666666666, 1\right) \cdot re\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\right) \cdot im \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 59.3

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

   9. Applied rewrites 59.3%

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

   if 3.99999999999999977e-157 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 58.8%

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

   2. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right)\right)} \]
   3. 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 83.2

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

   4. Applied rewrites 83.2%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 44.3

          \[\leadsto \left(\left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(re \cdot re\right) - 0.16666666666666666, re \cdot re, 1\right) \cdot re\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\right) \cdot im \]

   7. Applied rewrites 44.3%

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

Alternative 8: 54.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.16666666666666666, 1\right) \cdot re\right) \cdot \left(\mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(e^{-im} - e^{im}\right) \cdot 0.5\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) 4e-6)
   (*
    (* (fma (* re re) -0.16666666666666666 1.0) re)
    (* (fma (* -0.16666666666666666 im) im -1.0) im))
   (* (* (- (exp (- im)) (exp im)) 0.5) re)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= 4e-6) {
		tmp = (fma((re * re), -0.16666666666666666, 1.0) * re) * (fma((-0.16666666666666666 * im), im, -1.0) * im);
	} else {
		tmp = ((exp(-im) - exp(im)) * 0.5) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= 4e-6)
		tmp = Float64(Float64(fma(Float64(re * re), -0.16666666666666666, 1.0) * re) * Float64(fma(Float64(-0.16666666666666666 * im), im, -1.0) * im));
	else
		tmp = Float64(Float64(Float64(exp(Float64(-im)) - exp(im)) * 0.5) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], 4e-6], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(-0.16666666666666666 * im), $MachinePrecision] * im + -1.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < 3.99999999999999982e-6

   1. Initial program 68.7%

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

   2. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right)\right)} \]
   3. 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 80.4

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

   4. Applied rewrites 80.4%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 59.7

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

   7. Applied rewrites 59.7%

      \[\leadsto \left(\left(\mathsf{fma}\left(re \cdot re, -0.16666666666666666, 1\right) \cdot re\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\right) \cdot im \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 63.6

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

   9. Applied rewrites 63.6%

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

   if 3.99999999999999982e-6 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 56.2%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift--.f64 28.4

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

   4. Applied rewrites 28.4%

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

Alternative 9: 53.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\\ \mathbf{if}\;0.5 \cdot \sin re \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.16666666666666666, 1\right) \cdot re\right) \cdot \left(t\_0 \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right) \cdot 0.008333333333333333\right) \cdot re\right) \cdot t\_0\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (fma (* -0.16666666666666666 im) im -1.0)))
   (if (<= (* 0.5 (sin re)) 4e-6)
     (* (* (fma (* re re) -0.16666666666666666 1.0) re) (* t_0 im))
     (* (* (* (* (* (* re re) (* re re)) 0.008333333333333333) re) t_0) im))))

C

double code(double re, double im) {
	double t_0 = fma((-0.16666666666666666 * im), im, -1.0);
	double tmp;
	if ((0.5 * sin(re)) <= 4e-6) {
		tmp = (fma((re * re), -0.16666666666666666, 1.0) * re) * (t_0 * im);
	} else {
		tmp = (((((re * re) * (re * re)) * 0.008333333333333333) * re) * t_0) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(Float64(-0.16666666666666666 * im), im, -1.0)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= 4e-6)
		tmp = Float64(Float64(fma(Float64(re * re), -0.16666666666666666, 1.0) * re) * Float64(t_0 * im));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(re * re) * Float64(re * re)) * 0.008333333333333333) * re) * t_0) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(-0.16666666666666666 * im), $MachinePrecision] * im + -1.0), $MachinePrecision]}, If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], 4e-6], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * re), $MachinePrecision] * N[(t$95$0 * im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(re * re), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] * re), $MachinePrecision] * t$95$0), $MachinePrecision] * im), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < 3.99999999999999982e-6

   1. Initial program 68.7%

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

   2. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right)\right)} \]
   3. 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 80.4

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

   4. Applied rewrites 80.4%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 59.7

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

   7. Applied rewrites 59.7%

      \[\leadsto \left(\left(\mathsf{fma}\left(re \cdot re, -0.16666666666666666, 1\right) \cdot re\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\right) \cdot im \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 63.6

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

   9. Applied rewrites 63.6%

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

   if 3.99999999999999982e-6 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 56.2%

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

   2. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right)\right)} \]
   3. 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 83.6

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

   4. Applied rewrites 83.6%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 26.6

          \[\leadsto \left(\left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(re \cdot re\right) - 0.16666666666666666, re \cdot re, 1\right) \cdot re\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\right) \cdot im \]

   7. Applied rewrites 26.6%

      \[\leadsto \left(\left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(re \cdot re\right) - 0.16666666666666666, re \cdot re, 1\right) \cdot re\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\right) \cdot im \]

   8. Taylor expanded in re around inf

      \[\leadsto \left(\left(\left(\frac{1}{120} \cdot {re}^{4}\right) \cdot re\right) \cdot \mathsf{fma}\left(\frac{-1}{6} \cdot im, im, -1\right)\right) \cdot im \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\left({re}^{4} \cdot \frac{1}{120}\right) \cdot re\right) \cdot \mathsf{fma}\left(\frac{-1}{6} \cdot im, im, -1\right)\right) \cdot im \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\left({re}^{4} \cdot \frac{1}{120}\right) \cdot re\right) \cdot \mathsf{fma}\left(\frac{-1}{6} \cdot im, im, -1\right)\right) \cdot im \]

      3. metadata-eval N/A

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

      4. pow-prod-up N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 25.9

         \[\leadsto \left(\left(\left(\left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right) \cdot 0.008333333333333333\right) \cdot re\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\right) \cdot im \]

   10. Applied rewrites 25.9%

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

Alternative 10: 53.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq 0.0004:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.16666666666666666, 1\right) \cdot re\right) \cdot \left(\mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(-0.008333333333333333, re \cdot re, 0.16666666666666666\right) \cdot \left(re \cdot re\right) - 1\right) \cdot re\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) 0.0004)
   (*
    (* (fma (* re re) -0.16666666666666666 1.0) re)
    (* (fma (* -0.16666666666666666 im) im -1.0) im))
   (*
    (*
     (-
      (* (fma -0.008333333333333333 (* re re) 0.16666666666666666) (* re re))
      1.0)
     re)
    im)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= 0.0004) {
		tmp = (fma((re * re), -0.16666666666666666, 1.0) * re) * (fma((-0.16666666666666666 * im), im, -1.0) * im);
	} else {
		tmp = (((fma(-0.008333333333333333, (re * re), 0.16666666666666666) * (re * re)) - 1.0) * re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= 0.0004)
		tmp = Float64(Float64(fma(Float64(re * re), -0.16666666666666666, 1.0) * re) * Float64(fma(Float64(-0.16666666666666666 * im), im, -1.0) * im));
	else
		tmp = Float64(Float64(Float64(Float64(fma(-0.008333333333333333, Float64(re * re), 0.16666666666666666) * Float64(re * re)) - 1.0) * re) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], 0.0004], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(-0.16666666666666666 * im), $MachinePrecision] * im + -1.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(-0.008333333333333333 * N[(re * re), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] * re), $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < 4.00000000000000019e-4

   1. Initial program 68.7%

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

   2. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right)\right)} \]
   3. 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 80.4

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

   4. Applied rewrites 80.4%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 59.8

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

   7. Applied rewrites 59.8%

      \[\leadsto \left(\left(\mathsf{fma}\left(re \cdot re, -0.16666666666666666, 1\right) \cdot re\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\right) \cdot im \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 63.6

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

   9. Applied rewrites 63.6%

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

   if 4.00000000000000019e-4 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 56.2%

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

   2. Taylor expanded in im around 0

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

      1. *-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 50.1

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

   4. Applied rewrites 50.1%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 24.2

          \[\leadsto \left(\left(\mathsf{fma}\left(-0.008333333333333333, re \cdot re, 0.16666666666666666\right) \cdot \left(re \cdot re\right) - 1\right) \cdot re\right) \cdot im \]

   7. Applied rewrites 24.2%

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

Alternative 11: 53.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.001:\\ \;\;\;\;\left(\left(\left(\left(re \cdot re\right) \cdot -0.16666666666666666\right) \cdot re\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(im \cdot im\right) \cdot -0.16666666666666666 - 1\right) \cdot im\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.001)
   (*
    (*
     (* (* (* re re) -0.16666666666666666) re)
     (fma (* -0.16666666666666666 im) im -1.0))
    im)
   (* (* (- (* (* im im) -0.16666666666666666) 1.0) im) re)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.001) {
		tmp = ((((re * re) * -0.16666666666666666) * re) * fma((-0.16666666666666666 * im), im, -1.0)) * im;
	} else {
		tmp = ((((im * im) * -0.16666666666666666) - 1.0) * im) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.001)
		tmp = Float64(Float64(Float64(Float64(Float64(re * re) * -0.16666666666666666) * re) * fma(Float64(-0.16666666666666666 * im), im, -1.0)) * im);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(im * im) * -0.16666666666666666) - 1.0) * im) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.001], N[(N[(N[(N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * re), $MachinePrecision] * N[(N[(-0.16666666666666666 * im), $MachinePrecision] * im + -1.0), $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - 1.0), $MachinePrecision] * im), $MachinePrecision] * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < -1e-3

   1. Initial program 54.4%

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

   2. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right)\right)} \]
   3. 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 84.1

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

   4. Applied rewrites 84.1%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 24.7

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

   7. Applied rewrites 24.7%

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

   8. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 24.1

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

   10. Applied rewrites 24.1%

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

   if -1e-3 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 69.5%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift--.f64 60.1

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

   4. Applied rewrites 60.1%

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

   5. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 63.8

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

   7. Applied rewrites 63.8%

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

Alternative 12: 52.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.001:\\ \;\;\;\;\mathsf{fma}\left(re \cdot \left(im \cdot re\right), 0.16666666666666666, -im\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(im \cdot im\right) \cdot -0.16666666666666666 - 1\right) \cdot im\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.001)
   (* (fma (* re (* im re)) 0.16666666666666666 (- im)) re)
   (* (* (- (* (* im im) -0.16666666666666666) 1.0) im) re)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.001) {
		tmp = fma((re * (im * re)), 0.16666666666666666, -im) * re;
	} else {
		tmp = ((((im * im) * -0.16666666666666666) - 1.0) * im) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.001)
		tmp = Float64(fma(Float64(re * Float64(im * re)), 0.16666666666666666, Float64(-im)) * re);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(im * im) * -0.16666666666666666) - 1.0) * im) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.001], N[(N[(N[(re * N[(im * re), $MachinePrecision]), $MachinePrecision] * 0.16666666666666666 + (-im)), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - 1.0), $MachinePrecision] * im), $MachinePrecision] * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < -1e-3

   1. Initial program 54.4%

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

   2. Taylor expanded in im around 0

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

      1. *-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 51.7

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

   4. Applied rewrites 51.7%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lift-neg.f64 21.5

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

   7. Applied rewrites 21.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 21.5

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

   9. Applied rewrites 21.5%

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

   if -1e-3 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 69.5%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift--.f64 60.1

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

   4. Applied rewrites 60.1%

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

   5. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 63.8

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

   7. Applied rewrites 63.8%

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

Alternative 13: 52.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.001:\\ \;\;\;\;\left(\left(\left(re \cdot re\right) \cdot im\right) \cdot 0.16666666666666666\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(im \cdot im\right) \cdot -0.16666666666666666 - 1\right) \cdot im\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.001)
   (* (* (* (* re re) im) 0.16666666666666666) re)
   (* (* (- (* (* im im) -0.16666666666666666) 1.0) im) re)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.001) {
		tmp = (((re * re) * im) * 0.16666666666666666) * re;
	} else {
		tmp = ((((im * im) * -0.16666666666666666) - 1.0) * im) * re;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((0.5d0 * sin(re)) <= (-0.001d0)) then
        tmp = (((re * re) * im) * 0.16666666666666666d0) * re
    else
        tmp = ((((im * im) * (-0.16666666666666666d0)) - 1.0d0) * im) * re
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((0.5 * Math.sin(re)) <= -0.001) {
		tmp = (((re * re) * im) * 0.16666666666666666) * re;
	} else {
		tmp = ((((im * im) * -0.16666666666666666) - 1.0) * im) * re;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (0.5 * math.sin(re)) <= -0.001:
		tmp = (((re * re) * im) * 0.16666666666666666) * re
	else:
		tmp = ((((im * im) * -0.16666666666666666) - 1.0) * im) * re
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.001)
		tmp = Float64(Float64(Float64(Float64(re * re) * im) * 0.16666666666666666) * re);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(im * im) * -0.16666666666666666) - 1.0) * im) * re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((0.5 * sin(re)) <= -0.001)
		tmp = (((re * re) * im) * 0.16666666666666666) * re;
	else
		tmp = ((((im * im) * -0.16666666666666666) - 1.0) * im) * re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.001], N[(N[(N[(N[(re * re), $MachinePrecision] * im), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - 1.0), $MachinePrecision] * im), $MachinePrecision] * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < -1e-3

   1. Initial program 54.4%

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

   2. Taylor expanded in im around 0

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

      1. *-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 51.7

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

   4. Applied rewrites 51.7%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lift-neg.f64 21.5

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

   7. Applied rewrites 21.5%

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

   8. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 21.3

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

   10. Applied rewrites 21.3%

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

   if -1e-3 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 69.5%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift--.f64 60.1

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

   4. Applied rewrites 60.1%

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

   5. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 63.8

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

   7. Applied rewrites 63.8%

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

Alternative 14: 49.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.001:\\ \;\;\;\;\left(\left(\left(re \cdot re\right) \cdot im\right) \cdot 0.16666666666666666\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(im \cdot im\right) \cdot re\right) \cdot -0.16666666666666666 - re\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.001)
   (* (* (* (* re re) im) 0.16666666666666666) re)
   (* (- (* (* (* im im) re) -0.16666666666666666) re) im)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.001) {
		tmp = (((re * re) * im) * 0.16666666666666666) * re;
	} else {
		tmp = ((((im * im) * re) * -0.16666666666666666) - re) * im;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((0.5d0 * sin(re)) <= (-0.001d0)) then
        tmp = (((re * re) * im) * 0.16666666666666666d0) * re
    else
        tmp = ((((im * im) * re) * (-0.16666666666666666d0)) - re) * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((0.5 * Math.sin(re)) <= -0.001) {
		tmp = (((re * re) * im) * 0.16666666666666666) * re;
	} else {
		tmp = ((((im * im) * re) * -0.16666666666666666) - re) * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (0.5 * math.sin(re)) <= -0.001:
		tmp = (((re * re) * im) * 0.16666666666666666) * re
	else:
		tmp = ((((im * im) * re) * -0.16666666666666666) - re) * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.001)
		tmp = Float64(Float64(Float64(Float64(re * re) * im) * 0.16666666666666666) * re);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(im * im) * re) * -0.16666666666666666) - re) * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((0.5 * sin(re)) <= -0.001)
		tmp = (((re * re) * im) * 0.16666666666666666) * re;
	else
		tmp = ((((im * im) * re) * -0.16666666666666666) - re) * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.001], N[(N[(N[(N[(re * re), $MachinePrecision] * im), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(N[(N[(im * im), $MachinePrecision] * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - re), $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < -1e-3

   1. Initial program 54.4%

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

   2. Taylor expanded in im around 0

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

      1. *-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 51.7

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

   4. Applied rewrites 51.7%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lift-neg.f64 21.5

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

   7. Applied rewrites 21.5%

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

   8. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 21.3

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

   10. Applied rewrites 21.3%

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

   if -1e-3 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 69.5%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift--.f64 60.1

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

   4. Applied rewrites 60.1%

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

   5. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 60.0

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

   7. Applied rewrites 60.0%

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

   8. Taylor expanded in im around 0

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

      1. lower--.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 60.0

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

   10. Applied rewrites 60.0%

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

Alternative 15: 49.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.001:\\ \;\;\;\;\left(\left(\left(re \cdot re\right) \cdot im\right) \cdot 0.16666666666666666\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.001)
   (* (* (* (* re re) im) 0.16666666666666666) re)
   (* (* re (fma (* -0.16666666666666666 im) im -1.0)) im)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.001) {
		tmp = (((re * re) * im) * 0.16666666666666666) * re;
	} else {
		tmp = (re * fma((-0.16666666666666666 * im), im, -1.0)) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.001)
		tmp = Float64(Float64(Float64(Float64(re * re) * im) * 0.16666666666666666) * re);
	else
		tmp = Float64(Float64(re * fma(Float64(-0.16666666666666666 * im), im, -1.0)) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.001], N[(N[(N[(N[(re * re), $MachinePrecision] * im), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * re), $MachinePrecision], N[(N[(re * N[(N[(-0.16666666666666666 * im), $MachinePrecision] * im + -1.0), $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < -1e-3

   1. Initial program 54.4%

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

   2. Taylor expanded in im around 0

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

      1. *-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 51.7

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

   4. Applied rewrites 51.7%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lift-neg.f64 21.5

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

   7. Applied rewrites 21.5%

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

   8. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 21.3

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

   10. Applied rewrites 21.3%

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

   if -1e-3 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 69.5%

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

   2. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right)\right)} \]
   3. 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 80.2

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

   4. Applied rewrites 80.2%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 59.9%

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

Alternative 16: 41.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.001:\\ \;\;\;\;\left(\left(\left(re \cdot re\right) \cdot im\right) \cdot 0.16666666666666666\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot im\right) \cdot \left(-0.16666666666666666 \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.001)
   (* (* (* (* re re) im) 0.16666666666666666) re)
   (* (* (* im im) im) (* -0.16666666666666666 re))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.001) {
		tmp = (((re * re) * im) * 0.16666666666666666) * re;
	} else {
		tmp = ((im * im) * im) * (-0.16666666666666666 * re);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((0.5d0 * sin(re)) <= (-0.001d0)) then
        tmp = (((re * re) * im) * 0.16666666666666666d0) * re
    else
        tmp = ((im * im) * im) * ((-0.16666666666666666d0) * re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((0.5 * Math.sin(re)) <= -0.001) {
		tmp = (((re * re) * im) * 0.16666666666666666) * re;
	} else {
		tmp = ((im * im) * im) * (-0.16666666666666666 * re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (0.5 * math.sin(re)) <= -0.001:
		tmp = (((re * re) * im) * 0.16666666666666666) * re
	else:
		tmp = ((im * im) * im) * (-0.16666666666666666 * re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.001)
		tmp = Float64(Float64(Float64(Float64(re * re) * im) * 0.16666666666666666) * re);
	else
		tmp = Float64(Float64(Float64(im * im) * im) * Float64(-0.16666666666666666 * re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((0.5 * sin(re)) <= -0.001)
		tmp = (((re * re) * im) * 0.16666666666666666) * re;
	else
		tmp = ((im * im) * im) * (-0.16666666666666666 * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.001], N[(N[(N[(N[(re * re), $MachinePrecision] * im), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(im * im), $MachinePrecision] * im), $MachinePrecision] * N[(-0.16666666666666666 * re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < -1e-3

   1. Initial program 54.4%

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

   2. Taylor expanded in im around 0

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

      1. *-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 51.7

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

   4. Applied rewrites 51.7%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lift-neg.f64 21.5

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

   7. Applied rewrites 21.5%

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

   8. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 21.3

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

   10. Applied rewrites 21.3%

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

   if -1e-3 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 69.5%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift--.f64 60.1

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

   4. Applied rewrites 60.1%

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

   5. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 60.0

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

   7. Applied rewrites 60.0%

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

   8. Taylor expanded in im around inf

      \[\leadsto \frac{-1}{6} \cdot \left({im}^{3} \cdot \color{blue}{re}\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({im}^{3} \cdot re\right) \cdot \frac{-1}{6} \]

      2. lower-*.f64 N/A

         \[\leadsto \left({im}^{3} \cdot re\right) \cdot \frac{-1}{6} \]

      3. lower-*.f64 N/A

         \[\leadsto \left({im}^{3} \cdot re\right) \cdot \frac{-1}{6} \]

      4. unpow3 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 47.9

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

   10. Applied rewrites 47.9%

       \[\leadsto \left(\left(\left(im \cdot im\right) \cdot im\right) \cdot re\right) \cdot -0.16666666666666666 \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. pow3 N/A

          \[\leadsto {im}^{3} \cdot \left(re \cdot \frac{-1}{6}\right) \]

       7. *-commutative N/A

          \[\leadsto {im}^{3} \cdot \left(\frac{-1}{6} \cdot re\right) \]

       8. lower-*.f64 N/A

          \[\leadsto {im}^{3} \cdot \left(\frac{-1}{6} \cdot re\right) \]

       9. pow3 N/A

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

       10. lift-*.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. lower-*.f64 47.9

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

   12. Applied rewrites 47.9%

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

Alternative 17: 35.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.001:\\ \;\;\;\;\left(\left(\left(re \cdot re\right) \cdot im\right) \cdot 0.16666666666666666\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(-re\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.001)
   (* (* (* (* re re) im) 0.16666666666666666) re)
   (* (- re) im)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.001) {
		tmp = (((re * re) * im) * 0.16666666666666666) * re;
	} else {
		tmp = -re * im;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((0.5d0 * sin(re)) <= (-0.001d0)) then
        tmp = (((re * re) * im) * 0.16666666666666666d0) * re
    else
        tmp = -re * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((0.5 * Math.sin(re)) <= -0.001) {
		tmp = (((re * re) * im) * 0.16666666666666666) * re;
	} else {
		tmp = -re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (0.5 * math.sin(re)) <= -0.001:
		tmp = (((re * re) * im) * 0.16666666666666666) * re
	else:
		tmp = -re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.001)
		tmp = Float64(Float64(Float64(Float64(re * re) * im) * 0.16666666666666666) * re);
	else
		tmp = Float64(Float64(-re) * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((0.5 * sin(re)) <= -0.001)
		tmp = (((re * re) * im) * 0.16666666666666666) * re;
	else
		tmp = -re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.001], N[(N[(N[(N[(re * re), $MachinePrecision] * im), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * re), $MachinePrecision], N[((-re) * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < -1e-3

   1. Initial program 54.4%

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

   2. Taylor expanded in im around 0

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

      1. *-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 51.7

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

   4. Applied rewrites 51.7%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lift-neg.f64 21.5

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

   7. Applied rewrites 21.5%

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

   8. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 21.3

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

   10. Applied rewrites 21.3%

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

   if -1e-3 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 69.5%

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

   2. Taylor expanded in im around 0

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

      1. *-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 51.5

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

   4. Applied rewrites 51.5%

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

   5. Taylor expanded in re around 0

      \[\leadsto \left(-re\right) \cdot im \]
   6. Step-by-step derivation

   7. Applied rewrites 39.8%

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

Alternative 18: 35.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.001:\\ \;\;\;\;\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot im\right) \cdot 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\left(-re\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.001)
   (* (* (* (* re re) re) im) 0.16666666666666666)
   (* (- re) im)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.001) {
		tmp = (((re * re) * re) * im) * 0.16666666666666666;
	} else {
		tmp = -re * im;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((0.5d0 * sin(re)) <= (-0.001d0)) then
        tmp = (((re * re) * re) * im) * 0.16666666666666666d0
    else
        tmp = -re * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((0.5 * Math.sin(re)) <= -0.001) {
		tmp = (((re * re) * re) * im) * 0.16666666666666666;
	} else {
		tmp = -re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (0.5 * math.sin(re)) <= -0.001:
		tmp = (((re * re) * re) * im) * 0.16666666666666666
	else:
		tmp = -re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.001)
		tmp = Float64(Float64(Float64(Float64(re * re) * re) * im) * 0.16666666666666666);
	else
		tmp = Float64(Float64(-re) * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((0.5 * sin(re)) <= -0.001)
		tmp = (((re * re) * re) * im) * 0.16666666666666666;
	else
		tmp = -re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.001], N[(N[(N[(N[(re * re), $MachinePrecision] * re), $MachinePrecision] * im), $MachinePrecision] * 0.16666666666666666), $MachinePrecision], N[((-re) * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < -1e-3

   1. Initial program 54.4%

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

   2. Taylor expanded in im around 0

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

      1. *-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 51.7

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

   4. Applied rewrites 51.7%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lift-neg.f64 21.5

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

   7. Applied rewrites 21.5%

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

   8. Taylor expanded in re around inf

      \[\leadsto \frac{1}{6} \cdot \left(im \cdot \color{blue}{{re}^{3}}\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(im \cdot {re}^{3}\right) \cdot \frac{1}{6} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(im \cdot {re}^{3}\right) \cdot \frac{1}{6} \]

      3. *-commutative N/A

         \[\leadsto \left({re}^{3} \cdot im\right) \cdot \frac{1}{6} \]

      4. lower-*.f64 N/A

         \[\leadsto \left({re}^{3} \cdot im\right) \cdot \frac{1}{6} \]

      5. unpow3 N/A

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

      6. pow2 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 21.3

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

   10. Applied rewrites 21.3%

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

   if -1e-3 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 69.5%

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

   2. Taylor expanded in im around 0

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

      1. *-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 51.5

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

   4. Applied rewrites 51.5%

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

   5. Taylor expanded in re around 0

      \[\leadsto \left(-re\right) \cdot im \]
   6. Step-by-step derivation

   7. Applied rewrites 39.8%

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

Alternative 19: 33.3% accurate, 12.7× 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 65.6%

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

2. Taylor expanded in im around 0

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

   1. *-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 51.5

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

4. Applied rewrites 51.5%

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

5. Taylor expanded in re around 0

   \[\leadsto \left(-re\right) \cdot im \]
6. Step-by-step derivation

7. Applied rewrites 33.3%

   \[\leadsto \left(-re\right) \cdot im \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025117 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"
  :precision binary64
  (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))