math.cos on complex, imaginary part

Percentage Accurate: 65.1% → 99.9%

Time: 10.1s

Alternatives: 14

Speedup: 2.5×

• 49.3% 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: 65.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

4. Applied rewrites 99.9%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. associate-/l* N/A

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

   10. *-commutative N/A

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

   11. lift-sinh.f64 N/A

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

   12. sinh-undef-rev N/A

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

   13. sinh-def N/A

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

   14. lift-sinh.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 99.9

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

6. Applied rewrites 99.9%

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

Alternative 2: 58.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.01:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(im \cdot re\right) \cdot re, -0.16666666666666666, im\right) \cdot re\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333, im \cdot im, -2\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.01)
   (*
    (* (fma (* (* im re) re) -0.16666666666666666 im) re)
    (fma (* -0.16666666666666666 im) im -1.0))
   (*
    (* 0.5 re)
    (*
     (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 tmp;
	if ((0.5 * sin(re)) <= -0.01) {
		tmp = (fma(((im * re) * re), -0.16666666666666666, im) * re) * fma((-0.16666666666666666 * im), im, -1.0);
	} else {
		tmp = (0.5 * re) * (fma((((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)) <= -0.01)
		tmp = Float64(Float64(fma(Float64(Float64(im * re) * re), -0.16666666666666666, im) * re) * fma(Float64(-0.16666666666666666 * im), im, -1.0));
	else
		tmp = Float64(Float64(0.5 * re) * Float64(fma(Float64(Float64(Float64(fma(-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], -0.01], N[(N[(N[(N[(N[(im * re), $MachinePrecision] * re), $MachinePrecision] * -0.16666666666666666 + im), $MachinePrecision] * re), $MachinePrecision] * N[(N[(-0.16666666666666666 * im), $MachinePrecision] * im + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im * im), $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)) < -0.0100000000000000002

   1. Initial program 56.8%

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

   3. Taylor expanded in im around 0

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

      1. distribute-lft-in N/A

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

      2. mul-1-neg N/A

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

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

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

      4. mul-1-neg N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

      16. *-commutative N/A

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

      17. distribute-rgt-out N/A

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

      18. +-commutative N/A

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

   5. Applied rewrites 74.1%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 32.8%

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

   if -0.0100000000000000002 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 65.7%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   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 93.1%

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

   7. Applied rewrites 93.1%

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

   8. Taylor expanded in re around 0

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

      1. lower-*.f64 68.5

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

   10. Applied rewrites 68.5%

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

   11. Taylor expanded in im around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   13. Applied rewrites 68.5%

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

Alternative 3: 93.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \sin re\right) \cdot \left(\mathsf{fma}\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333, im \cdot im, -2\right) \cdot im\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (*
  (* 0.5 (sin re))
  (*
   (fma
    (-
     (*
      (* (- (* -0.0003968253968253968 (* im im)) 0.016666666666666666) im)
      im)
     0.3333333333333333)
    (* im im)
    -2.0)
   im)))

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (fma((((((-0.0003968253968253968 * (im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333), (im * im), -2.0) * im);
}

Julia

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

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $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. Initial program 63.6%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
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 92.1%

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

Alternative 4: 57.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.01:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(im \cdot re\right) \cdot re, -0.16666666666666666, im\right) \cdot re\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.008333333333333333, -0.16666666666666666\right), im \cdot im, -1\right) \cdot im\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.01)
   (*
    (* (fma (* (* im re) re) -0.16666666666666666 im) re)
    (fma (* -0.16666666666666666 im) im -1.0))
   (*
    (*
     (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)) <= -0.01) {
		tmp = (fma(((im * re) * re), -0.16666666666666666, im) * re) * fma((-0.16666666666666666 * im), im, -1.0);
	} 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)) <= -0.01)
		tmp = Float64(Float64(fma(Float64(Float64(im * re) * re), -0.16666666666666666, im) * re) * fma(Float64(-0.16666666666666666 * im), im, -1.0));
	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], -0.01], N[(N[(N[(N[(N[(im * re), $MachinePrecision] * re), $MachinePrecision] * -0.16666666666666666 + im), $MachinePrecision] * re), $MachinePrecision] * N[(N[(-0.16666666666666666 * im), $MachinePrecision] * im + -1.0), $MachinePrecision]), $MachinePrecision], 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)) < -0.0100000000000000002

   1. Initial program 56.8%

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

   3. Taylor expanded in im around 0

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

      1. distribute-lft-in N/A

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

      2. mul-1-neg N/A

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

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

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

      4. mul-1-neg N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

      16. *-commutative N/A

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

      17. distribute-rgt-out N/A

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

      18. +-commutative N/A

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

   5. Applied rewrites 74.1%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 32.8%

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

   if -0.0100000000000000002 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 65.7%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   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 85.9%

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

   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 64.6%

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

Alternative 5: 75.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 10^{-9}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333, im \cdot im, -2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(\mathsf{fma}\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333, im \cdot im, -2\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1e-9)
   (*
    (* 0.5 re)
    (*
     (fma
      (-
       (*
        (* (fma -0.0003968253968253968 (* im im) -0.016666666666666666) im)
        im)
       0.3333333333333333)
      (* im im)
      -2.0)
     im))
   (*
    (* 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 <= 1e-9) {
		tmp = (0.5 * re) * (fma((((fma(-0.0003968253968253968, (im * im), -0.016666666666666666) * im) * im) - 0.3333333333333333), (im * im), -2.0) * im);
	} 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 <= 1e-9)
		tmp = Float64(Float64(0.5 * re) * Float64(fma(Float64(Float64(Float64(fma(-0.0003968253968253968, Float64(im * im), -0.016666666666666666) * im) * im) - 0.3333333333333333), Float64(im * im), -2.0) * im));
	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, 1e-9], N[(N[(0.5 * re), $MachinePrecision] * N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im * im), $MachinePrecision] + -0.016666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + -2.0), $MachinePrecision] * im), $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 < 1.00000000000000006e-9

   1. Initial program 65.8%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      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 93.9%

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

   7. Applied rewrites 93.9%

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

   8. Taylor expanded in re around 0

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

      1. lower-*.f64 69.4

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

   10. Applied rewrites 69.4%

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

   11. Taylor expanded in im around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   13. Applied rewrites 69.4%

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

   if 1.00000000000000006e-9 < re

   1. Initial program 57.3%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)} \]
   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 82.3

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

   5. Applied rewrites 82.3%

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

Alternative 6: 56.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.01:\\ \;\;\;\;\left(\left(\left(re \cdot im\right) \cdot re\right) \cdot 0.16666666666666666\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.008333333333333333, -0.16666666666666666\right), im \cdot im, -1\right) \cdot im\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.01)
   (* (* (* (* re im) re) 0.16666666666666666) 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)) <= -0.01) {
		tmp = (((re * im) * re) * 0.16666666666666666) * 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)) <= -0.01)
		tmp = Float64(Float64(Float64(Float64(re * im) * re) * 0.16666666666666666) * 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], -0.01], N[(N[(N[(N[(re * im), $MachinePrecision] * re), $MachinePrecision] * 0.16666666666666666), $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)) < -0.0100000000000000002

   1. Initial program 56.8%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      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 48.2

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

   5. Applied rewrites 48.2%

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

   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 31.2%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 30.3%

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

   if -0.0100000000000000002 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 65.7%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   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 85.9%

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

   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 64.6%

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

Alternative 7: 75.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 10^{-9}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333, im \cdot im, -2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right), -1\right)\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1e-9)
   (*
    (* 0.5 re)
    (*
     (fma
      (-
       (*
        (* (fma -0.0003968253968253968 (* im im) -0.016666666666666666) im)
        im)
       0.3333333333333333)
      (* im im)
      -2.0)
     im))
   (*
    (*
     (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 <= 1e-9) {
		tmp = (0.5 * re) * (fma((((fma(-0.0003968253968253968, (im * im), -0.016666666666666666) * im) * im) - 0.3333333333333333), (im * im), -2.0) * im);
	} 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 <= 1e-9)
		tmp = Float64(Float64(0.5 * re) * Float64(fma(Float64(Float64(Float64(fma(-0.0003968253968253968, Float64(im * im), -0.016666666666666666) * im) * im) - 0.3333333333333333), Float64(im * im), -2.0) * im));
	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, 1e-9], N[(N[(0.5 * re), $MachinePrecision] * N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im * im), $MachinePrecision] + -0.016666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + -2.0), $MachinePrecision] * im), $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 < 1.00000000000000006e-9

   1. Initial program 65.8%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      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 93.9%

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

   7. Applied rewrites 93.9%

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

   8. Taylor expanded in re around 0

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

      1. lower-*.f64 69.4

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

   10. Applied rewrites 69.4%

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

   11. Taylor expanded in im around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   13. Applied rewrites 69.4%

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

   if 1.00000000000000006e-9 < re

   1. Initial program 57.3%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   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 81.0%

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

Alternative 8: 52.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.01:\\ \;\;\;\;\left(\left(\left(re \cdot im\right) \cdot re\right) \cdot 0.16666666666666666\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(im \cdot im, -0.16666666666666666, -1\right) \cdot im\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.01)
   (* (* (* (* re im) re) 0.16666666666666666) re)
   (* (* (fma (* im im) -0.16666666666666666 -1.0) im) re)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.01) {
		tmp = (((re * im) * re) * 0.16666666666666666) * re;
	} else {
		tmp = (fma((im * im), -0.16666666666666666, -1.0) * im) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.01)
		tmp = Float64(Float64(Float64(Float64(re * im) * re) * 0.16666666666666666) * re);
	else
		tmp = Float64(Float64(fma(Float64(im * im), -0.16666666666666666, -1.0) * im) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.01], N[(N[(N[(N[(re * im), $MachinePrecision] * re), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666 + -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)) < -0.0100000000000000002

   1. Initial program 56.8%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      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 48.2

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

   5. Applied rewrites 48.2%

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

   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 31.2%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 30.3%

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

   if -0.0100000000000000002 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 65.7%

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

   3. Taylor expanded in im around 0

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

      1. distribute-lft-in N/A

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

      2. mul-1-neg N/A

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

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

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

      4. mul-1-neg N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

      16. *-commutative N/A

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

      17. distribute-rgt-out N/A

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

      18. +-commutative N/A

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

   5. Applied rewrites 76.1%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 58.2%

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

Alternative 9: 35.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.01:\\ \;\;\;\;\left(\left(\left(re \cdot im\right) \cdot re\right) \cdot 0.16666666666666666\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(-2 \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.01)
   (* (* (* (* re im) re) 0.16666666666666666) re)
   (* (* 0.5 re) (* -2.0 im))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.01) {
		tmp = (((re * im) * re) * 0.16666666666666666) * re;
	} else {
		tmp = (0.5 * re) * (-2.0 * im);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((0.5d0 * sin(re)) <= (-0.01d0)) then
        tmp = (((re * im) * re) * 0.16666666666666666d0) * re
    else
        tmp = (0.5d0 * re) * ((-2.0d0) * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((0.5 * Math.sin(re)) <= -0.01) {
		tmp = (((re * im) * re) * 0.16666666666666666) * re;
	} else {
		tmp = (0.5 * re) * (-2.0 * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (0.5 * math.sin(re)) <= -0.01:
		tmp = (((re * im) * re) * 0.16666666666666666) * re
	else:
		tmp = (0.5 * re) * (-2.0 * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.01)
		tmp = Float64(Float64(Float64(Float64(re * im) * re) * 0.16666666666666666) * re);
	else
		tmp = Float64(Float64(0.5 * re) * Float64(-2.0 * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((0.5 * sin(re)) <= -0.01)
		tmp = (((re * im) * re) * 0.16666666666666666) * re;
	else
		tmp = (0.5 * re) * (-2.0 * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.01], N[(N[(N[(N[(re * im), $MachinePrecision] * re), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * re), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(-2.0 * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < -0.0100000000000000002

   1. Initial program 56.8%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      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 48.2

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

   5. Applied rewrites 48.2%

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

   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 31.2%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 30.3%

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

   if -0.0100000000000000002 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 65.7%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   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 93.1%

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

   7. Applied rewrites 93.1%

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

   8. Taylor expanded in re around 0

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

      1. lower-*.f64 68.5

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

   10. Applied rewrites 68.5%

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

   11. Taylor expanded in im around 0

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

       1. lower-*.f64 38.0

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

   13. Applied rewrites 38.0%

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

Alternative 10: 35.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.01:\\ \;\;\;\;\left(\left(0.16666666666666666 \cdot im\right) \cdot \left(re \cdot re\right)\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(-2 \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.01)
   (* (* (* 0.16666666666666666 im) (* re re)) re)
   (* (* 0.5 re) (* -2.0 im))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.01) {
		tmp = ((0.16666666666666666 * im) * (re * re)) * re;
	} else {
		tmp = (0.5 * re) * (-2.0 * im);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((0.5d0 * sin(re)) <= (-0.01d0)) then
        tmp = ((0.16666666666666666d0 * im) * (re * re)) * re
    else
        tmp = (0.5d0 * re) * ((-2.0d0) * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((0.5 * Math.sin(re)) <= -0.01) {
		tmp = ((0.16666666666666666 * im) * (re * re)) * re;
	} else {
		tmp = (0.5 * re) * (-2.0 * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (0.5 * math.sin(re)) <= -0.01:
		tmp = ((0.16666666666666666 * im) * (re * re)) * re
	else:
		tmp = (0.5 * re) * (-2.0 * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.01)
		tmp = Float64(Float64(Float64(0.16666666666666666 * im) * Float64(re * re)) * re);
	else
		tmp = Float64(Float64(0.5 * re) * Float64(-2.0 * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((0.5 * sin(re)) <= -0.01)
		tmp = ((0.16666666666666666 * im) * (re * re)) * re;
	else
		tmp = (0.5 * re) * (-2.0 * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.01], N[(N[(N[(0.16666666666666666 * im), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(-2.0 * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < -0.0100000000000000002

   1. Initial program 56.8%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      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 48.2

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

   5. Applied rewrites 48.2%

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

   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 31.2%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 30.3%

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

   13. Applied rewrites 30.3%

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

   if -0.0100000000000000002 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 65.7%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   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 93.1%

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

   7. Applied rewrites 93.1%

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

   8. Taylor expanded in re around 0

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

      1. lower-*.f64 68.5

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

   10. Applied rewrites 68.5%

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

   11. Taylor expanded in im around 0

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

       1. lower-*.f64 38.0

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

   13. Applied rewrites 38.0%

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

Alternative 11: 73.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 0.0064:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333, im \cdot im, -2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin re \cdot im\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 0.0064)
   (*
    (* 0.5 re)
    (*
     (fma
      (-
       (*
        (* (fma -0.0003968253968253968 (* im im) -0.016666666666666666) im)
        im)
       0.3333333333333333)
      (* im im)
      -2.0)
     im))
   (* (* (sin re) im) (fma (* -0.16666666666666666 im) im -1.0))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 0.0064) {
		tmp = (0.5 * re) * (fma((((fma(-0.0003968253968253968, (im * im), -0.016666666666666666) * im) * im) - 0.3333333333333333), (im * im), -2.0) * im);
	} else {
		tmp = (sin(re) * im) * fma((-0.16666666666666666 * im), im, -1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 0.0064)
		tmp = Float64(Float64(0.5 * re) * Float64(fma(Float64(Float64(Float64(fma(-0.0003968253968253968, Float64(im * im), -0.016666666666666666) * im) * im) - 0.3333333333333333), Float64(im * im), -2.0) * im));
	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, 0.0064], N[(N[(0.5 * re), $MachinePrecision] * N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im * im), $MachinePrecision] + -0.016666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + -2.0), $MachinePrecision] * im), $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 < 0.00640000000000000031

   1. Initial program 65.6%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   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 94.0%

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

   7. Applied rewrites 94.0%

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

   8. Taylor expanded in re around 0

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

      1. lower-*.f64 69.5

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

   10. Applied rewrites 69.5%

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

   11. Taylor expanded in im around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   13. Applied rewrites 69.5%

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

   if 0.00640000000000000031 < re

   1. Initial program 57.4%

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

   3. Taylor expanded in im around 0

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

      1. distribute-lft-in N/A

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

      2. mul-1-neg N/A

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

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

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

      4. mul-1-neg N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

      16. *-commutative N/A

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

      17. distribute-rgt-out N/A

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

      18. +-commutative N/A

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

   5. Applied rewrites 75.5%

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

Alternative 12: 82.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -5 \cdot 10^{+233}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(im \cdot re\right) \cdot re, -0.16666666666666666, im\right) \cdot re\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\\ \mathbf{elif}\;im \leq -5 \cdot 10^{+21} \lor \neg \left(im \leq 430\right):\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333, im \cdot im, -2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\sin re\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -5e+233)
   (*
    (* (fma (* (* im re) re) -0.16666666666666666 im) re)
    (fma (* -0.16666666666666666 im) im -1.0))
   (if (or (<= im -5e+21) (not (<= im 430.0)))
     (*
      (* 0.5 re)
      (*
       (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 tmp;
	if (im <= -5e+233) {
		tmp = (fma(((im * re) * re), -0.16666666666666666, im) * re) * fma((-0.16666666666666666 * im), im, -1.0);
	} else if ((im <= -5e+21) || !(im <= 430.0)) {
		tmp = (0.5 * re) * (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)
	tmp = 0.0
	if (im <= -5e+233)
		tmp = Float64(Float64(fma(Float64(Float64(im * re) * re), -0.16666666666666666, im) * re) * fma(Float64(-0.16666666666666666 * im), im, -1.0));
	elseif ((im <= -5e+21) || !(im <= 430.0))
		tmp = Float64(Float64(0.5 * re) * Float64(fma(Float64(Float64(Float64(fma(-0.0003968253968253968, Float64(im * im), -0.016666666666666666) * 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_] := If[LessEqual[im, -5e+233], N[(N[(N[(N[(N[(im * re), $MachinePrecision] * re), $MachinePrecision] * -0.16666666666666666 + im), $MachinePrecision] * re), $MachinePrecision] * N[(N[(-0.16666666666666666 * im), $MachinePrecision] * im + -1.0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[im, -5e+21], N[Not[LessEqual[im, 430.0]], $MachinePrecision]], N[(N[(0.5 * re), $MachinePrecision] * N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im * im), $MachinePrecision] + -0.016666666666666666), $MachinePrecision] * im), $MachinePrecision] * 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 3 regimes

2. if im < -5.00000000000000009e233

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. distribute-lft-in N/A

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

      2. mul-1-neg N/A

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

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

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

      4. mul-1-neg N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

      16. *-commutative N/A

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

      17. distribute-rgt-out N/A

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

      18. +-commutative N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 75.0%

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

   if -5.00000000000000009e233 < im < -5e21 or 430 < 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 84.9%

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

   7. Applied rewrites 84.9%

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

   8. Taylor expanded in re around 0

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

      1. lower-*.f64 71.4

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

   10. Applied rewrites 71.4%

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

   11. Taylor expanded in im around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   13. Applied rewrites 71.4%

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

   if -5e21 < im < 430

   1. Initial program 27.8%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      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 96.5

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

   5. Applied rewrites 96.5%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -5 \cdot 10^{+233}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(im \cdot re\right) \cdot re, -0.16666666666666666, im\right) \cdot re\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\\ \mathbf{elif}\;im \leq -5 \cdot 10^{+21} \lor \neg \left(im \leq 430\right):\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333, im \cdot im, -2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\sin re\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 33.2% accurate, 19.8× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot re\right) \cdot \left(-2 \cdot im\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (* 0.5 re) (* -2.0 im)))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.6%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
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 92.1%

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

7. Applied rewrites 92.1%

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

8. Taylor expanded in re around 0

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

   1. lower-*.f64 58.7

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

10. Applied rewrites 58.7%

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

11. Taylor expanded in im around 0

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

    1. lower-*.f64 32.4

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

13. Applied rewrites 32.4%

    \[\leadsto \left(0.5 \cdot re\right) \cdot \color{blue}{\left(-2 \cdot im\right)} \]
14. Add Preprocessing

Alternative 14: 33.2% 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 63.6%

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

3. Taylor expanded in im around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. mul-1-neg N/A

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

   5. lower-neg.f64 N/A

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

   6. lower-sin.f64 50.8

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

5. Applied rewrites 50.8%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 32.0%

   \[\leadsto \left(-re\right) \cdot im \]
9. Add Preprocessing

Developer Target 1: 99.7% 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 2024357 
(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))))