math.cos on complex, imaginary part

Percentage Accurate: 65.1% → 99.9%

Time: 9.9s

Alternatives: 18

Speedup: 1.5×

• 49.0% 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: 65.1% 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: 99.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 62.5

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

   7. lift--.f64 N/A

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

   8. lift-exp.f64 N/A

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

   9. rem-exp-log N/A

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

   10. lift-exp.f64 N/A

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

   11. rem-log-exp N/A

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

   12. remove-double-neg N/A

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

   13. lift-neg.f64 N/A

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

   14. sinh-undef N/A

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

   15. lower-*.f64 N/A

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

   16. lower-sinh.f64 99.9

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

4. Applied rewrites 99.9%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. associate-/l* N/A

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

   10. *-commutative N/A

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

   11. lift-sinh.f64 N/A

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

   12. sinh-undef-rev N/A

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

   13. sinh-def N/A

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

   14. lift-sinh.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 99.9

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

6. Applied rewrites 99.9%

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

Alternative 2: 75.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\left(-2\right) \cdot \sinh im\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;t\_0 \leq 0.002:\\ \;\;\;\;\left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right), -1\right)\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left({re}^{7} \cdot 0.0001984126984126984\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (- (exp (- im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (* (* (- 2.0) (sinh im)) (* 0.5 re))
     (if (<= t_0 0.002)
       (*
        (*
         (sin re)
         (fma
          (* im im)
          (fma -0.008333333333333333 (* im im) -0.16666666666666666)
          -1.0))
        im)
       (* (* (pow re 7.0) 0.0001984126984126984) im)))))

C

double code(double re, double im) {
	double t_0 = (0.5 * sin(re)) * (exp(-im) - exp(im));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (-2.0 * sinh(im)) * (0.5 * re);
	} else if (t_0 <= 0.002) {
		tmp = (sin(re) * fma((im * im), fma(-0.008333333333333333, (im * im), -0.16666666666666666), -1.0)) * im;
	} else {
		tmp = (pow(re, 7.0) * 0.0001984126984126984) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - exp(im)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(-2.0) * sinh(im)) * Float64(0.5 * re));
	elseif (t_0 <= 0.002)
		tmp = Float64(Float64(sin(re) * fma(Float64(im * im), fma(-0.008333333333333333, Float64(im * im), -0.16666666666666666), -1.0)) * im);
	else
		tmp = Float64(Float64((re ^ 7.0) * 0.0001984126984126984) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[((-2.0) * N[Sinh[im], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.002], N[(N[(N[Sin[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(-0.008333333333333333 * N[(im * im), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision], N[(N[(N[Power[re, 7.0], $MachinePrecision] * 0.0001984126984126984), $MachinePrecision] * im), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-*.f64 79.3

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

   5. Applied rewrites 79.3%

      \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot \left(e^{-im} - e^{im}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. remove-double-neg N/A

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

      10. lift-neg.f64 N/A

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

      11. sinh-undef-rev N/A

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

      12. lift-neg.f64 N/A

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

      13. sinh-neg N/A

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

      14. distribute-rgt-neg-out N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-sinh.f64 79.3

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

   7. Applied rewrites 79.3%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 2e-3

   1. Initial program 28.3%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.8%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sin.f64 4.3

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

   5. Applied rewrites 4.3%

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

   6. Taylor expanded in re around 0

      \[\leadsto \left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{6} + {re}^{2} \cdot \left(\frac{1}{5040} \cdot {re}^{2} - \frac{1}{120}\right)\right) - 1\right)\right) \cdot im \]
   7. Step-by-step derivation

   8. Applied rewrites 24.9%

      \[\leadsto \left(\mathsf{fma}\left({re}^{4}, 0.0001984126984126984 \cdot \left(re \cdot re\right) - 0.008333333333333333, 0.16666666666666666 \cdot \left(re \cdot re\right) - 1\right) \cdot re\right) \cdot im \]

   9. Taylor expanded in re around inf

      \[\leadsto \left(\frac{1}{5040} \cdot {re}^{7}\right) \cdot im \]
   10. Step-by-step derivation

   11. Applied rewrites 24.1%

       \[\leadsto \left({re}^{7} \cdot 0.0001984126984126984\right) \cdot im \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \leq -\infty:\\ \;\;\;\;\left(\left(-2\right) \cdot \sinh im\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0.002:\\ \;\;\;\;\left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right), -1\right)\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left({re}^{7} \cdot 0.0001984126984126984\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\left(-2\right) \cdot \sinh im\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;t\_0 \leq 0.002:\\ \;\;\;\;\left(\sin re \cdot im\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left({re}^{7} \cdot 0.0001984126984126984\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (- (exp (- im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (* (* (- 2.0) (sinh im)) (* 0.5 re))
     (if (<= t_0 0.002)
       (* (* (sin re) im) (fma (* -0.16666666666666666 im) im -1.0))
       (* (* (pow re 7.0) 0.0001984126984126984) im)))))

C

double code(double re, double im) {
	double t_0 = (0.5 * sin(re)) * (exp(-im) - exp(im));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (-2.0 * sinh(im)) * (0.5 * re);
	} else if (t_0 <= 0.002) {
		tmp = (sin(re) * im) * fma((-0.16666666666666666 * im), im, -1.0);
	} else {
		tmp = (pow(re, 7.0) * 0.0001984126984126984) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - exp(im)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(-2.0) * sinh(im)) * Float64(0.5 * re));
	elseif (t_0 <= 0.002)
		tmp = Float64(Float64(sin(re) * im) * fma(Float64(-0.16666666666666666 * im), im, -1.0));
	else
		tmp = Float64(Float64((re ^ 7.0) * 0.0001984126984126984) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[((-2.0) * N[Sinh[im], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.002], N[(N[(N[Sin[re], $MachinePrecision] * im), $MachinePrecision] * N[(N[(-0.16666666666666666 * im), $MachinePrecision] * im + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[re, 7.0], $MachinePrecision] * 0.0001984126984126984), $MachinePrecision] * im), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-*.f64 79.3

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

   5. Applied rewrites 79.3%

      \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot \left(e^{-im} - e^{im}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. remove-double-neg N/A

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

      10. lift-neg.f64 N/A

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

      11. sinh-undef-rev N/A

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

      12. lift-neg.f64 N/A

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

      13. sinh-neg N/A

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

      14. distribute-rgt-neg-out N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-sinh.f64 79.3

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

   7. Applied rewrites 79.3%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 2e-3

   1. Initial program 28.3%

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

   3. Taylor expanded in im around 0

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

      1. distribute-lft-in N/A

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

      2. mul-1-neg N/A

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

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

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

      4. mul-1-neg N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. associate-*l* N/A

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

      18. *-commutative N/A

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

   5. Applied rewrites 99.6%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sin.f64 4.3

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

   5. Applied rewrites 4.3%

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

   6. Taylor expanded in re around 0

      \[\leadsto \left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{6} + {re}^{2} \cdot \left(\frac{1}{5040} \cdot {re}^{2} - \frac{1}{120}\right)\right) - 1\right)\right) \cdot im \]
   7. Step-by-step derivation

   8. Applied rewrites 24.9%

      \[\leadsto \left(\mathsf{fma}\left({re}^{4}, 0.0001984126984126984 \cdot \left(re \cdot re\right) - 0.008333333333333333, 0.16666666666666666 \cdot \left(re \cdot re\right) - 1\right) \cdot re\right) \cdot im \]

   9. Taylor expanded in re around inf

      \[\leadsto \left(\frac{1}{5040} \cdot {re}^{7}\right) \cdot im \]
   10. Step-by-step derivation

   11. Applied rewrites 24.1%

       \[\leadsto \left({re}^{7} \cdot 0.0001984126984126984\right) \cdot im \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \leq -\infty:\\ \;\;\;\;\left(\left(-2\right) \cdot \sinh im\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0.002:\\ \;\;\;\;\left(\sin re \cdot im\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left({re}^{7} \cdot 0.0001984126984126984\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\left(-2\right) \cdot \sinh im\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;t\_0 \leq 0.002:\\ \;\;\;\;\left(-\sin re\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left({re}^{7} \cdot 0.0001984126984126984\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (- (exp (- im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (* (* (- 2.0) (sinh im)) (* 0.5 re))
     (if (<= t_0 0.002)
       (* (- (sin re)) im)
       (* (* (pow re 7.0) 0.0001984126984126984) im)))))

C

double code(double re, double im) {
	double t_0 = (0.5 * sin(re)) * (exp(-im) - exp(im));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (-2.0 * sinh(im)) * (0.5 * re);
	} else if (t_0 <= 0.002) {
		tmp = -sin(re) * im;
	} else {
		tmp = (pow(re, 7.0) * 0.0001984126984126984) * im;
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = (0.5 * Math.sin(re)) * (Math.exp(-im) - Math.exp(im));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = (-2.0 * Math.sinh(im)) * (0.5 * re);
	} else if (t_0 <= 0.002) {
		tmp = -Math.sin(re) * im;
	} else {
		tmp = (Math.pow(re, 7.0) * 0.0001984126984126984) * im;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (0.5 * math.sin(re)) * (math.exp(-im) - math.exp(im))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = (-2.0 * math.sinh(im)) * (0.5 * re)
	elif t_0 <= 0.002:
		tmp = -math.sin(re) * im
	else:
		tmp = (math.pow(re, 7.0) * 0.0001984126984126984) * im
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - exp(im)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(-2.0) * sinh(im)) * Float64(0.5 * re));
	elseif (t_0 <= 0.002)
		tmp = Float64(Float64(-sin(re)) * im);
	else
		tmp = Float64(Float64((re ^ 7.0) * 0.0001984126984126984) * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (0.5 * sin(re)) * (exp(-im) - exp(im));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = (-2.0 * sinh(im)) * (0.5 * re);
	elseif (t_0 <= 0.002)
		tmp = -sin(re) * im;
	else
		tmp = ((re ^ 7.0) * 0.0001984126984126984) * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[((-2.0) * N[Sinh[im], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.002], N[((-N[Sin[re], $MachinePrecision]) * im), $MachinePrecision], N[(N[(N[Power[re, 7.0], $MachinePrecision] * 0.0001984126984126984), $MachinePrecision] * im), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-*.f64 79.3

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

   5. Applied rewrites 79.3%

      \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot \left(e^{-im} - e^{im}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. remove-double-neg N/A

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

      10. lift-neg.f64 N/A

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

      11. sinh-undef-rev N/A

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

      12. lift-neg.f64 N/A

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

      13. sinh-neg N/A

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

      14. distribute-rgt-neg-out N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-sinh.f64 79.3

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

   7. Applied rewrites 79.3%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 2e-3

   1. Initial program 28.3%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sin.f64 99.4

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

   5. Applied rewrites 99.4%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sin.f64 4.3

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

   5. Applied rewrites 4.3%

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

   6. Taylor expanded in re around 0

      \[\leadsto \left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{6} + {re}^{2} \cdot \left(\frac{1}{5040} \cdot {re}^{2} - \frac{1}{120}\right)\right) - 1\right)\right) \cdot im \]
   7. Step-by-step derivation

   8. Applied rewrites 24.9%

      \[\leadsto \left(\mathsf{fma}\left({re}^{4}, 0.0001984126984126984 \cdot \left(re \cdot re\right) - 0.008333333333333333, 0.16666666666666666 \cdot \left(re \cdot re\right) - 1\right) \cdot re\right) \cdot im \]

   9. Taylor expanded in re around inf

      \[\leadsto \left(\frac{1}{5040} \cdot {re}^{7}\right) \cdot im \]
   10. Step-by-step derivation

   11. Applied rewrites 24.1%

       \[\leadsto \left({re}^{7} \cdot 0.0001984126984126984\right) \cdot im \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \leq -\infty:\\ \;\;\;\;\left(\left(-2\right) \cdot \sinh im\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0.002:\\ \;\;\;\;\left(-\sin re\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left({re}^{7} \cdot 0.0001984126984126984\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(0.004166666666666667 \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right)\\ \mathbf{elif}\;t\_0 \leq 0.002:\\ \;\;\;\;\left(-\sin re\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left({re}^{7} \cdot 0.0001984126984126984\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (- (exp (- im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (*
      (*
       (fma
        (- (* 0.004166666666666667 (* re re)) 0.08333333333333333)
        (* re re)
        0.5)
       re)
      (*
       (-
        (*
         (* (- (* -0.016666666666666666 (* im im)) 0.3333333333333333) im)
         im)
        2.0)
       im))
     (if (<= t_0 0.002)
       (* (- (sin re)) im)
       (* (* (pow re 7.0) 0.0001984126984126984) im)))))

C

double code(double re, double im) {
	double t_0 = (0.5 * sin(re)) * (exp(-im) - exp(im));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma(((0.004166666666666667 * (re * re)) - 0.08333333333333333), (re * re), 0.5) * re) * ((((((-0.016666666666666666 * (im * im)) - 0.3333333333333333) * im) * im) - 2.0) * im);
	} else if (t_0 <= 0.002) {
		tmp = -sin(re) * im;
	} else {
		tmp = (pow(re, 7.0) * 0.0001984126984126984) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - exp(im)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(fma(Float64(Float64(0.004166666666666667 * Float64(re * re)) - 0.08333333333333333), Float64(re * re), 0.5) * re) * Float64(Float64(Float64(Float64(Float64(Float64(-0.016666666666666666 * Float64(im * im)) - 0.3333333333333333) * im) * im) - 2.0) * im));
	elseif (t_0 <= 0.002)
		tmp = Float64(Float64(-sin(re)) * im);
	else
		tmp = Float64(Float64((re ^ 7.0) * 0.0001984126984126984) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(0.004166666666666667 * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.08333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(N[(N[(N[(-0.016666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.002], N[((-N[Sin[re], $MachinePrecision]) * im), $MachinePrecision], N[(N[(N[Power[re, 7.0], $MachinePrecision] * 0.0001984126984126984), $MachinePrecision] * im), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 80.2

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

   5. Applied rewrites 80.2%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 66.0

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

   8. Applied rewrites 66.0%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 2e-3

   1. Initial program 28.3%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sin.f64 99.4

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

   5. Applied rewrites 99.4%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sin.f64 4.3

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

   5. Applied rewrites 4.3%

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

   6. Taylor expanded in re around 0

      \[\leadsto \left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{6} + {re}^{2} \cdot \left(\frac{1}{5040} \cdot {re}^{2} - \frac{1}{120}\right)\right) - 1\right)\right) \cdot im \]
   7. Step-by-step derivation

   8. Applied rewrites 24.9%

      \[\leadsto \left(\mathsf{fma}\left({re}^{4}, 0.0001984126984126984 \cdot \left(re \cdot re\right) - 0.008333333333333333, 0.16666666666666666 \cdot \left(re \cdot re\right) - 1\right) \cdot re\right) \cdot im \]

   9. Taylor expanded in re around inf

      \[\leadsto \left(\frac{1}{5040} \cdot {re}^{7}\right) \cdot im \]
   10. Step-by-step derivation

   11. Applied rewrites 24.1%

       \[\leadsto \left({re}^{7} \cdot 0.0001984126984126984\right) \cdot im \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 82.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(0.004166666666666667 \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right)\\ \mathbf{elif}\;t\_0 \leq 0.002:\\ \;\;\;\;\left(-\sin re\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (- (exp (- im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (*
      (*
       (fma
        (- (* 0.004166666666666667 (* re re)) 0.08333333333333333)
        (* re re)
        0.5)
       re)
      (*
       (-
        (*
         (* (- (* -0.016666666666666666 (* im im)) 0.3333333333333333) im)
         im)
        2.0)
       im))
     (if (<= t_0 0.002)
       (* (- (sin re)) im)
       (*
        (* (fma (* re re) -0.08333333333333333 0.5) re)
        (*
         (-
          (*
           (-
            (*
             (*
              (- (* -0.0003968253968253968 (* im im)) 0.016666666666666666)
              im)
             im)
            0.3333333333333333)
           (* im im))
          2.0)
         im))))))

C

double code(double re, double im) {
	double t_0 = (0.5 * sin(re)) * (exp(-im) - exp(im));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma(((0.004166666666666667 * (re * re)) - 0.08333333333333333), (re * re), 0.5) * re) * ((((((-0.016666666666666666 * (im * im)) - 0.3333333333333333) * im) * im) - 2.0) * im);
	} else if (t_0 <= 0.002) {
		tmp = -sin(re) * im;
	} else {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * re) * ((((((((-0.0003968253968253968 * (im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - exp(im)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(fma(Float64(Float64(0.004166666666666667 * Float64(re * re)) - 0.08333333333333333), Float64(re * re), 0.5) * re) * Float64(Float64(Float64(Float64(Float64(Float64(-0.016666666666666666 * Float64(im * im)) - 0.3333333333333333) * im) * im) - 2.0) * im));
	elseif (t_0 <= 0.002)
		tmp = Float64(Float64(-sin(re)) * im);
	else
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * Float64(im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * Float64(im * im)) - 2.0) * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(0.004166666666666667 * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.08333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(N[(N[(N[(-0.016666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.002], N[((-N[Sin[re], $MachinePrecision]) * im), $MachinePrecision], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.016666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 80.2

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

   5. Applied rewrites 80.2%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 66.0

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

   8. Applied rewrites 66.0%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 2e-3

   1. Initial program 28.3%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sin.f64 99.4

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

   5. Applied rewrites 99.4%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 52.0

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

   5. Applied rewrites 52.0%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 43.3

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

   8. Applied rewrites 43.3%

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

   9. Taylor expanded in im around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   11. Applied rewrites 58.4%

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

Alternative 7: 54.2% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))) -1e-10)
   (*
    (* 0.5 re)
    (*
     (-
      (* (* (- (* -0.016666666666666666 (* im im)) 0.3333333333333333) im) im)
      2.0)
     im))
   (*
    (* (fma (* (* im re) re) -0.16666666666666666 im) re)
    (fma (* -0.16666666666666666 im) im -1.0))))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(-im) - exp(im))) <= -1e-10) {
		tmp = (0.5 * re) * ((((((-0.016666666666666666 * (im * im)) - 0.3333333333333333) * im) * im) - 2.0) * im);
	} else {
		tmp = (fma(((im * re) * re), -0.16666666666666666, im) * re) * fma((-0.16666666666666666 * im), im, -1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - exp(im))) <= -1e-10)
		tmp = Float64(Float64(0.5 * re) * Float64(Float64(Float64(Float64(Float64(Float64(-0.016666666666666666 * Float64(im * im)) - 0.3333333333333333) * im) * im) - 2.0) * im));
	else
		tmp = Float64(Float64(fma(Float64(Float64(im * re) * re), -0.16666666666666666, im) * re) * fma(Float64(-0.16666666666666666 * im), im, -1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-10], N[(N[(0.5 * re), $MachinePrecision] * N[(N[(N[(N[(N[(N[(-0.016666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(im * re), $MachinePrecision] * re), $MachinePrecision] * -0.16666666666666666 + im), $MachinePrecision] * re), $MachinePrecision] * N[(N[(-0.16666666666666666 * im), $MachinePrecision] * im + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -1.00000000000000004e-10

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 80.2

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

   5. Applied rewrites 80.2%

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

   6. Taylor expanded in re around 0

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

      1. lower-*.f64 66.0

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

   8. Applied rewrites 66.0%

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

   if -1.00000000000000004e-10 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 51.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 51.5

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

      7. lift--.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. rem-exp-log N/A

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

      10. lift-exp.f64 N/A

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

      11. rem-log-exp N/A

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

      12. remove-double-neg N/A

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

      13. lift-neg.f64 N/A

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

      14. sinh-undef N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sinh.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. mul-1-neg N/A

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

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

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

      13. mul-1-neg N/A

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

      14. distribute-rgt-out N/A

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

      15. +-commutative N/A

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

      16. lower-*.f64 N/A

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

   7. Applied rewrites 82.8%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 46.6%

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

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \leq -1 \cdot 10^{-10}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(im \cdot re\right) \cdot re, -0.16666666666666666, im\right) \cdot re\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 53.7% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))) -1e-10)
   (*
    (* im re)
    (-
     (* (* (- (* -0.008333333333333333 (* im im)) 0.16666666666666666) im) im)
     1.0))
   (*
    (* (fma (* (* im re) re) -0.16666666666666666 im) re)
    (fma (* -0.16666666666666666 im) im -1.0))))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(-im) - exp(im))) <= -1e-10) {
		tmp = (im * re) * (((((-0.008333333333333333 * (im * im)) - 0.16666666666666666) * im) * im) - 1.0);
	} else {
		tmp = (fma(((im * re) * re), -0.16666666666666666, im) * re) * fma((-0.16666666666666666 * im), im, -1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - exp(im))) <= -1e-10)
		tmp = Float64(Float64(im * re) * Float64(Float64(Float64(Float64(Float64(-0.008333333333333333 * Float64(im * im)) - 0.16666666666666666) * im) * im) - 1.0));
	else
		tmp = Float64(Float64(fma(Float64(Float64(im * re) * re), -0.16666666666666666, im) * re) * fma(Float64(-0.16666666666666666 * im), im, -1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-10], N[(N[(im * re), $MachinePrecision] * N[(N[(N[(N[(N[(-0.008333333333333333 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(im * re), $MachinePrecision] * re), $MachinePrecision] * -0.16666666666666666 + im), $MachinePrecision] * re), $MachinePrecision] * N[(N[(-0.16666666666666666 * im), $MachinePrecision] * im + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -1.00000000000000004e-10

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 80.2%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 66.0%

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

   if -1.00000000000000004e-10 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 51.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 51.5

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

      7. lift--.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. rem-exp-log N/A

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

      10. lift-exp.f64 N/A

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

      11. rem-log-exp N/A

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

      12. remove-double-neg N/A

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

      13. lift-neg.f64 N/A

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

      14. sinh-undef N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sinh.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. mul-1-neg N/A

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

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

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

      13. mul-1-neg N/A

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

      14. distribute-rgt-out N/A

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

      15. +-commutative N/A

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

      16. lower-*.f64 N/A

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

   7. Applied rewrites 82.8%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 46.6%

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

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \leq -1 \cdot 10^{-10}:\\ \;\;\;\;\left(im \cdot re\right) \cdot \left(\left(\left(-0.008333333333333333 \cdot \left(im \cdot im\right) - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(im \cdot re\right) \cdot re, -0.16666666666666666, im\right) \cdot re\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 46.7% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))) -1e-10)
   (*
    (* im re)
    (-
     (* (* (- (* -0.008333333333333333 (* im im)) 0.16666666666666666) im) im)
     1.0))
   (* (* im (fma (* 0.16666666666666666 re) re -1.0)) re)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(-im) - exp(im))) <= -1e-10) {
		tmp = (im * re) * (((((-0.008333333333333333 * (im * im)) - 0.16666666666666666) * im) * im) - 1.0);
	} else {
		tmp = (im * fma((0.16666666666666666 * re), re, -1.0)) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - exp(im))) <= -1e-10)
		tmp = Float64(Float64(im * re) * Float64(Float64(Float64(Float64(Float64(-0.008333333333333333 * Float64(im * im)) - 0.16666666666666666) * im) * im) - 1.0));
	else
		tmp = Float64(Float64(im * fma(Float64(0.16666666666666666 * re), re, -1.0)) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-10], N[(N[(im * re), $MachinePrecision] * N[(N[(N[(N[(N[(-0.008333333333333333 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(im * N[(N[(0.16666666666666666 * re), $MachinePrecision] * re + -1.0), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -1.00000000000000004e-10

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 80.2%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 66.0%

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

   if -1.00000000000000004e-10 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 51.5%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sin.f64 68.7

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

   5. Applied rewrites 68.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 40.6%

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

Alternative 10: 45.4% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))) -1e-10)
   (* (* (- (* -0.16666666666666666 (* im im)) 1.0) im) re)
   (* (* im (fma (* 0.16666666666666666 re) re -1.0)) re)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(-im) - exp(im))) <= -1e-10) {
		tmp = (((-0.16666666666666666 * (im * im)) - 1.0) * im) * re;
	} else {
		tmp = (im * fma((0.16666666666666666 * re), re, -1.0)) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - exp(im))) <= -1e-10)
		tmp = Float64(Float64(Float64(Float64(-0.16666666666666666 * Float64(im * im)) - 1.0) * im) * re);
	else
		tmp = Float64(Float64(im * fma(Float64(0.16666666666666666 * re), re, -1.0)) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-10], N[(N[(N[(N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] * im), $MachinePrecision] * re), $MachinePrecision], N[(N[(im * N[(N[(0.16666666666666666 * re), $MachinePrecision] * re + -1.0), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -1.00000000000000004e-10

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 100.0

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

      7. lift--.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. rem-exp-log N/A

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

      10. lift-exp.f64 N/A

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

      11. rem-log-exp N/A

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

      12. remove-double-neg N/A

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

      13. lift-neg.f64 N/A

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

      14. sinh-undef N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sinh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. mul-1-neg N/A

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

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

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

      13. mul-1-neg N/A

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

      14. distribute-rgt-out N/A

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

      15. +-commutative N/A

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

      16. lower-*.f64 N/A

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

   7. Applied rewrites 60.7%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 62.7%

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

   if -1.00000000000000004e-10 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 51.5%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sin.f64 68.7

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

   5. Applied rewrites 68.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 40.6%

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

4. Final simplification 45.6%

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

Alternative 11: 43.9% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))) -1e-10)
   (* (* im re) (fma (* -0.16666666666666666 im) im -1.0))
   (* (* im (fma (* 0.16666666666666666 re) re -1.0)) re)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(-im) - exp(im))) <= -1e-10) {
		tmp = (im * re) * fma((-0.16666666666666666 * im), im, -1.0);
	} else {
		tmp = (im * fma((0.16666666666666666 * re), re, -1.0)) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - exp(im))) <= -1e-10)
		tmp = Float64(Float64(im * re) * fma(Float64(-0.16666666666666666 * im), im, -1.0));
	else
		tmp = Float64(Float64(im * fma(Float64(0.16666666666666666 * re), re, -1.0)) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-10], N[(N[(im * re), $MachinePrecision] * N[(N[(-0.16666666666666666 * im), $MachinePrecision] * im + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(im * N[(N[(0.16666666666666666 * re), $MachinePrecision] * re + -1.0), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -1.00000000000000004e-10

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 100.0

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

      7. lift--.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. rem-exp-log N/A

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

      10. lift-exp.f64 N/A

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

      11. rem-log-exp N/A

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

      12. remove-double-neg N/A

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

      13. lift-neg.f64 N/A

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

      14. sinh-undef N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sinh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. mul-1-neg N/A

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

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

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

      13. mul-1-neg N/A

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

      14. distribute-rgt-out N/A

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

      15. +-commutative N/A

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

      16. lower-*.f64 N/A

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

   7. Applied rewrites 60.7%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 56.2%

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

   if -1.00000000000000004e-10 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 51.5%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sin.f64 68.7

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

   5. Applied rewrites 68.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 40.6%

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

4. Final simplification 44.2%

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

Alternative 12: 58.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot re\right) \cdot re, 0.008333333333333333, -0.16666666666666666 \cdot im\right), re \cdot re, im\right) \cdot re\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) 5e-8)
   (*
    (* (fma (* re re) -0.08333333333333333 0.5) re)
    (*
     (-
      (*
       (-
        (*
         (* (- (* -0.0003968253968253968 (* im im)) 0.016666666666666666) im)
         im)
        0.3333333333333333)
       (* im im))
      2.0)
     im))
   (*
    (*
     (fma
      (fma (* (* im re) re) 0.008333333333333333 (* -0.16666666666666666 im))
      (* re re)
      im)
     re)
    (fma (* -0.16666666666666666 im) im -1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= 5e-8) {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * re) * ((((((((-0.0003968253968253968 * (im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	} else {
		tmp = (fma(fma(((im * re) * re), 0.008333333333333333, (-0.16666666666666666 * im)), (re * re), im) * re) * fma((-0.16666666666666666 * im), im, -1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= 5e-8)
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * Float64(im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * Float64(im * im)) - 2.0) * im));
	else
		tmp = Float64(Float64(fma(fma(Float64(Float64(im * re) * re), 0.008333333333333333, Float64(-0.16666666666666666 * im)), Float64(re * re), im) * re) * fma(Float64(-0.16666666666666666 * im), im, -1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], 5e-8], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.016666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(im * re), $MachinePrecision] * re), $MachinePrecision] * 0.008333333333333333 + N[(-0.16666666666666666 * im), $MachinePrecision]), $MachinePrecision] * N[(re * re), $MachinePrecision] + im), $MachinePrecision] * re), $MachinePrecision] * N[(N[(-0.16666666666666666 * im), $MachinePrecision] * im + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 64.7%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 80.0

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

   5. Applied rewrites 80.0%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 58.2

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

   8. Applied rewrites 58.2%

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

   9. Taylor expanded in im around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   11. Applied rewrites 64.2%

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

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

   1. Initial program 55.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 55.6

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

      7. lift--.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. rem-exp-log N/A

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

      10. lift-exp.f64 N/A

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

      11. rem-log-exp N/A

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

      12. remove-double-neg N/A

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

      13. lift-neg.f64 N/A

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

      14. sinh-undef N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sinh.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. mul-1-neg N/A

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

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

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

      13. mul-1-neg N/A

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

      14. distribute-rgt-out N/A

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

      15. +-commutative N/A

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

      16. lower-*.f64 N/A

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

   7. Applied rewrites 81.3%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 27.8%

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

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot re\right) \cdot re, 0.008333333333333333, -0.16666666666666666 \cdot im\right), re \cdot re, im\right) \cdot re\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 56.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot re\right) \cdot re, 0.008333333333333333, -0.16666666666666666 \cdot im\right), re \cdot re, im\right) \cdot re\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) 5e-8)
   (*
    (* (fma (* re re) -0.08333333333333333 0.5) re)
    (*
     (-
      (* (* (- (* -0.016666666666666666 (* im im)) 0.3333333333333333) im) im)
      2.0)
     im))
   (*
    (*
     (fma
      (fma (* (* im re) re) 0.008333333333333333 (* -0.16666666666666666 im))
      (* re re)
      im)
     re)
    (fma (* -0.16666666666666666 im) im -1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= 5e-8) {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * re) * ((((((-0.016666666666666666 * (im * im)) - 0.3333333333333333) * im) * im) - 2.0) * im);
	} else {
		tmp = (fma(fma(((im * re) * re), 0.008333333333333333, (-0.16666666666666666 * im)), (re * re), im) * re) * fma((-0.16666666666666666 * im), im, -1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= 5e-8)
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re) * Float64(Float64(Float64(Float64(Float64(Float64(-0.016666666666666666 * Float64(im * im)) - 0.3333333333333333) * im) * im) - 2.0) * im));
	else
		tmp = Float64(Float64(fma(fma(Float64(Float64(im * re) * re), 0.008333333333333333, Float64(-0.16666666666666666 * im)), Float64(re * re), im) * re) * fma(Float64(-0.16666666666666666 * im), im, -1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], 5e-8], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(N[(N[(N[(-0.016666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(im * re), $MachinePrecision] * re), $MachinePrecision] * 0.008333333333333333 + N[(-0.16666666666666666 * im), $MachinePrecision]), $MachinePrecision] * N[(re * re), $MachinePrecision] + im), $MachinePrecision] * re), $MachinePrecision] * N[(N[(-0.16666666666666666 * im), $MachinePrecision] * im + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 64.7%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 87.1

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

   5. Applied rewrites 87.1%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 61.7

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

   8. Applied rewrites 61.7%

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

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

   1. Initial program 55.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 55.6

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

      7. lift--.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. rem-exp-log N/A

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

      10. lift-exp.f64 N/A

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

      11. rem-log-exp N/A

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

      12. remove-double-neg N/A

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

      13. lift-neg.f64 N/A

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

      14. sinh-undef N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sinh.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. mul-1-neg N/A

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

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

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

      13. mul-1-neg N/A

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

      14. distribute-rgt-out N/A

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

      15. +-commutative N/A

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

      16. lower-*.f64 N/A

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

   7. Applied rewrites 81.3%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 27.8%

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

4. Final simplification 53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot re\right) \cdot re, 0.008333333333333333, -0.16666666666666666 \cdot im\right), re \cdot re, im\right) \cdot re\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 56.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.004166666666666667 \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \cdot \left(\left(\left(im \cdot im\right) \cdot -0.3333333333333333 - 2\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) 5e-8)
   (*
    (* (fma (* re re) -0.08333333333333333 0.5) re)
    (*
     (-
      (* (* (- (* -0.016666666666666666 (* im im)) 0.3333333333333333) im) im)
      2.0)
     im))
   (*
    (*
     (fma
      (- (* 0.004166666666666667 (* re re)) 0.08333333333333333)
      (* re re)
      0.5)
     re)
    (* (- (* (* im im) -0.3333333333333333) 2.0) im))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= 5e-8) {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * re) * ((((((-0.016666666666666666 * (im * im)) - 0.3333333333333333) * im) * im) - 2.0) * im);
	} else {
		tmp = (fma(((0.004166666666666667 * (re * re)) - 0.08333333333333333), (re * re), 0.5) * re) * ((((im * im) * -0.3333333333333333) - 2.0) * im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= 5e-8)
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re) * Float64(Float64(Float64(Float64(Float64(Float64(-0.016666666666666666 * Float64(im * im)) - 0.3333333333333333) * im) * im) - 2.0) * im));
	else
		tmp = Float64(Float64(fma(Float64(Float64(0.004166666666666667 * Float64(re * re)) - 0.08333333333333333), Float64(re * re), 0.5) * re) * Float64(Float64(Float64(Float64(im * im) * -0.3333333333333333) - 2.0) * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], 5e-8], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(N[(N[(N[(-0.016666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.004166666666666667 * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.08333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(N[(im * im), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 64.7%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 87.1

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

   5. Applied rewrites 87.1%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 61.7

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

   8. Applied rewrites 61.7%

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

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

   1. Initial program 55.6%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 81.3

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

   5. Applied rewrites 81.3%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 27.8

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

   8. Applied rewrites 27.8%

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

Alternative 15: 56.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq 0.0002:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot re\right) \cdot \left(\left(\left(-0.008333333333333333 \cdot \left(im \cdot im\right) - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) 0.0002)
   (*
    (* (fma (* re re) -0.08333333333333333 0.5) re)
    (*
     (-
      (* (* (- (* -0.016666666666666666 (* im im)) 0.3333333333333333) im) im)
      2.0)
     im))
   (*
    (* im re)
    (-
     (* (* (- (* -0.008333333333333333 (* im im)) 0.16666666666666666) im) im)
     1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= 0.0002) {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * re) * ((((((-0.016666666666666666 * (im * im)) - 0.3333333333333333) * im) * im) - 2.0) * im);
	} else {
		tmp = (im * re) * (((((-0.008333333333333333 * (im * im)) - 0.16666666666666666) * im) * im) - 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= 0.0002)
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re) * Float64(Float64(Float64(Float64(Float64(Float64(-0.016666666666666666 * Float64(im * im)) - 0.3333333333333333) * im) * im) - 2.0) * im));
	else
		tmp = Float64(Float64(im * re) * Float64(Float64(Float64(Float64(Float64(-0.008333333333333333 * Float64(im * im)) - 0.16666666666666666) * im) * im) - 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], 0.0002], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(N[(N[(N[(-0.016666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[(im * re), $MachinePrecision] * N[(N[(N[(N[(N[(-0.008333333333333333 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < 2.0000000000000001e-4

   1. Initial program 64.9%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 87.1

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

   5. Applied rewrites 87.1%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 61.9

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

   8. Applied rewrites 61.9%

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

   if 2.0000000000000001e-4 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 54.9%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 82.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 22.3%

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

Alternative 16: 34.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq 0.0002:\\ \;\;\;\;\left(im \cdot \mathsf{fma}\left(0.16666666666666666 \cdot re, re, -1\right)\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(-re\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) 0.0002)
   (* (* im (fma (* 0.16666666666666666 re) re -1.0)) re)
   (* (- re) im)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= 0.0002) {
		tmp = (im * fma((0.16666666666666666 * re), re, -1.0)) * re;
	} else {
		tmp = -re * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= 0.0002)
		tmp = Float64(Float64(im * fma(Float64(0.16666666666666666 * re), re, -1.0)) * re);
	else
		tmp = Float64(Float64(-re) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], 0.0002], N[(N[(im * N[(N[(0.16666666666666666 * re), $MachinePrecision] * re + -1.0), $MachinePrecision]), $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)) < 2.0000000000000001e-4

   1. Initial program 64.9%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sin.f64 55.3

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

   5. Applied rewrites 55.3%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 39.1%

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

   if 2.0000000000000001e-4 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 54.9%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sin.f64 50.1

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

   5. Applied rewrites 50.1%

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

   6. Taylor expanded in re around 0

      \[\leadsto \left(-1 \cdot re\right) \cdot im \]
   7. Step-by-step derivation

   8. Applied rewrites 17.5%

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

Alternative 17: 34.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.01:\\ \;\;\;\;\left(\left(\left(im \cdot re\right) \cdot re\right) \cdot 0.16666666666666666\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(-re\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.01)
   (* (* (* (* im re) re) 0.16666666666666666) re)
   (* (- re) im)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.01) {
		tmp = (((im * re) * re) * 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)) <= (-0.01d0)) then
        tmp = (((im * re) * re) * 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)) <= -0.01) {
		tmp = (((im * re) * re) * 0.16666666666666666) * re;
	} else {
		tmp = -re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (0.5 * math.sin(re)) <= -0.01:
		tmp = (((im * re) * re) * 0.16666666666666666) * re
	else:
		tmp = -re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.01)
		tmp = Float64(Float64(Float64(Float64(im * re) * re) * 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)) <= -0.01)
		tmp = (((im * re) * re) * 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], -0.01], N[(N[(N[(N[(im * re), $MachinePrecision] * re), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * re), $MachinePrecision], N[((-re) * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < -0.0100000000000000002

   1. Initial program 46.6%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sin.f64 60.1

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

   5. Applied rewrites 60.1%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 13.6%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 13.7%

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

   if -0.0100000000000000002 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 68.2%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sin.f64 51.9

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

   5. Applied rewrites 51.9%

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

   6. Taylor expanded in re around 0

      \[\leadsto \left(-1 \cdot re\right) \cdot im \]
   7. Step-by-step derivation

   8. Applied rewrites 40.9%

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

Alternative 18: 33.4% accurate, 39.5× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

3. Taylor expanded in im around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. mul-1-neg N/A

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

   5. lower-neg.f64 N/A

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

   6. lower-sin.f64 54.1

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

5. Applied rewrites 54.1%

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

6. Taylor expanded in re around 0

   \[\leadsto \left(-1 \cdot re\right) \cdot im \]
7. Step-by-step derivation

8. Applied rewrites 34.3%

   \[\leadsto \left(-re\right) \cdot im \]
9. Add Preprocessing

Developer Target 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|im\right| < 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(0.16666666666666666 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (< (fabs im) 1.0)
   (-
    (*
     (sin re)
     (+
      (+ im (* (* (* 0.16666666666666666 im) im) im))
      (* (* (* (* (* 0.008333333333333333 im) im) im) im) im))))
   (* (* 0.5 (sin re)) (- (exp (- im)) (exp im)))))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (abs(im) < 1.0d0) then
        tmp = -(sin(re) * ((im + (((0.16666666666666666d0 * im) * im) * im)) + (((((0.008333333333333333d0 * im) * im) * im) * im) * im)))
    else
        tmp = (0.5d0 * sin(re)) * (exp(-im) - exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.abs(im) < 1.0) {
		tmp = -(Math.sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * Math.sin(re)) * (Math.exp(-im) - Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.fabs(im) < 1.0:
		tmp = -(math.sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)))
	else:
		tmp = (0.5 * math.sin(re)) * (math.exp(-im) - math.exp(im))
	return tmp

Julia

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

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (abs(im) < 1.0)
		tmp = -(sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	else
		tmp = (0.5 * sin(re)) * (exp(-im) - exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

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

Reproduce

herbie shell --seed 2024363 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (fabs im) 1) (- (* (sin re) (+ im (* 1/6 im im im) (* 1/120 im im im im im)))) (* (* 1/2 (sin re)) (- (exp (- im)) (exp im)))))

  (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))