math.cos on complex, imaginary part

Percentage Accurate: 66.4% → 99.9%

Time: 8.1s

Alternatives: 21

Speedup: 2.5×

• 50.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.4% 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 61.7%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
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. *-commutative N/A

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

   7. lower-*.f64 61.7

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

   8. lift--.f64 N/A

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

   9. lift-exp.f64 N/A

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

   10. remove-double-neg N/A

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

   11. lift-neg.f64 N/A

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

   12. lift-exp.f64 N/A

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

   13. sinh-undef N/A

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

   14. lower-*.f64 N/A

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

   15. lower-sinh.f64 99.9

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

4. Applied rewrites 99.9%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. associate-*r* N/A

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

   7. metadata-eval N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 99.9

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

6. Applied rewrites 99.9%

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

   1. lift-*.f64 N/A

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

   2. *-lft-identity 99.9

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 99.9

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

8. Applied rewrites 99.9%

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

Alternative 2: 83.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t\_0 \leq -0.002 \lor \neg \left(t\_0 \leq 2 \cdot 10^{-6}\right):\\ \;\;\;\;\left(0.5 \cdot \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(-0.0001984126984126984, re \cdot re, 0.008333333333333333\right) \cdot re\right) \cdot re - 0.16666666666666666\right) \cdot \left(re \cdot re\right), re, re\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right), im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\sin re\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 -0.002) (not (<= t_0 2e-6)))
     (*
      (*
       0.5
       (fma
        (*
         (-
          (*
           (* (fma -0.0001984126984126984 (* re re) 0.008333333333333333) re)
           re)
          0.16666666666666666)
         (* re re))
        re
        re))
      (*
       (fma
        (fma
         (fma -0.0003968253968253968 (* im im) -0.016666666666666666)
         (* im im)
         -0.3333333333333333)
        (* im im)
        -2.0)
       im))
     (* (- (sin re)) im))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double tmp;
	if ((t_0 <= -0.002) || !(t_0 <= 2e-6)) {
		tmp = (0.5 * fma(((((fma(-0.0001984126984126984, (re * re), 0.008333333333333333) * re) * re) - 0.16666666666666666) * (re * re)), re, re)) * (fma(fma(fma(-0.0003968253968253968, (im * im), -0.016666666666666666), (im * im), -0.3333333333333333), (im * im), -2.0) * im);
	} else {
		tmp = -sin(re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	tmp = 0.0
	if ((t_0 <= -0.002) || !(t_0 <= 2e-6))
		tmp = Float64(Float64(0.5 * fma(Float64(Float64(Float64(Float64(fma(-0.0001984126984126984, Float64(re * re), 0.008333333333333333) * re) * re) - 0.16666666666666666) * Float64(re * re)), re, re)) * Float64(fma(fma(fma(-0.0003968253968253968, Float64(im * im), -0.016666666666666666), Float64(im * im), -0.3333333333333333), Float64(im * im), -2.0) * im));
	else
		tmp = Float64(Float64(-sin(re)) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -0.002], N[Not[LessEqual[t$95$0, 2e-6]], $MachinePrecision]], N[(N[(0.5 * N[(N[(N[(N[(N[(N[(-0.0001984126984126984 * N[(re * re), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] * re + re), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(-0.0003968253968253968 * N[(im * im), $MachinePrecision] + -0.016666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + -2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[((-N[Sin[re], $MachinePrecision]) * im), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -2e-3 or 1.99999999999999991e-6 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 84.7%

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

   6. Taylor expanded in re around 0

      \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(re \cdot \left(1 + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)\right)\right)}\right) \cdot \left(\mathsf{fma}\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}, im \cdot im, -2\right) \cdot im\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(re \cdot \color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right) + 1\right)}\right)\right) \cdot \left(\mathsf{fma}\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}, im \cdot im, -2\right) \cdot im\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)\right) \cdot re + 1 \cdot re\right)}\right) \cdot \left(\mathsf{fma}\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}, im \cdot im, -2\right) \cdot im\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\color{blue}{re \cdot \left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)\right)} + 1 \cdot re\right)\right) \cdot \left(\mathsf{fma}\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}, im \cdot im, -2\right) \cdot im\right) \]

      4. *-lft-identity N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(re \cdot \left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)\right) + \color{blue}{re}\right)\right) \cdot \left(\mathsf{fma}\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}, im \cdot im, -2\right) \cdot im\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\color{blue}{\left(re \cdot {re}^{2}\right) \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)} + re\right)\right) \cdot \left(\mathsf{fma}\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}, im \cdot im, -2\right) \cdot im\right) \]

      6. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\left(re \cdot \color{blue}{\left(re \cdot re\right)}\right) \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right) + re\right)\right) \cdot \left(\mathsf{fma}\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}, im \cdot im, -2\right) \cdot im\right) \]

      7. cube-unmult N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\color{blue}{{re}^{3}} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right) + re\right)\right) \cdot \left(\mathsf{fma}\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}, im \cdot im, -2\right) \cdot im\right) \]

      8. metadata-eval N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left({re}^{\color{blue}{\left(2 + 1\right)}} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right) + re\right)\right) \cdot \left(\mathsf{fma}\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}, im \cdot im, -2\right) \cdot im\right) \]

      9. pow-plus N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\color{blue}{\left({re}^{2} \cdot re\right)} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right) + re\right)\right) \cdot \left(\mathsf{fma}\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}, im \cdot im, -2\right) \cdot im\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\mathsf{fma}\left({re}^{2} \cdot re, {re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}, re\right)}\right) \cdot \left(\mathsf{fma}\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}, im \cdot im, -2\right) \cdot im\right) \]

   8. Applied rewrites 72.8%

      \[\leadsto \left(0.5 \cdot \color{blue}{\mathsf{fma}\left({re}^{3}, \left(\mathsf{fma}\left(-0.0001984126984126984, re \cdot re, 0.008333333333333333\right) \cdot re\right) \cdot re - 0.16666666666666666, re\right)}\right) \cdot \left(\mathsf{fma}\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333, im \cdot im, -2\right) \cdot im\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 72.8%

       \[\leadsto \left(0.5 \cdot \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(-0.0001984126984126984, re \cdot re, 0.008333333333333333\right) \cdot re\right) \cdot re - 0.16666666666666666\right) \cdot \left(re \cdot re\right), \color{blue}{re}, re\right)\right) \cdot \left(\mathsf{fma}\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333, im \cdot im, -2\right) \cdot im\right) \]

   11. Taylor expanded in im around 0

       \[\leadsto \left(\frac{1}{2} \cdot \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\frac{-1}{5040}, re \cdot re, \frac{1}{120}\right) \cdot re\right) \cdot re - \frac{1}{6}\right) \cdot \left(re \cdot re\right), re, re\right)\right) \cdot \left(\mathsf{fma}\left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}, im \cdot im, -2\right) \cdot im\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 72.8%

       \[\leadsto \left(0.5 \cdot \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(-0.0001984126984126984, re \cdot re, 0.008333333333333333\right) \cdot re\right) \cdot re - 0.16666666666666666\right) \cdot \left(re \cdot re\right), re, re\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right), im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \]

   if -2e-3 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < 1.99999999999999991e-6

   1. Initial program 28.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   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 99.8

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

   5. Applied rewrites 99.8%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -0.002 \lor \neg \left(e^{-im} - e^{im} \leq 2 \cdot 10^{-6}\right):\\ \;\;\;\;\left(0.5 \cdot \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(-0.0001984126984126984, re \cdot re, 0.008333333333333333\right) \cdot re\right) \cdot re - 0.16666666666666666\right) \cdot \left(re \cdot re\right), re, re\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right), im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\sin re\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t\_0 \leq -0.002:\\ \;\;\;\;\left(2 \cdot \sinh \left(-im\right)\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\left(-\sin re\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(-0.0001984126984126984, re \cdot re, 0.008333333333333333\right) \cdot re\right) \cdot re - 0.16666666666666666\right) \cdot \left(re \cdot re\right), re, re\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right), im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (<= t_0 -0.002)
     (* (* 2.0 (sinh (- im))) (* 0.5 re))
     (if (<= t_0 2e-6)
       (* (- (sin re)) im)
       (*
        (*
         0.5
         (fma
          (*
           (-
            (*
             (* (fma -0.0001984126984126984 (* re re) 0.008333333333333333) re)
             re)
            0.16666666666666666)
           (* re re))
          re
          re))
        (*
         (fma
          (fma
           (fma -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 = exp(-im) - exp(im);
	double tmp;
	if (t_0 <= -0.002) {
		tmp = (2.0 * sinh(-im)) * (0.5 * re);
	} else if (t_0 <= 2e-6) {
		tmp = -sin(re) * im;
	} else {
		tmp = (0.5 * fma(((((fma(-0.0001984126984126984, (re * re), 0.008333333333333333) * re) * re) - 0.16666666666666666) * (re * re)), re, re)) * (fma(fma(fma(-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(exp(Float64(-im)) - exp(im))
	tmp = 0.0
	if (t_0 <= -0.002)
		tmp = Float64(Float64(2.0 * sinh(Float64(-im))) * Float64(0.5 * re));
	elseif (t_0 <= 2e-6)
		tmp = Float64(Float64(-sin(re)) * im);
	else
		tmp = Float64(Float64(0.5 * fma(Float64(Float64(Float64(Float64(fma(-0.0001984126984126984, Float64(re * re), 0.008333333333333333) * re) * re) - 0.16666666666666666) * Float64(re * re)), re, re)) * Float64(fma(fma(fma(-0.0003968253968253968, Float64(im * im), -0.016666666666666666), Float64(im * im), -0.3333333333333333), Float64(im * im), -2.0) * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.002], N[(N[(2.0 * N[Sinh[(-im)], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-6], N[((-N[Sin[re], $MachinePrecision]) * im), $MachinePrecision], N[(N[(0.5 * N[(N[(N[(N[(N[(N[(-0.0001984126984126984 * N[(re * re), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] * re + re), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(-0.0003968253968253968 * N[(im * im), $MachinePrecision] + -0.016666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + -2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -2e-3

   1. Initial program 99.8%

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

      1. lift-exp.f64 N/A

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

      2. remove-double-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. lift-exp.f64 N/A

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

      6. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in re around 0

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

      1. lower-*.f64 80.8

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

   7. Applied rewrites 80.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. exp-neg N/A

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

      8. remove-double-div N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. lower-*.f64 N/A

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

   9. Applied rewrites 80.9%

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

   if -2e-3 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < 1.99999999999999991e-6

   1. Initial program 28.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   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 99.8

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

   5. Applied rewrites 99.8%

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

   if 1.99999999999999991e-6 < (-.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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 87.3%

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

   6. Taylor expanded in re around 0

      \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(re \cdot \left(1 + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)\right)\right)}\right) \cdot \left(\mathsf{fma}\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}, im \cdot im, -2\right) \cdot im\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(re \cdot \color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right) + 1\right)}\right)\right) \cdot \left(\mathsf{fma}\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}, im \cdot im, -2\right) \cdot im\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)\right) \cdot re + 1 \cdot re\right)}\right) \cdot \left(\mathsf{fma}\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}, im \cdot im, -2\right) \cdot im\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\color{blue}{re \cdot \left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)\right)} + 1 \cdot re\right)\right) \cdot \left(\mathsf{fma}\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}, im \cdot im, -2\right) \cdot im\right) \]

      4. *-lft-identity N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(re \cdot \left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)\right) + \color{blue}{re}\right)\right) \cdot \left(\mathsf{fma}\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}, im \cdot im, -2\right) \cdot im\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\color{blue}{\left(re \cdot {re}^{2}\right) \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)} + re\right)\right) \cdot \left(\mathsf{fma}\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}, im \cdot im, -2\right) \cdot im\right) \]

      6. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\left(re \cdot \color{blue}{\left(re \cdot re\right)}\right) \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right) + re\right)\right) \cdot \left(\mathsf{fma}\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}, im \cdot im, -2\right) \cdot im\right) \]

      7. cube-unmult N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\color{blue}{{re}^{3}} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right) + re\right)\right) \cdot \left(\mathsf{fma}\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}, im \cdot im, -2\right) \cdot im\right) \]

      8. metadata-eval N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left({re}^{\color{blue}{\left(2 + 1\right)}} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right) + re\right)\right) \cdot \left(\mathsf{fma}\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}, im \cdot im, -2\right) \cdot im\right) \]

      9. pow-plus N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\color{blue}{\left({re}^{2} \cdot re\right)} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right) + re\right)\right) \cdot \left(\mathsf{fma}\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}, im \cdot im, -2\right) \cdot im\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\mathsf{fma}\left({re}^{2} \cdot re, {re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}, re\right)}\right) \cdot \left(\mathsf{fma}\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}, im \cdot im, -2\right) \cdot im\right) \]

   8. Applied rewrites 76.4%

      \[\leadsto \left(0.5 \cdot \color{blue}{\mathsf{fma}\left({re}^{3}, \left(\mathsf{fma}\left(-0.0001984126984126984, re \cdot re, 0.008333333333333333\right) \cdot re\right) \cdot re - 0.16666666666666666, re\right)}\right) \cdot \left(\mathsf{fma}\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333, im \cdot im, -2\right) \cdot im\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 76.4%

       \[\leadsto \left(0.5 \cdot \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(-0.0001984126984126984, re \cdot re, 0.008333333333333333\right) \cdot re\right) \cdot re - 0.16666666666666666\right) \cdot \left(re \cdot re\right), \color{blue}{re}, re\right)\right) \cdot \left(\mathsf{fma}\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333, im \cdot im, -2\right) \cdot im\right) \]

   11. Taylor expanded in im around 0

       \[\leadsto \left(\frac{1}{2} \cdot \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\frac{-1}{5040}, re \cdot re, \frac{1}{120}\right) \cdot re\right) \cdot re - \frac{1}{6}\right) \cdot \left(re \cdot re\right), re, re\right)\right) \cdot \left(\mathsf{fma}\left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}, im \cdot im, -2\right) \cdot im\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 76.4%

       \[\leadsto \left(0.5 \cdot \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(-0.0001984126984126984, re \cdot re, 0.008333333333333333\right) \cdot re\right) \cdot re - 0.16666666666666666\right) \cdot \left(re \cdot re\right), re, re\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right), im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 59.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq 10^{-6}:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right), im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(\mathsf{fma}\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333, im \cdot im, -2\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) 1e-6)
   (*
    (* (fma (* re re) -0.08333333333333333 0.5) re)
    (*
     (fma
      (fma
       (fma -0.0003968253968253968 (* im im) -0.016666666666666666)
       (* im im)
       -0.3333333333333333)
      (* im im)
      -2.0)
     im))
   (*
    (* 0.5 re)
    (*
     (fma
      (-
       (*
        (* (- (* -0.0003968253968253968 (* im im)) 0.016666666666666666) im)
        im)
       0.3333333333333333)
      (* im im)
      -2.0)
     im))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= 1e-6) {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * re) * (fma(fma(fma(-0.0003968253968253968, (im * im), -0.016666666666666666), (im * im), -0.3333333333333333), (im * im), -2.0) * im);
	} else {
		tmp = (0.5 * re) * (fma((((((-0.0003968253968253968 * (im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333), (im * im), -2.0) * im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= 1e-6)
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re) * Float64(fma(fma(fma(-0.0003968253968253968, Float64(im * im), -0.016666666666666666), Float64(im * im), -0.3333333333333333), Float64(im * im), -2.0) * im));
	else
		tmp = Float64(Float64(0.5 * re) * Float64(fma(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * Float64(im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333), Float64(im * im), -2.0) * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], 1e-6], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(N[(-0.0003968253968253968 * N[(im * im), $MachinePrecision] + -0.016666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + -2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * 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] + -2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < 9.99999999999999955e-7

   1. Initial program 67.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   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. metadata-eval N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 84.7

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

   5. Applied rewrites 84.7%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 63.5

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

   8. Applied rewrites 63.5%

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

   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 69.6%

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

   if 9.99999999999999955e-7 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 43.7%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 91.9%

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

   6. Taylor expanded in re around 0

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

      1. lower-*.f64 19.4

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

   8. Applied rewrites 19.4%

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

Alternative 5: 52.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq 5 \cdot 10^{-295}:\\ \;\;\;\;\left(\left(\left(\left(\frac{0.5}{re \cdot re} - 0.08333333333333333\right) \cdot re\right) \cdot re\right) \cdot re\right) \cdot \left(\mathsf{fma}\left(-0.3333333333333333, im \cdot im, -2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(\mathsf{fma}\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333, im \cdot im, -2\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) 5e-295)
   (*
    (* (* (* (- (/ 0.5 (* re re)) 0.08333333333333333) re) re) re)
    (* (fma -0.3333333333333333 (* im im) -2.0) im))
   (*
    (* 0.5 re)
    (*
     (fma
      (-
       (*
        (* (- (* -0.0003968253968253968 (* im im)) 0.016666666666666666) im)
        im)
       0.3333333333333333)
      (* im im)
      -2.0)
     im))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= 5e-295) {
		tmp = (((((0.5 / (re * re)) - 0.08333333333333333) * re) * re) * re) * (fma(-0.3333333333333333, (im * im), -2.0) * im);
	} else {
		tmp = (0.5 * re) * (fma((((((-0.0003968253968253968 * (im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333), (im * im), -2.0) * im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= 5e-295)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(0.5 / Float64(re * re)) - 0.08333333333333333) * re) * re) * re) * Float64(fma(-0.3333333333333333, Float64(im * im), -2.0) * im));
	else
		tmp = Float64(Float64(0.5 * re) * Float64(fma(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * Float64(im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333), Float64(im * im), -2.0) * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], 5e-295], N[(N[(N[(N[(N[(N[(0.5 / N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.08333333333333333), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision] * N[(N[(-0.3333333333333333 * N[(im * im), $MachinePrecision] + -2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * 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] + -2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < 5.00000000000000008e-295

   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

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 2\right)\right)} \]
   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. metadata-eval N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 84.0

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

   5. Applied rewrites 84.0%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 54.2

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

   8. Applied rewrites 54.2%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 47.0%

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

   if 5.00000000000000008e-295 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 60.6%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 92.6%

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

   6. Taylor expanded in re around 0

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

      1. lower-*.f64 55.7

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

   8. Applied rewrites 55.7%

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

Alternative 6: 79.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 0.56:\\ \;\;\;\;\left(2 \cdot \sinh \left(-im\right)\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot im\right) \cdot im, im, -2 \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 0.56)
   (* (* 2.0 (sinh (- im))) (* 0.5 re))
   (*
    (* 0.5 (sin re))
    (fma
     (*
      (*
       (-
        (*
         (* (- (* -0.0003968253968253968 (* im im)) 0.016666666666666666) im)
         im)
        0.3333333333333333)
       im)
      im)
     im
     (* -2.0 im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 0.56) {
		tmp = (2.0 * sinh(-im)) * (0.5 * re);
	} else {
		tmp = (0.5 * sin(re)) * fma((((((((-0.0003968253968253968 * (im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * im) * im), im, (-2.0 * im));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 0.56)
		tmp = Float64(Float64(2.0 * sinh(Float64(-im))) * Float64(0.5 * re));
	else
		tmp = Float64(Float64(0.5 * sin(re)) * fma(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * Float64(im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * im) * im), im, Float64(-2.0 * im)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, 0.56], N[(N[(2.0 * N[Sinh[(-im)], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $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] * im), $MachinePrecision] * im), $MachinePrecision] * im + N[(-2.0 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 0.56000000000000005

   1. Initial program 66.9%

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

      1. lift-exp.f64 N/A

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

      2. remove-double-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. lift-exp.f64 N/A

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

      6. lower-/.f64 66.9

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

   4. Applied rewrites 66.9%

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

   5. Taylor expanded in re around 0

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

      1. lower-*.f64 60.8

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

   7. Applied rewrites 60.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. exp-neg N/A

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

      8. remove-double-div N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. lower-*.f64 N/A

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

   9. Applied rewrites 75.1%

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

   if 0.56000000000000005 < re

   1. Initial program 45.9%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 92.3%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333, im \cdot im, -2\right) \cdot im\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 92.3%

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

Alternative 7: 59.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.002:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\mathsf{fma}\left(-0.3333333333333333, im \cdot im, -2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(\mathsf{fma}\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333, im \cdot im, -2\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.002], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(-0.3333333333333333 * N[(im * im), $MachinePrecision] + -2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * 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] + -2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 46.1%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      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. metadata-eval N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 83.7

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

   5. Applied rewrites 83.7%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 22.3

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

   8. Applied rewrites 22.3%

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

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

   1. Initial program 67.3%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 93.2%

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

   6. Taylor expanded in re around 0

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

      1. lower-*.f64 70.4

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

   8. Applied rewrites 70.4%

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

Alternative 8: 79.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 0.56:\\ \;\;\;\;\left(2 \cdot \sinh \left(-im\right)\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right) \cdot im\right) \cdot im - 0.16666666666666666, im \cdot im, -1\right) \cdot im\right) \cdot \sin re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 0.56)
   (* (* 2.0 (sinh (- im))) (* 0.5 re))
   (*
    (*
     (fma
      (-
       (*
        (* (fma (* im im) -0.0001984126984126984 -0.008333333333333333) im)
        im)
       0.16666666666666666)
      (* im im)
      -1.0)
     im)
    (sin re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 0.56) {
		tmp = (2.0 * sinh(-im)) * (0.5 * re);
	} else {
		tmp = (fma((((fma((im * im), -0.0001984126984126984, -0.008333333333333333) * im) * im) - 0.16666666666666666), (im * im), -1.0) * im) * sin(re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 0.56)
		tmp = Float64(Float64(2.0 * sinh(Float64(-im))) * Float64(0.5 * re));
	else
		tmp = Float64(Float64(fma(Float64(Float64(Float64(fma(Float64(im * im), -0.0001984126984126984, -0.008333333333333333) * im) * im) - 0.16666666666666666), Float64(im * im), -1.0) * im) * sin(re));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, 0.56], N[(N[(2.0 * N[Sinh[(-im)], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(N[(im * im), $MachinePrecision] * -0.0001984126984126984 + -0.008333333333333333), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision] + -1.0), $MachinePrecision] * im), $MachinePrecision] * N[Sin[re], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 0.56000000000000005

   1. Initial program 66.9%

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

      1. lift-exp.f64 N/A

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

      2. remove-double-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. lift-exp.f64 N/A

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

      6. lower-/.f64 66.9

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

   4. Applied rewrites 66.9%

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

   5. Taylor expanded in re around 0

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

      1. lower-*.f64 60.8

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

   7. Applied rewrites 60.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. exp-neg N/A

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

      8. remove-double-div N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. lower-*.f64 N/A

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

   9. Applied rewrites 75.1%

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

   if 0.56000000000000005 < re

   1. Initial program 45.9%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   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. *-commutative N/A

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

      7. lower-*.f64 45.9

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

      8. lift--.f64 N/A

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

      9. lift-exp.f64 N/A

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

      10. remove-double-neg N/A

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

      11. lift-neg.f64 N/A

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

      12. lift-exp.f64 N/A

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

      13. sinh-undef N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sinh.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.8

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

   6. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 99.8

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 99.8

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

   8. Applied rewrites 99.8%

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

   9. Taylor expanded in im around 0

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

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) - 1\right) \cdot im\right)} \cdot \sin re \]

       2. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) - 1\right) \cdot im\right)} \cdot \sin re \]

   11. Applied rewrites 92.3%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right) \cdot im\right) \cdot im - 0.16666666666666666, im \cdot im, -1\right) \cdot im\right)} \cdot \sin re \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 57.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.002:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\mathsf{fma}\left(-0.3333333333333333, im \cdot im, -2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.008333333333333333, -0.16666666666666666\right), im \cdot im, -1\right) \cdot im\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.002)
   (*
    (* (fma (* re re) -0.08333333333333333 0.5) re)
    (* (fma -0.3333333333333333 (* im im) -2.0) im))
   (*
    (*
     (fma
      (fma (* im im) -0.008333333333333333 -0.16666666666666666)
      (* im im)
      -1.0)
     im)
    re)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.002) {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * re) * (fma(-0.3333333333333333, (im * im), -2.0) * im);
	} else {
		tmp = (fma(fma((im * im), -0.008333333333333333, -0.16666666666666666), (im * im), -1.0) * im) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.002)
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re) * Float64(fma(-0.3333333333333333, Float64(im * im), -2.0) * im));
	else
		tmp = Float64(Float64(fma(fma(Float64(im * im), -0.008333333333333333, -0.16666666666666666), Float64(im * im), -1.0) * im) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.002], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(-0.3333333333333333 * N[(im * im), $MachinePrecision] + -2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(im * im), $MachinePrecision] * -0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision] + -1.0), $MachinePrecision] * im), $MachinePrecision] * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 46.1%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      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. metadata-eval N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 83.7

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

   5. Applied rewrites 83.7%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 22.3

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

   8. Applied rewrites 22.3%

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

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

   1. Initial program 67.3%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   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 88.0%

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

   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 67.7%

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

Alternative 10: 57.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.002:\\ \;\;\;\;\left(\left(\left(re \cdot re\right) \cdot -0.08333333333333333\right) \cdot re\right) \cdot \left(\left(-0.3333333333333333 \cdot \left(im \cdot im\right)\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.008333333333333333, -0.16666666666666666\right), im \cdot im, -1\right) \cdot im\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.002)
   (*
    (* (* (* re re) -0.08333333333333333) re)
    (* (* -0.3333333333333333 (* im im)) im))
   (*
    (*
     (fma
      (fma (* im im) -0.008333333333333333 -0.16666666666666666)
      (* im im)
      -1.0)
     im)
    re)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.002) {
		tmp = (((re * re) * -0.08333333333333333) * re) * ((-0.3333333333333333 * (im * im)) * im);
	} else {
		tmp = (fma(fma((im * im), -0.008333333333333333, -0.16666666666666666), (im * im), -1.0) * im) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.002)
		tmp = Float64(Float64(Float64(Float64(re * re) * -0.08333333333333333) * re) * Float64(Float64(-0.3333333333333333 * Float64(im * im)) * im));
	else
		tmp = Float64(Float64(fma(fma(Float64(im * im), -0.008333333333333333, -0.16666666666666666), Float64(im * im), -1.0) * im) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.002], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333), $MachinePrecision] * re), $MachinePrecision] * N[(N[(-0.3333333333333333 * N[(im * im), $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(im * im), $MachinePrecision] * -0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision] + -1.0), $MachinePrecision] * im), $MachinePrecision] * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 46.1%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      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. metadata-eval N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 83.7

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

   5. Applied rewrites 83.7%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 22.3

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

   8. Applied rewrites 22.3%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 22.2%

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

   12. Taylor expanded in re around inf

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

   14. Applied rewrites 22.1%

       \[\leadsto \left(\left(\left(re \cdot re\right) \cdot -0.08333333333333333\right) \cdot re\right) \cdot \left(\left(-0.3333333333333333 \cdot \left(im \cdot im\right)\right) \cdot im\right) \]

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

   1. Initial program 67.3%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   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 88.0%

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

   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 67.7%

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

Alternative 11: 78.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 0.56:\\ \;\;\;\;\left(2 \cdot \sinh \left(-im\right)\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(\mathsf{fma}\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333, im \cdot im, -2\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 0.56)
   (* (* 2.0 (sinh (- im))) (* 0.5 re))
   (*
    (* 0.5 (sin re))
    (*
     (fma
      (- (* -0.016666666666666666 (* im im)) 0.3333333333333333)
      (* im im)
      -2.0)
     im))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 0.56) {
		tmp = (2.0 * sinh(-im)) * (0.5 * re);
	} else {
		tmp = (0.5 * sin(re)) * (fma(((-0.016666666666666666 * (im * im)) - 0.3333333333333333), (im * im), -2.0) * im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 0.56)
		tmp = Float64(Float64(2.0 * sinh(Float64(-im))) * Float64(0.5 * re));
	else
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(fma(Float64(Float64(-0.016666666666666666 * Float64(im * im)) - 0.3333333333333333), Float64(im * im), -2.0) * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, 0.56], N[(N[(2.0 * N[Sinh[(-im)], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(-0.016666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + -2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 0.56000000000000005

   1. Initial program 66.9%

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

      1. lift-exp.f64 N/A

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

      2. remove-double-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. lift-exp.f64 N/A

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

      6. lower-/.f64 66.9

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

   4. Applied rewrites 66.9%

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

   5. Taylor expanded in re around 0

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

      1. lower-*.f64 60.8

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

   7. Applied rewrites 60.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. exp-neg N/A

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

      8. remove-double-div N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. lower-*.f64 N/A

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

   9. Applied rewrites 75.1%

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

   if 0.56000000000000005 < re

   1. Initial program 45.9%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   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. metadata-eval N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 88.9

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

   5. Applied rewrites 88.9%

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

Alternative 12: 56.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.002:\\ \;\;\;\;\mathsf{fma}\left(\left(re \cdot im\right) \cdot re, 0.16666666666666666, -im\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.008333333333333333, -0.16666666666666666\right), im \cdot im, -1\right) \cdot im\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.002)
   (* (fma (* (* re im) re) 0.16666666666666666 (- im)) re)
   (*
    (*
     (fma
      (fma (* im im) -0.008333333333333333 -0.16666666666666666)
      (* im im)
      -1.0)
     im)
    re)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.002) {
		tmp = fma(((re * im) * re), 0.16666666666666666, -im) * re;
	} else {
		tmp = (fma(fma((im * im), -0.008333333333333333, -0.16666666666666666), (im * im), -1.0) * im) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.002)
		tmp = Float64(fma(Float64(Float64(re * im) * re), 0.16666666666666666, Float64(-im)) * re);
	else
		tmp = Float64(Float64(fma(fma(Float64(im * im), -0.008333333333333333, -0.16666666666666666), Float64(im * im), -1.0) * im) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.002], N[(N[(N[(N[(re * im), $MachinePrecision] * re), $MachinePrecision] * 0.16666666666666666 + (-im)), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(N[(N[(im * im), $MachinePrecision] * -0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision] + -1.0), $MachinePrecision] * im), $MachinePrecision] * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 46.1%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      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 60.0

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

   5. Applied rewrites 60.0%

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

   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 20.8%

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

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

   1. Initial program 67.3%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   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 88.0%

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

   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 67.7%

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

Alternative 13: 78.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 0.56:\\ \;\;\;\;\left(2 \cdot \sinh \left(-im\right)\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right), -1\right)\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 0.56)
   (* (* 2.0 (sinh (- im))) (* 0.5 re))
   (*
    (*
     (sin re)
     (fma
      (* im im)
      (fma -0.008333333333333333 (* im im) -0.16666666666666666)
      -1.0))
    im)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 0.56) {
		tmp = (2.0 * sinh(-im)) * (0.5 * re);
	} else {
		tmp = (sin(re) * fma((im * im), fma(-0.008333333333333333, (im * im), -0.16666666666666666), -1.0)) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 0.56)
		tmp = Float64(Float64(2.0 * sinh(Float64(-im))) * Float64(0.5 * re));
	else
		tmp = Float64(Float64(sin(re) * fma(Float64(im * im), fma(-0.008333333333333333, Float64(im * im), -0.16666666666666666), -1.0)) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, 0.56], N[(N[(2.0 * N[Sinh[(-im)], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], 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]]

Derivation

1. Split input into 2 regimes

2. if re < 0.56000000000000005

   1. Initial program 66.9%

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

      1. lift-exp.f64 N/A

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

      2. remove-double-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. lift-exp.f64 N/A

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

      6. lower-/.f64 66.9

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

   4. Applied rewrites 66.9%

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

   5. Taylor expanded in re around 0

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

      1. lower-*.f64 60.8

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

   7. Applied rewrites 60.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. exp-neg N/A

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

      8. remove-double-div N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. lower-*.f64 N/A

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

   9. Applied rewrites 75.1%

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

   if 0.56000000000000005 < re

   1. Initial program 45.9%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   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 88.9%

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

Alternative 14: 53.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.002:\\ \;\;\;\;\mathsf{fma}\left(\left(re \cdot im\right) \cdot re, 0.16666666666666666, -im\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(\mathsf{fma}\left(-0.3333333333333333, im \cdot im, -2\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.002)
   (* (fma (* (* re im) re) 0.16666666666666666 (- im)) re)
   (* (* 0.5 re) (* (fma -0.3333333333333333 (* im im) -2.0) im))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.002) {
		tmp = fma(((re * im) * re), 0.16666666666666666, -im) * re;
	} else {
		tmp = (0.5 * re) * (fma(-0.3333333333333333, (im * im), -2.0) * im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.002)
		tmp = Float64(fma(Float64(Float64(re * im) * re), 0.16666666666666666, Float64(-im)) * re);
	else
		tmp = Float64(Float64(0.5 * re) * Float64(fma(-0.3333333333333333, Float64(im * im), -2.0) * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.002], N[(N[(N[(N[(re * im), $MachinePrecision] * re), $MachinePrecision] * 0.16666666666666666 + (-im)), $MachinePrecision] * re), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(N[(-0.3333333333333333 * N[(im * im), $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)) < -2e-3

   1. Initial program 46.1%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      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 60.0

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

   5. Applied rewrites 60.0%

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

   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 20.8%

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

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

   1. Initial program 67.3%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 2\right)\right)} \]
   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. metadata-eval N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 84.8

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

   5. Applied rewrites 84.8%

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

   6. Taylor expanded in re around 0

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

      1. lower-*.f64 64.0

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

   8. Applied rewrites 64.0%

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

Alternative 15: 34.9% accurate, 2.3× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) 1e-6)
   (* (fma (* (* re im) re) 0.16666666666666666 (- im)) re)
   (* (- re) im)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= 1e-6) {
		tmp = fma(((re * im) * re), 0.16666666666666666, -im) * re;
	} else {
		tmp = -re * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= 1e-6)
		tmp = Float64(fma(Float64(Float64(re * im) * re), 0.16666666666666666, Float64(-im)) * re);
	else
		tmp = Float64(Float64(-re) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], 1e-6], N[(N[(N[(N[(re * im), $MachinePrecision] * re), $MachinePrecision] * 0.16666666666666666 + (-im)), $MachinePrecision] * re), $MachinePrecision], N[((-re) * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < 9.99999999999999955e-7

   1. Initial program 67.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   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 53.3

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

   5. Applied rewrites 53.3%

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

   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 39.8%

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

   if 9.99999999999999955e-7 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 43.7%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      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 62.5

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

   5. Applied rewrites 62.5%

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

   6. Taylor expanded in re around 0

      \[\leadsto \left(-1 \cdot re\right) \cdot im \]
   7. Step-by-step derivation

   8. Applied rewrites 8.3%

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

Alternative 16: 34.9% accurate, 2.4× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) 1e-6)
   (* (* (fma 0.16666666666666666 (* re re) -1.0) re) im)
   (* (- re) im)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= 1e-6) {
		tmp = (fma(0.16666666666666666, (re * re), -1.0) * re) * im;
	} else {
		tmp = -re * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= 1e-6)
		tmp = Float64(Float64(fma(0.16666666666666666, Float64(re * re), -1.0) * re) * im);
	else
		tmp = Float64(Float64(-re) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], 1e-6], N[(N[(N[(0.16666666666666666 * N[(re * re), $MachinePrecision] + -1.0), $MachinePrecision] * re), $MachinePrecision] * im), $MachinePrecision], N[((-re) * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < 9.99999999999999955e-7

   1. Initial program 67.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   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 53.3

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

   5. Applied rewrites 53.3%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 39.7%

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

   if 9.99999999999999955e-7 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 43.7%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      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 62.5

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

   5. Applied rewrites 62.5%

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

   6. Taylor expanded in re around 0

      \[\leadsto \left(-1 \cdot re\right) \cdot im \]
   7. Step-by-step derivation

   8. Applied rewrites 8.3%

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

Alternative 17: 34.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.002:\\ \;\;\;\;\left(\left(0.16666666666666666 \cdot \left(re \cdot re\right)\right) \cdot re\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(-re\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.002)
   (* (* (* 0.16666666666666666 (* re re)) re) im)
   (* (- re) im)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.002) {
		tmp = ((0.16666666666666666 * (re * re)) * re) * im;
	} 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)) <= (-0.002d0)) then
        tmp = ((0.16666666666666666d0 * (re * re)) * re) * im
    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)) <= -0.002) {
		tmp = ((0.16666666666666666 * (re * re)) * re) * im;
	} else {
		tmp = -re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (0.5 * math.sin(re)) <= -0.002:
		tmp = ((0.16666666666666666 * (re * re)) * re) * im
	else:
		tmp = -re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.002)
		tmp = Float64(Float64(Float64(0.16666666666666666 * Float64(re * re)) * re) * im);
	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)) <= -0.002)
		tmp = ((0.16666666666666666 * (re * re)) * re) * im;
	else
		tmp = -re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.002], N[(N[(N[(0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision] * im), $MachinePrecision], N[((-re) * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 46.1%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      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 60.0

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

   5. Applied rewrites 60.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 20.7%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 20.7%

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

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

   1. Initial program 67.3%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   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 53.8

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

   5. Applied rewrites 53.8%

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

   6. Taylor expanded in re around 0

      \[\leadsto \left(-1 \cdot re\right) \cdot im \]
   7. Step-by-step derivation

   8. Applied rewrites 36.7%

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

Alternative 18: 77.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 2.5:\\ \;\;\;\;\left(2 \cdot \sinh \left(-im\right)\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(im \cdot im, -0.16666666666666666, -1\right) \cdot im\right) \cdot \sin re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 2.5)
   (* (* 2.0 (sinh (- im))) (* 0.5 re))
   (* (* (fma (* im im) -0.16666666666666666 -1.0) im) (sin re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 2.5) {
		tmp = (2.0 * sinh(-im)) * (0.5 * re);
	} else {
		tmp = (fma((im * im), -0.16666666666666666, -1.0) * im) * sin(re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 2.5)
		tmp = Float64(Float64(2.0 * sinh(Float64(-im))) * Float64(0.5 * re));
	else
		tmp = Float64(Float64(fma(Float64(im * im), -0.16666666666666666, -1.0) * im) * sin(re));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, 2.5], N[(N[(2.0 * N[Sinh[(-im)], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666 + -1.0), $MachinePrecision] * im), $MachinePrecision] * N[Sin[re], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 2.5

   1. Initial program 66.9%

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

      1. lift-exp.f64 N/A

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

      2. remove-double-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. lift-exp.f64 N/A

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

      6. lower-/.f64 66.9

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

   4. Applied rewrites 66.9%

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

   5. Taylor expanded in re around 0

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

      1. lower-*.f64 60.8

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

   7. Applied rewrites 60.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. exp-neg N/A

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

      8. remove-double-div N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. lower-*.f64 N/A

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

   9. Applied rewrites 75.1%

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

   if 2.5 < re

   1. Initial program 45.9%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   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. *-commutative N/A

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

      7. lower-*.f64 45.9

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

      8. lift--.f64 N/A

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

      9. lift-exp.f64 N/A

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

      10. remove-double-neg N/A

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

      11. lift-neg.f64 N/A

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

      12. lift-exp.f64 N/A

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

      13. sinh-undef N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sinh.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.8

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

   6. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 99.8

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 99.8

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

   8. Applied rewrites 99.8%

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

   9. Taylor expanded in im around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. metadata-eval N/A

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

       4. fp-cancel-sub-sign-inv N/A

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

       5. metadata-eval N/A

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

       6. *-commutative N/A

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

       7. metadata-eval N/A

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

       8. lower-fma.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 84.2

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

   11. Applied rewrites 84.2%

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

Alternative 19: 76.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 2.5:\\ \;\;\;\;\left(2 \cdot \sinh \left(-im\right)\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin re \cdot im\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 2.5)
   (* (* 2.0 (sinh (- im))) (* 0.5 re))
   (* (* (sin re) im) (fma (* -0.16666666666666666 im) im -1.0))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 2.5) {
		tmp = (2.0 * sinh(-im)) * (0.5 * re);
	} else {
		tmp = (sin(re) * im) * fma((-0.16666666666666666 * im), im, -1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 2.5)
		tmp = Float64(Float64(2.0 * sinh(Float64(-im))) * Float64(0.5 * re));
	else
		tmp = Float64(Float64(sin(re) * im) * fma(Float64(-0.16666666666666666 * im), im, -1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, 2.5], N[(N[(2.0 * N[Sinh[(-im)], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[re], $MachinePrecision] * im), $MachinePrecision] * N[(N[(-0.16666666666666666 * im), $MachinePrecision] * im + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 2.5

   1. Initial program 66.9%

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

      1. lift-exp.f64 N/A

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

      2. remove-double-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. lift-exp.f64 N/A

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

      6. lower-/.f64 66.9

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

   4. Applied rewrites 66.9%

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

   5. Taylor expanded in re around 0

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

      1. lower-*.f64 60.8

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

   7. Applied rewrites 60.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. exp-neg N/A

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

      8. remove-double-div N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. lower-*.f64 N/A

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

   9. Applied rewrites 75.1%

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

   if 2.5 < re

   1. Initial program 45.9%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right)\right)} \]
   4. 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. mul-1-neg N/A

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

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

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

      8. mul-1-neg N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. distribute-rgt-out N/A

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

   5. Applied rewrites 84.2%

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

Alternative 20: 51.4% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -3.9 \cdot 10^{+153} \lor \neg \left(im \leq 1.5 \cdot 10^{+80}\right):\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(\left(-0.3333333333333333 \cdot \left(im \cdot im\right)\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(re \cdot im\right) \cdot re, 0.16666666666666666, -im\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -3.9e+153) (not (<= im 1.5e+80)))
   (* (* 0.5 re) (* (* -0.3333333333333333 (* im im)) im))
   (* (fma (* (* re im) re) 0.16666666666666666 (- im)) re)))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -3.9e+153) || !(im <= 1.5e+80)) {
		tmp = (0.5 * re) * ((-0.3333333333333333 * (im * im)) * im);
	} else {
		tmp = fma(((re * im) * re), 0.16666666666666666, -im) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -3.9e+153) || !(im <= 1.5e+80))
		tmp = Float64(Float64(0.5 * re) * Float64(Float64(-0.3333333333333333 * Float64(im * im)) * im));
	else
		tmp = Float64(fma(Float64(Float64(re * im) * re), 0.16666666666666666, Float64(-im)) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -3.9e+153], N[Not[LessEqual[im, 1.5e+80]], $MachinePrecision]], N[(N[(0.5 * re), $MachinePrecision] * N[(N[(-0.3333333333333333 * N[(im * im), $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(re * im), $MachinePrecision] * re), $MachinePrecision] * 0.16666666666666666 + (-im)), $MachinePrecision] * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -3.89999999999999983e153 or 1.49999999999999993e80 < 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. metadata-eval N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 96.1

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

   5. Applied rewrites 96.1%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 79.2

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

   8. Applied rewrites 79.2%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 79.2%

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

   12. Taylor expanded in re around 0

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

   14. Applied rewrites 80.6%

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

   if -3.89999999999999983e153 < im < 1.49999999999999993e80

   1. Initial program 46.7%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      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 75.2

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

   5. Applied rewrites 75.2%

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

   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 41.6%

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

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3.9 \cdot 10^{+153} \lor \neg \left(im \leq 1.5 \cdot 10^{+80}\right):\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(\left(-0.3333333333333333 \cdot \left(im \cdot im\right)\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(re \cdot im\right) \cdot re, 0.16666666666666666, -im\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 33.0% 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 61.7%

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

3. Taylor expanded in im around 0

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

   1. *-commutative N/A

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

   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 55.4

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

5. Applied rewrites 55.4%

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

6. Taylor expanded in re around 0

   \[\leadsto \left(-1 \cdot re\right) \cdot im \]
7. Step-by-step derivation

8. Applied rewrites 29.3%

   \[\leadsto \left(-re\right) \cdot im \]
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 2025017 
(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))))