math.cos on complex, imaginary part

Percentage Accurate: 65.5% → 99.9%

Time: 10.3s

Alternatives: 21

Speedup: 1.5×

• 48.9% 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: 65.5% 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 64.1%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 64.1

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

   7. lift--.f64 N/A

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

   8. lift-exp.f64 N/A

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

   9. rem-exp-log N/A

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

   10. lift-exp.f64 N/A

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

   11. rem-log-exp N/A

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

   12. remove-double-neg N/A

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

   13. lift-neg.f64 N/A

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

   14. sinh-undef N/A

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

   15. lower-*.f64 N/A

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

   16. lower-sinh.f64 99.9

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

4. Applied rewrites 99.9%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. associate-/l* N/A

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

   10. *-commutative N/A

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

   11. lift-sinh.f64 N/A

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

   12. sinh-undef-rev N/A

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

   13. sinh-def N/A

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

   14. lift-sinh.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 99.9

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

6. Applied rewrites 99.9%

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

Alternative 2: 85.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ t_1 := t\_0 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(\left(-2\right) \cdot \sinh im\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+46}:\\ \;\;\;\;t\_0 \cdot \left(\left(\left(im \cdot im\right) \cdot -0.3333333333333333 - 2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re))) (t_1 (* t_0 (- (exp (- im)) (exp im)))))
   (if (<= t_1 (- INFINITY))
     (* (* (- 2.0) (sinh im)) (* 0.5 re))
     (if (<= t_1 5e+46)
       (* t_0 (* (- (* (* im im) -0.3333333333333333) 2.0) im))
       (*
        (* (fma (* re re) -0.08333333333333333 0.5) re)
        (*
         (-
          (*
           (-
            (*
             (*
              (- (* -0.0003968253968253968 (* im im)) 0.016666666666666666)
              im)
             im)
            0.3333333333333333)
           (* im im))
          2.0)
         im))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sin(re);
	double t_1 = t_0 * (exp(-im) - exp(im));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (-2.0 * sinh(im)) * (0.5 * re);
	} else if (t_1 <= 5e+46) {
		tmp = t_0 * ((((im * im) * -0.3333333333333333) - 2.0) * im);
	} else {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * re) * ((((((((-0.0003968253968253968 * (im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * sin(re))
	t_1 = Float64(t_0 * Float64(exp(Float64(-im)) - exp(im)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(-2.0) * sinh(im)) * Float64(0.5 * re));
	elseif (t_1 <= 5e+46)
		tmp = Float64(t_0 * Float64(Float64(Float64(Float64(im * im) * -0.3333333333333333) - 2.0) * im));
	else
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * Float64(im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * Float64(im * im)) - 2.0) * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[((-2.0) * N[Sinh[im], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+46], N[(t$95$0 * N[(N[(N[(N[(im * im), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.016666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-*.f64 78.0

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

   5. Applied rewrites 78.0%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. remove-double-neg N/A

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

      10. lift-neg.f64 N/A

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

      11. sinh-undef-rev N/A

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

      12. lift-neg.f64 N/A

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

      13. sinh-neg N/A

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

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

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-sinh.f64 78.0

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

   7. Applied rewrites 78.0%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 5.0000000000000002e46

   1. Initial program 32.9%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 98.5

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

   5. Applied rewrites 98.5%

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

   if 5.0000000000000002e46 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 63.5

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

   5. Applied rewrites 63.5%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 55.5

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

   8. Applied rewrites 55.5%

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

   9. Taylor expanded in im around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   11. Applied rewrites 61.9%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \leq -\infty:\\ \;\;\;\;\left(\left(-2\right) \cdot \sinh im\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 5 \cdot 10^{+46}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(\left(\left(im \cdot im\right) \cdot -0.3333333333333333 - 2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\left(-2\right) \cdot \sinh im\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+46}:\\ \;\;\;\;\left(\sin re \cdot im\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (- (exp (- im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (* (* (- 2.0) (sinh im)) (* 0.5 re))
     (if (<= t_0 5e+46)
       (* (* (sin re) im) (fma (* -0.16666666666666666 im) im -1.0))
       (*
        (* (fma (* re re) -0.08333333333333333 0.5) re)
        (*
         (-
          (*
           (-
            (*
             (*
              (- (* -0.0003968253968253968 (* im im)) 0.016666666666666666)
              im)
             im)
            0.3333333333333333)
           (* im im))
          2.0)
         im))))))

C

double code(double re, double im) {
	double t_0 = (0.5 * sin(re)) * (exp(-im) - exp(im));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (-2.0 * sinh(im)) * (0.5 * re);
	} else if (t_0 <= 5e+46) {
		tmp = (sin(re) * im) * fma((-0.16666666666666666 * im), im, -1.0);
	} else {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * re) * ((((((((-0.0003968253968253968 * (im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - exp(im)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(-2.0) * sinh(im)) * Float64(0.5 * re));
	elseif (t_0 <= 5e+46)
		tmp = Float64(Float64(sin(re) * im) * fma(Float64(-0.16666666666666666 * im), im, -1.0));
	else
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * Float64(im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * Float64(im * im)) - 2.0) * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[((-2.0) * N[Sinh[im], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+46], N[(N[(N[Sin[re], $MachinePrecision] * im), $MachinePrecision] * N[(N[(-0.16666666666666666 * im), $MachinePrecision] * im + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.016666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-*.f64 78.0

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

   5. Applied rewrites 78.0%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. remove-double-neg N/A

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

      10. lift-neg.f64 N/A

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

      11. sinh-undef-rev N/A

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

      12. lift-neg.f64 N/A

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

      13. sinh-neg N/A

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

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

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-sinh.f64 78.0

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

   7. Applied rewrites 78.0%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 5.0000000000000002e46

   1. Initial program 32.9%

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

   3. Taylor expanded in im around 0

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

      1. 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. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. distribute-rgt-out N/A

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

      16. lower-*.f64 N/A

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

   5. Applied rewrites 98.5%

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

   if 5.0000000000000002e46 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 63.5

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

   5. Applied rewrites 63.5%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 55.5

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

   8. Applied rewrites 55.5%

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

   9. Taylor expanded in im around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   11. Applied rewrites 61.9%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \leq -\infty:\\ \;\;\;\;\left(\left(-2\right) \cdot \sinh im\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 5 \cdot 10^{+46}:\\ \;\;\;\;\left(\sin re \cdot im\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\left(-2\right) \cdot \sinh im\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+46}:\\ \;\;\;\;\left(-\sin re\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (- (exp (- im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (* (* (- 2.0) (sinh im)) (* 0.5 re))
     (if (<= t_0 5e+46)
       (* (- (sin re)) im)
       (*
        (* (fma (* re re) -0.08333333333333333 0.5) re)
        (*
         (-
          (*
           (-
            (*
             (*
              (- (* -0.0003968253968253968 (* im im)) 0.016666666666666666)
              im)
             im)
            0.3333333333333333)
           (* im im))
          2.0)
         im))))))

C

double code(double re, double im) {
	double t_0 = (0.5 * sin(re)) * (exp(-im) - exp(im));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (-2.0 * sinh(im)) * (0.5 * re);
	} else if (t_0 <= 5e+46) {
		tmp = -sin(re) * im;
	} else {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * re) * ((((((((-0.0003968253968253968 * (im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - exp(im)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(-2.0) * sinh(im)) * Float64(0.5 * re));
	elseif (t_0 <= 5e+46)
		tmp = Float64(Float64(-sin(re)) * im);
	else
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * Float64(im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * Float64(im * im)) - 2.0) * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[((-2.0) * N[Sinh[im], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+46], N[((-N[Sin[re], $MachinePrecision]) * im), $MachinePrecision], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.016666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-*.f64 78.0

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

   5. Applied rewrites 78.0%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. remove-double-neg N/A

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

      10. lift-neg.f64 N/A

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

      11. sinh-undef-rev N/A

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

      12. lift-neg.f64 N/A

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

      13. sinh-neg N/A

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

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

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-sinh.f64 78.0

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

   7. Applied rewrites 78.0%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 5.0000000000000002e46

   1. Initial program 32.9%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sin.f64 97.9

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

   5. Applied rewrites 97.9%

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

   if 5.0000000000000002e46 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 63.5

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

   5. Applied rewrites 63.5%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 55.5

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

   8. Applied rewrites 55.5%

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

   9. Taylor expanded in im around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   11. Applied rewrites 61.9%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \leq -\infty:\\ \;\;\;\;\left(\left(-2\right) \cdot \sinh im\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 5 \cdot 10^{+46}:\\ \;\;\;\;\left(-\sin re\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(0.004166666666666667 \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+46}:\\ \;\;\;\;\left(-\sin re\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (- (exp (- im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (*
      (*
       (fma
        (- (* 0.004166666666666667 (* re re)) 0.08333333333333333)
        (* re re)
        0.5)
       re)
      (*
       (-
        (*
         (* (- (* -0.016666666666666666 (* im im)) 0.3333333333333333) im)
         im)
        2.0)
       im))
     (if (<= t_0 5e+46)
       (* (- (sin re)) im)
       (*
        (* (fma (* re re) -0.08333333333333333 0.5) re)
        (*
         (-
          (*
           (-
            (*
             (*
              (- (* -0.0003968253968253968 (* im im)) 0.016666666666666666)
              im)
             im)
            0.3333333333333333)
           (* im im))
          2.0)
         im))))))

C

double code(double re, double im) {
	double t_0 = (0.5 * sin(re)) * (exp(-im) - exp(im));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma(((0.004166666666666667 * (re * re)) - 0.08333333333333333), (re * re), 0.5) * re) * ((((((-0.016666666666666666 * (im * im)) - 0.3333333333333333) * im) * im) - 2.0) * im);
	} else if (t_0 <= 5e+46) {
		tmp = -sin(re) * im;
	} else {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * re) * ((((((((-0.0003968253968253968 * (im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - exp(im)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(fma(Float64(Float64(0.004166666666666667 * Float64(re * re)) - 0.08333333333333333), Float64(re * re), 0.5) * re) * Float64(Float64(Float64(Float64(Float64(Float64(-0.016666666666666666 * Float64(im * im)) - 0.3333333333333333) * im) * im) - 2.0) * im));
	elseif (t_0 <= 5e+46)
		tmp = Float64(Float64(-sin(re)) * im);
	else
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * Float64(im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * Float64(im * im)) - 2.0) * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(0.004166666666666667 * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.08333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(N[(N[(N[(-0.016666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+46], N[((-N[Sin[re], $MachinePrecision]) * im), $MachinePrecision], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.016666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 83.8

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

   5. Applied rewrites 83.8%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 68.2

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

   8. Applied rewrites 68.2%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 5.0000000000000002e46

   1. Initial program 32.9%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sin.f64 97.9

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

   5. Applied rewrites 97.9%

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

   if 5.0000000000000002e46 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 63.5

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

   5. Applied rewrites 63.5%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 55.5

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

   8. Applied rewrites 55.5%

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

   9. Taylor expanded in im around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   11. Applied rewrites 61.9%

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

Alternative 6: 90.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double re, double im) {
	double tmp;
	if (((0.5 * Math.sin(re)) * (Math.exp(-im) - Math.exp(im))) <= -Double.POSITIVE_INFINITY) {
		tmp = (-2.0 * Math.sinh(im)) * (0.5 * re);
	} else {
		tmp = (((((((((im * im) * -0.0001984126984126984) - 0.008333333333333333) * im) * im) - 0.16666666666666666) * (im * im)) - 1.0) * im) * Math.sin(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if ((0.5 * math.sin(re)) * (math.exp(-im) - math.exp(im))) <= -math.inf:
		tmp = (-2.0 * math.sinh(im)) * (0.5 * re)
	else:
		tmp = (((((((((im * im) * -0.0001984126984126984) - 0.008333333333333333) * im) * im) - 0.16666666666666666) * (im * im)) - 1.0) * im) * math.sin(re)
	return tmp

Julia

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

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (((0.5 * sin(re)) * (exp(-im) - exp(im))) <= -Inf)
		tmp = (-2.0 * sinh(im)) * (0.5 * re);
	else
		tmp = (((((((((im * im) * -0.0001984126984126984) - 0.008333333333333333) * im) * im) - 0.16666666666666666) * (im * im)) - 1.0) * im) * sin(re);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-*.f64 78.0

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

   5. Applied rewrites 78.0%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. remove-double-neg N/A

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

      10. lift-neg.f64 N/A

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

      11. sinh-undef-rev N/A

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

      12. lift-neg.f64 N/A

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

      13. sinh-neg N/A

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

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

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-sinh.f64 78.0

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

   7. Applied rewrites 78.0%

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

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

   1. Initial program 53.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 53.3

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

      7. lift--.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. rem-exp-log N/A

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

      10. lift-exp.f64 N/A

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

      11. rem-log-exp N/A

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

      12. remove-double-neg N/A

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

      13. lift-neg.f64 N/A

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

      14. sinh-undef N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sinh.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lift-sinh.f64 N/A

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

      12. sinh-undef-rev N/A

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

      13. sinh-def N/A

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

      14. lift-sinh.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 99.9

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   9. Applied rewrites 94.1%

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

4. Final simplification 90.4%

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

Alternative 7: 88.8% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re))))
   (if (<= (* t_0 (- (exp (- im)) (exp im))) (- INFINITY))
     (* (* (- 2.0) (sinh im)) (* 0.5 re))
     (*
      t_0
      (*
       (-
        (*
         (* (- (* -0.016666666666666666 (* im im)) 0.3333333333333333) im)
         im)
        2.0)
       im)))))

C

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

Java

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

Python

def code(re, im):
	t_0 = 0.5 * math.sin(re)
	tmp = 0
	if (t_0 * (math.exp(-im) - math.exp(im))) <= -math.inf:
		tmp = (-2.0 * math.sinh(im)) * (0.5 * re)
	else:
		tmp = t_0 * ((((((-0.016666666666666666 * (im * im)) - 0.3333333333333333) * im) * im) - 2.0) * im)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sin(re))
	tmp = 0.0
	if (Float64(t_0 * Float64(exp(Float64(-im)) - exp(im))) <= Float64(-Inf))
		tmp = Float64(Float64(Float64(-2.0) * sinh(im)) * Float64(0.5 * re));
	else
		tmp = Float64(t_0 * Float64(Float64(Float64(Float64(Float64(Float64(-0.016666666666666666 * Float64(im * im)) - 0.3333333333333333) * im) * im) - 2.0) * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * sin(re);
	tmp = 0.0;
	if ((t_0 * (exp(-im) - exp(im))) <= -Inf)
		tmp = (-2.0 * sinh(im)) * (0.5 * re);
	else
		tmp = t_0 * ((((((-0.016666666666666666 * (im * im)) - 0.3333333333333333) * im) * im) - 2.0) * im);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-*.f64 78.0

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

   5. Applied rewrites 78.0%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. remove-double-neg N/A

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

      10. lift-neg.f64 N/A

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

      11. sinh-undef-rev N/A

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

      12. lift-neg.f64 N/A

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

      13. sinh-neg N/A

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

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

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-sinh.f64 78.0

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

   7. Applied rewrites 78.0%

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

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

   1. Initial program 53.3%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   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. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 92.7

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

   5. Applied rewrites 92.7%

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

4. Final simplification 89.3%

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

Alternative 8: 88.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-*.f64 78.0

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

   5. Applied rewrites 78.0%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. remove-double-neg N/A

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

      10. lift-neg.f64 N/A

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

      11. sinh-undef-rev N/A

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

      12. lift-neg.f64 N/A

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

      13. sinh-neg N/A

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

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

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-sinh.f64 78.0

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

   7. Applied rewrites 78.0%

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

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

   1. Initial program 53.3%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   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 92.6%

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

4. Final simplification 89.3%

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

Alternative 9: 59.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq 10^{-6}:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.004166666666666667 \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) 1e-6)
   (*
    (* (fma (* re re) -0.08333333333333333 0.5) re)
    (*
     (-
      (*
       (-
        (*
         (* (- (* -0.0003968253968253968 (* im im)) 0.016666666666666666) im)
         im)
        0.3333333333333333)
       (* im im))
      2.0)
     im))
   (*
    (*
     (fma
      (- (* 0.004166666666666667 (* re re)) 0.08333333333333333)
      (* re re)
      0.5)
     re)
    (*
     (-
      (* (* (- (* -0.016666666666666666 (* im im)) 0.3333333333333333) im) im)
      2.0)
     im))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= 1e-6) {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * re) * ((((((((-0.0003968253968253968 * (im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	} else {
		tmp = (fma(((0.004166666666666667 * (re * re)) - 0.08333333333333333), (re * re), 0.5) * re) * ((((((-0.016666666666666666 * (im * im)) - 0.3333333333333333) * im) * im) - 2.0) * im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= 1e-6)
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * Float64(im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * Float64(im * im)) - 2.0) * im));
	else
		tmp = Float64(Float64(fma(Float64(Float64(0.004166666666666667 * Float64(re * re)) - 0.08333333333333333), Float64(re * re), 0.5) * re) * Float64(Float64(Float64(Float64(Float64(Float64(-0.016666666666666666 * Float64(im * im)) - 0.3333333333333333) * im) * im) - 2.0) * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], 1e-6], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.016666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.004166666666666667 * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.08333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(N[(N[(N[(-0.016666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 66.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 85.0

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

   5. Applied rewrites 85.0%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 66.1

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

   8. Applied rewrites 66.1%

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

   9. Taylor expanded in im around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   11. Applied rewrites 70.1%

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

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

   1. Initial program 58.4%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 92.5

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

   5. Applied rewrites 92.5%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 38.6

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

   8. Applied rewrites 38.6%

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

Alternative 10: 59.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq 10^{-6}:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.004166666666666667 \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \cdot \left(\left(\left(im \cdot im\right) \cdot -0.3333333333333333 - 2\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) 1e-6)
   (*
    (* (fma (* re re) -0.08333333333333333 0.5) re)
    (*
     (-
      (*
       (-
        (*
         (* (- (* -0.0003968253968253968 (* im im)) 0.016666666666666666) im)
         im)
        0.3333333333333333)
       (* im im))
      2.0)
     im))
   (*
    (*
     (fma
      (- (* 0.004166666666666667 (* re re)) 0.08333333333333333)
      (* re re)
      0.5)
     re)
    (* (- (* (* im im) -0.3333333333333333) 2.0) im))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= 1e-6) {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * re) * ((((((((-0.0003968253968253968 * (im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	} else {
		tmp = (fma(((0.004166666666666667 * (re * re)) - 0.08333333333333333), (re * re), 0.5) * re) * ((((im * im) * -0.3333333333333333) - 2.0) * im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= 1e-6)
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * Float64(im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * Float64(im * im)) - 2.0) * im));
	else
		tmp = Float64(Float64(fma(Float64(Float64(0.004166666666666667 * Float64(re * re)) - 0.08333333333333333), Float64(re * re), 0.5) * re) * Float64(Float64(Float64(Float64(im * im) * -0.3333333333333333) - 2.0) * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], 1e-6], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.016666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.004166666666666667 * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.08333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(N[(im * im), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 66.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 85.0

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

   5. Applied rewrites 85.0%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 66.1

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

   8. Applied rewrites 66.1%

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

   9. Taylor expanded in im around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   11. Applied rewrites 70.1%

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

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

   1. Initial program 58.4%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 85.3

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

   5. Applied rewrites 85.3%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 38.6

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

   8. Applied rewrites 38.6%

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

Alternative 11: 58.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(im \cdot im, -0.008333333333333333, -0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, re \cdot re, im\right)\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.004166666666666667 \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \cdot \left(\left(\left(im \cdot im\right) \cdot -0.3333333333333333 - 2\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) 2e-6)
   (*
    (*
     (-
      (* (* (fma (* im im) -0.008333333333333333 -0.16666666666666666) im) im)
      1.0)
     (fma (* -0.16666666666666666 im) (* re re) im))
    re)
   (*
    (*
     (fma
      (- (* 0.004166666666666667 (* re re)) 0.08333333333333333)
      (* re re)
      0.5)
     re)
    (* (- (* (* im im) -0.3333333333333333) 2.0) im))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= 2e-6) {
		tmp = ((((fma((im * im), -0.008333333333333333, -0.16666666666666666) * im) * im) - 1.0) * fma((-0.16666666666666666 * im), (re * re), im)) * re;
	} else {
		tmp = (fma(((0.004166666666666667 * (re * re)) - 0.08333333333333333), (re * re), 0.5) * re) * ((((im * im) * -0.3333333333333333) - 2.0) * im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= 2e-6)
		tmp = Float64(Float64(Float64(Float64(Float64(fma(Float64(im * im), -0.008333333333333333, -0.16666666666666666) * im) * im) - 1.0) * fma(Float64(-0.16666666666666666 * im), Float64(re * re), im)) * re);
	else
		tmp = Float64(Float64(fma(Float64(Float64(0.004166666666666667 * Float64(re * re)) - 0.08333333333333333), Float64(re * re), 0.5) * re) * Float64(Float64(Float64(Float64(im * im) * -0.3333333333333333) - 2.0) * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], 2e-6], N[(N[(N[(N[(N[(N[(N[(im * im), $MachinePrecision] * -0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 1.0), $MachinePrecision] * N[(N[(-0.16666666666666666 * im), $MachinePrecision] * N[(re * re), $MachinePrecision] + im), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(N[(N[(0.004166666666666667 * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.08333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(N[(im * im), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < 1.99999999999999991e-6

   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 90.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 68.8%

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

   if 1.99999999999999991e-6 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 59.2%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 85.0

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

   5. Applied rewrites 85.0%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 37.6

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

   8. Applied rewrites 37.6%

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

Alternative 12: 57.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\mathsf{fma}\left(im \cdot im, -0.008333333333333333, -0.16666666666666666\right) \cdot im\right) \cdot im - 1\\ \mathbf{if}\;0.5 \cdot \sin re \leq 0.01:\\ \;\;\;\;\left(t\_0 \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, re \cdot re, im\right)\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(re \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (-
          (*
           (* (fma (* im im) -0.008333333333333333 -0.16666666666666666) im)
           im)
          1.0)))
   (if (<= (* 0.5 (sin re)) 0.01)
     (* (* t_0 (fma (* -0.16666666666666666 im) (* re re) im)) re)
     (* t_0 (* re im)))))

C

double code(double re, double im) {
	double t_0 = ((fma((im * im), -0.008333333333333333, -0.16666666666666666) * im) * im) - 1.0;
	double tmp;
	if ((0.5 * sin(re)) <= 0.01) {
		tmp = (t_0 * fma((-0.16666666666666666 * im), (re * re), im)) * re;
	} else {
		tmp = t_0 * (re * im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(Float64(fma(Float64(im * im), -0.008333333333333333, -0.16666666666666666) * im) * im) - 1.0)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= 0.01)
		tmp = Float64(Float64(t_0 * fma(Float64(-0.16666666666666666 * im), Float64(re * re), im)) * re);
	else
		tmp = Float64(t_0 * Float64(re * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[(N[(N[(im * im), $MachinePrecision] * -0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], 0.01], N[(N[(t$95$0 * N[(N[(-0.16666666666666666 * im), $MachinePrecision] * N[(re * re), $MachinePrecision] + im), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision], N[(t$95$0 * N[(re * 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 65.5%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   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 90.2%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 68.7%

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

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

   1. Initial program 59.4%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   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 91.9%

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

   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 34.3%

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

Alternative 13: 56.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.04:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(\left(im \cdot im\right) \cdot -0.3333333333333333 - 2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(im \cdot im, -0.008333333333333333, -0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot \left(re \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.04)
   (*
    (* (fma (* re re) -0.08333333333333333 0.5) re)
    (* (- (* (* im im) -0.3333333333333333) 2.0) im))
   (*
    (-
     (* (* (fma (* im im) -0.008333333333333333 -0.16666666666666666) im) im)
     1.0)
    (* re im))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.04) {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * re) * ((((im * im) * -0.3333333333333333) - 2.0) * im);
	} else {
		tmp = (((fma((im * im), -0.008333333333333333, -0.16666666666666666) * im) * im) - 1.0) * (re * im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.04)
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re) * Float64(Float64(Float64(Float64(im * im) * -0.3333333333333333) - 2.0) * im));
	else
		tmp = Float64(Float64(Float64(Float64(fma(Float64(im * im), -0.008333333333333333, -0.16666666666666666) * im) * im) - 1.0) * Float64(re * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.04], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(N[(im * im), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(im * im), $MachinePrecision] * -0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 1.0), $MachinePrecision] * N[(re * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < -0.0400000000000000008

   1. Initial program 53.5%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 78.2

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

   5. Applied rewrites 78.2%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 22.5

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

   8. Applied rewrites 22.5%

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

   if -0.0400000000000000008 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 67.4%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   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 91.6%

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

   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 71.7%

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

Alternative 14: 56.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.04:\\ \;\;\;\;\left(\left(\left(re \cdot re\right) \cdot -0.08333333333333333\right) \cdot re\right) \cdot \left(\left(\left(im \cdot im\right) \cdot -0.3333333333333333 - 2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(im \cdot im, -0.008333333333333333, -0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot \left(re \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.04)
   (*
    (* (* (* re re) -0.08333333333333333) re)
    (* (- (* (* im im) -0.3333333333333333) 2.0) im))
   (*
    (-
     (* (* (fma (* im im) -0.008333333333333333 -0.16666666666666666) im) im)
     1.0)
    (* re im))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.04) {
		tmp = (((re * re) * -0.08333333333333333) * re) * ((((im * im) * -0.3333333333333333) - 2.0) * im);
	} else {
		tmp = (((fma((im * im), -0.008333333333333333, -0.16666666666666666) * im) * im) - 1.0) * (re * im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.04)
		tmp = Float64(Float64(Float64(Float64(re * re) * -0.08333333333333333) * re) * Float64(Float64(Float64(Float64(im * im) * -0.3333333333333333) - 2.0) * im));
	else
		tmp = Float64(Float64(Float64(Float64(fma(Float64(im * im), -0.008333333333333333, -0.16666666666666666) * im) * im) - 1.0) * Float64(re * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.04], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(N[(im * im), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(im * im), $MachinePrecision] * -0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 1.0), $MachinePrecision] * N[(re * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < -0.0400000000000000008

   1. Initial program 53.5%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 78.2

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

   5. Applied rewrites 78.2%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 22.5

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

   8. Applied rewrites 22.5%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 22.5%

       \[\leadsto \left(\left(\left(re \cdot re\right) \cdot -0.08333333333333333\right) \cdot re\right) \cdot \left(\left(\left(im \cdot im\right) \cdot -0.3333333333333333 - 2\right) \cdot im\right) \]

   if -0.0400000000000000008 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 67.4%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   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 91.6%

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

   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 71.7%

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

Alternative 15: 55.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.04:\\ \;\;\;\;\mathsf{fma}\left(im \cdot re, 0.16666666666666666 \cdot re, -im\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(im \cdot im, -0.008333333333333333, -0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot \left(re \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.04)
   (* (fma (* im re) (* 0.16666666666666666 re) (- im)) re)
   (*
    (-
     (* (* (fma (* im im) -0.008333333333333333 -0.16666666666666666) im) im)
     1.0)
    (* re im))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.04) {
		tmp = fma((im * re), (0.16666666666666666 * re), -im) * re;
	} else {
		tmp = (((fma((im * im), -0.008333333333333333, -0.16666666666666666) * im) * im) - 1.0) * (re * im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.04)
		tmp = Float64(fma(Float64(im * re), Float64(0.16666666666666666 * re), Float64(-im)) * re);
	else
		tmp = Float64(Float64(Float64(Float64(fma(Float64(im * im), -0.008333333333333333, -0.16666666666666666) * im) * im) - 1.0) * Float64(re * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.04], N[(N[(N[(im * re), $MachinePrecision] * N[(0.16666666666666666 * re), $MachinePrecision] + (-im)), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(N[(N[(N[(im * im), $MachinePrecision] * -0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 1.0), $MachinePrecision] * N[(re * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < -0.0400000000000000008

   1. Initial program 53.5%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sin.f64 51.6

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

   5. Applied rewrites 51.6%

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

   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 17.9%

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

   10. Applied rewrites 17.9%

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

   if -0.0400000000000000008 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 67.4%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   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 91.6%

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

   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 71.7%

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

Alternative 16: 53.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.04:\\ \;\;\;\;\mathsf{fma}\left(im \cdot re, 0.16666666666666666 \cdot re, -im\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(\left(\left(-0.3333333333333333 \cdot im\right) \cdot im - 2\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.04)
   (* (fma (* im re) (* 0.16666666666666666 re) (- im)) re)
   (* (* 0.5 re) (* (- (* (* -0.3333333333333333 im) im) 2.0) im))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.04) {
		tmp = fma((im * re), (0.16666666666666666 * re), -im) * re;
	} else {
		tmp = (0.5 * re) * ((((-0.3333333333333333 * im) * im) - 2.0) * im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.04)
		tmp = Float64(fma(Float64(im * re), Float64(0.16666666666666666 * re), Float64(-im)) * re);
	else
		tmp = Float64(Float64(0.5 * re) * Float64(Float64(Float64(Float64(-0.3333333333333333 * im) * im) - 2.0) * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.04], N[(N[(N[(im * re), $MachinePrecision] * N[(0.16666666666666666 * re), $MachinePrecision] + (-im)), $MachinePrecision] * re), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(N[(N[(N[(-0.3333333333333333 * im), $MachinePrecision] * 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)) < -0.0400000000000000008

   1. Initial program 53.5%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sin.f64 51.6

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

   5. Applied rewrites 51.6%

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

   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 17.9%

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

   10. Applied rewrites 17.9%

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

   if -0.0400000000000000008 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 67.4%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 91.6

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

   5. Applied rewrites 91.6%

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

   6. Taylor expanded in re around 0

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

      1. lower-*.f64 71.7

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

   8. Applied rewrites 71.7%

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

   9. Taylor expanded in im around 0

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

   11. Applied rewrites 69.7%

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

Alternative 17: 35.5% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 65.5%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sin.f64 56.2

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

   5. Applied rewrites 56.2%

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

   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 44.0%

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

   10. Applied rewrites 44.0%

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

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

   1. Initial program 59.4%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      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 48.1

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

   5. Applied rewrites 48.1%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 18.6%

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

Alternative 18: 35.5% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], 0.01], N[(N[(N[(0.16666666666666666 * re), $MachinePrecision] * re + -1.0), $MachinePrecision] * N[(im * re), $MachinePrecision]), $MachinePrecision], N[((-re) * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 65.5%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sin.f64 56.2

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

   5. Applied rewrites 56.2%

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

   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 44.0%

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

   10. Applied rewrites 44.0%

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

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

   1. Initial program 59.4%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      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 48.1

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

   5. Applied rewrites 48.1%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 18.6%

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

Alternative 19: 35.5% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], 0.01], N[(N[(im * N[(N[(0.16666666666666666 * re), $MachinePrecision] * re + -1.0), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision], N[((-re) * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 65.5%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sin.f64 56.2

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

   5. Applied rewrites 56.2%

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

   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 44.0%

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

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

   1. Initial program 59.4%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      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 48.1

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

   5. Applied rewrites 48.1%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 18.6%

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

Alternative 20: 35.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.04:\\ \;\;\;\;\left(\left(\left(im \cdot re\right) \cdot re\right) \cdot 0.16666666666666666\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(-re\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.04)
   (* (* (* (* im re) re) 0.16666666666666666) re)
   (* (- re) im)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.04) {
		tmp = (((im * re) * re) * 0.16666666666666666) * re;
	} else {
		tmp = -re * im;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((0.5d0 * sin(re)) <= (-0.04d0)) then
        tmp = (((im * re) * re) * 0.16666666666666666d0) * re
    else
        tmp = -re * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((0.5 * Math.sin(re)) <= -0.04) {
		tmp = (((im * re) * re) * 0.16666666666666666) * re;
	} else {
		tmp = -re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (0.5 * math.sin(re)) <= -0.04:
		tmp = (((im * re) * re) * 0.16666666666666666) * re
	else:
		tmp = -re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.04)
		tmp = Float64(Float64(Float64(Float64(im * re) * re) * 0.16666666666666666) * re);
	else
		tmp = Float64(Float64(-re) * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((0.5 * sin(re)) <= -0.04)
		tmp = (((im * re) * re) * 0.16666666666666666) * re;
	else
		tmp = -re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.04], N[(N[(N[(N[(im * re), $MachinePrecision] * re), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * re), $MachinePrecision], N[((-re) * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < -0.0400000000000000008

   1. Initial program 53.5%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sin.f64 51.6

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

   5. Applied rewrites 51.6%

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

   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 17.9%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 17.9%

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

   if -0.0400000000000000008 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 67.4%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sin.f64 55.3

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

   5. Applied rewrites 55.3%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 43.9%

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

Alternative 21: 33.5% 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 64.1%

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

3. Taylor expanded in im around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. mul-1-neg N/A

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

   5. lower-neg.f64 N/A

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

   6. lower-sin.f64 54.4

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

5. Applied rewrites 54.4%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 36.9%

   \[\leadsto \left(-re\right) \cdot im \]
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 2024364 
(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))))