math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 7.8s

Alternatives: 20

Speedup: 1.5×

• 49.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp((0.0 - 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((0.0d0 - im)) + exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (Math.exp((0.0 - im)) + Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.sin(re)) * (math.exp((0.0 - im)) + math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp((0.0 - 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((0.0d0 - im)) + exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (Math.exp((0.0 - im)) + Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.sin(re)) * (math.exp((0.0 - im)) + math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \sin re \cdot \cosh im \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (sin re) (cosh im)))

C

double code(double re, double im) {
	return sin(re) * cosh(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 = sin(re) * cosh(im)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

   4. associate-*l* N/A

      \[\leadsto \color{blue}{\sin re \cdot \left(\frac{1}{2} \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

   5. lift-+.f64 N/A

      \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)}\right) \]

   6. +-commutative N/A

      \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(e^{im} + e^{0 - im}\right)}\right) \]

   7. lift-exp.f64 N/A

      \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(\color{blue}{e^{im}} + e^{0 - im}\right)\right) \]

   8. lift-exp.f64 N/A

      \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + \color{blue}{e^{0 - im}}\right)\right) \]

   9. lift--.f64 N/A

      \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + e^{\color{blue}{0 - im}}\right)\right) \]

   10. sub0-neg N/A

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

   11. cosh-undef N/A

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

   12. associate-*r* N/A

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

   13. metadata-eval N/A

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

   14. cosh-0 N/A

       \[\leadsto \sin re \cdot \left(\color{blue}{\cosh 0} \cdot \cosh im\right) \]

   15. *-commutative N/A

       \[\leadsto \color{blue}{\left(\cosh 0 \cdot \cosh im\right) \cdot \sin re} \]

   16. lower-*.f64 N/A

       \[\leadsto \color{blue}{\left(\cosh 0 \cdot \cosh im\right) \cdot \sin re} \]

   17. cosh-0 N/A

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

   18. exp-0 N/A

       \[\leadsto \left(\color{blue}{e^{0}} \cdot \cosh im\right) \cdot \sin re \]

   19. lower-*.f64 N/A

       \[\leadsto \color{blue}{\left(e^{0} \cdot \cosh im\right)} \cdot \sin re \]

   20. exp-0 N/A

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

   21. lower-cosh.f64 100.0

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

4. Applied rewrites 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-lft-identity N/A

      \[\leadsto \color{blue}{\cosh im} \cdot \sin re \]

   4. *-commutative N/A

      \[\leadsto \color{blue}{\sin re \cdot \cosh im} \]

   5. lower-*.f64 100.0

      \[\leadsto \color{blue}{\sin re \cdot \cosh im} \]

6. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\sin re \cdot \cosh im} \]
7. Add Preprocessing

Alternative 2: 76.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(im, \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right), {re}^{4} \cdot \left(im \cdot \mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re \cdot re, 0.004166666666666667\right)\right)\right) \cdot re\right) \cdot im\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left({re}^{3}, 0.008333333333333333 \cdot \left(re \cdot re\right) - 0.16666666666666666, re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (*
      (*
       (fma
        im
        (fma (* re re) -0.08333333333333333 0.5)
        (*
         (pow re 4.0)
         (* im (fma -9.92063492063492e-5 (* re re) 0.004166666666666667))))
       re)
      im)
     (if (<= t_0 1.0)
       (*
        (fma (fma 0.041666666666666664 (* im im) 0.5) (* im im) 1.0)
        (sin re))
       (*
        (fma
         (fma
          (fma 0.001388888888888889 (* im im) 0.041666666666666664)
          (* im im)
          0.5)
         (* im im)
         1.0)
        (fma
         (pow re 3.0)
         (- (* 0.008333333333333333 (* re re)) 0.16666666666666666)
         re))))))

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(im, fma((re * re), -0.08333333333333333, 0.5), (pow(re, 4.0) * (im * fma(-9.92063492063492e-5, (re * re), 0.004166666666666667)))) * re) * im;
	} else if (t_0 <= 1.0) {
		tmp = fma(fma(0.041666666666666664, (im * im), 0.5), (im * im), 1.0) * sin(re);
	} else {
		tmp = fma(fma(fma(0.001388888888888889, (im * im), 0.041666666666666664), (im * im), 0.5), (im * im), 1.0) * fma(pow(re, 3.0), ((0.008333333333333333 * (re * re)) - 0.16666666666666666), re);
	}
	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(im, fma(Float64(re * re), -0.08333333333333333, 0.5), Float64((re ^ 4.0) * Float64(im * fma(-9.92063492063492e-5, Float64(re * re), 0.004166666666666667)))) * re) * im);
	elseif (t_0 <= 1.0)
		tmp = Float64(fma(fma(0.041666666666666664, Float64(im * im), 0.5), Float64(im * im), 1.0) * sin(re));
	else
		tmp = Float64(fma(fma(fma(0.001388888888888889, Float64(im * im), 0.041666666666666664), Float64(im * im), 0.5), Float64(im * im), 1.0) * fma((re ^ 3.0), Float64(Float64(0.008333333333333333 * Float64(re * re)) - 0.16666666666666666), re));
	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[(im * N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] + N[(N[Power[re, 4.0], $MachinePrecision] * N[(im * N[(-9.92063492063492e-5 * N[(re * re), $MachinePrecision] + 0.004166666666666667), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision] * im), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sin[re], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.001388888888888889 * N[(im * im), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[Power[re, 3.0], $MachinePrecision] * N[(N[(0.008333333333333333 * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 45.0

          \[\leadsto \mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \color{blue}{\sin re} \]

   5. Applied rewrites 45.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \sin re} \]

   6. Taylor expanded in im around inf

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

   8. Applied rewrites 36.7%

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

   9. Taylor expanded in re around 0

      \[\leadsto \left(re \cdot \left(\frac{1}{2} \cdot im + {re}^{2} \cdot \left(\frac{-1}{12} \cdot im + {re}^{2} \cdot \left(\frac{-1}{10080} \cdot \left(im \cdot {re}^{2}\right) + \frac{1}{240} \cdot im\right)\right)\right)\right) \cdot im \]
   10. Step-by-step derivation

   11. Applied rewrites 32.0%

       \[\leadsto \left(\mathsf{fma}\left(im, \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right), {re}^{4} \cdot \left(im \cdot \mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re \cdot re, 0.004166666666666667\right)\right)\right) \cdot re\right) \cdot im \]

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(\frac{1}{2} \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)}\right) \]

      6. +-commutative N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(e^{im} + e^{0 - im}\right)}\right) \]

      7. lift-exp.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(\color{blue}{e^{im}} + e^{0 - im}\right)\right) \]

      8. lift-exp.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + \color{blue}{e^{0 - im}}\right)\right) \]

      9. lift--.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + e^{\color{blue}{0 - im}}\right)\right) \]

      10. sub0-neg N/A

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

      11. cosh-undef N/A

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

      12. associate-*r* N/A

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

      13. metadata-eval N/A

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

      14. cosh-0 N/A

          \[\leadsto \sin re \cdot \left(\color{blue}{\cosh 0} \cdot \cosh im\right) \]

      15. *-commutative N/A

          \[\leadsto \color{blue}{\left(\cosh 0 \cdot \cosh im\right) \cdot \sin re} \]

      16. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\cosh 0 \cdot \cosh im\right) \cdot \sin re} \]

      17. cosh-0 N/A

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

      18. exp-0 N/A

          \[\leadsto \left(\color{blue}{e^{0}} \cdot \cosh im\right) \cdot \sin re \]

      19. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(e^{0} \cdot \cosh im\right)} \cdot \sin re \]

      20. exp-0 N/A

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

      21. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \cdot \sin re \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) + 1\right)} \cdot \sin re \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot {im}^{2}} + 1\right) \cdot \sin re \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}, {im}^{2}, 1\right)} \cdot \sin re \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}}, {im}^{2}, 1\right) \cdot \sin re \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {im}^{2}, \frac{1}{2}\right)}, {im}^{2}, 1\right) \cdot \sin re \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right), \color{blue}{im \cdot im}, 1\right) \cdot \sin re \]

      9. lower-*.f64 100.0

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), \color{blue}{im \cdot im}, 1\right) \cdot \sin re \]

   7. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right)} \cdot \sin re \]

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(\frac{1}{2} \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)}\right) \]

      6. +-commutative N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(e^{im} + e^{0 - im}\right)}\right) \]

      7. lift-exp.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(\color{blue}{e^{im}} + e^{0 - im}\right)\right) \]

      8. lift-exp.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + \color{blue}{e^{0 - im}}\right)\right) \]

      9. lift--.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + e^{\color{blue}{0 - im}}\right)\right) \]

      10. sub0-neg N/A

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

      11. cosh-undef N/A

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

      12. associate-*r* N/A

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

      13. metadata-eval N/A

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

      14. cosh-0 N/A

          \[\leadsto \sin re \cdot \left(\color{blue}{\cosh 0} \cdot \cosh im\right) \]

      15. *-commutative N/A

          \[\leadsto \color{blue}{\left(\cosh 0 \cdot \cosh im\right) \cdot \sin re} \]

      16. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\cosh 0 \cdot \cosh im\right) \cdot \sin re} \]

      17. cosh-0 N/A

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

      18. exp-0 N/A

          \[\leadsto \left(\color{blue}{e^{0}} \cdot \cosh im\right) \cdot \sin re \]

      19. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(e^{0} \cdot \cosh im\right)} \cdot \sin re \]

      20. exp-0 N/A

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

      21. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \cdot \sin re \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + 1\right)} \cdot \sin re \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) \cdot {im}^{2}} + 1\right) \cdot \sin re \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right), {im}^{2}, 1\right)} \cdot \sin re \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) + \frac{1}{2}}, {im}^{2}, 1\right) \cdot \sin re \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) \cdot {im}^{2}} + \frac{1}{2}, {im}^{2}, 1\right) \cdot \sin re \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}, {im}^{2}, \frac{1}{2}\right)}, {im}^{2}, 1\right) \cdot \sin re \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{720} \cdot {im}^{2} + \frac{1}{24}}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{720}, {im}^{2}, \frac{1}{24}\right)}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{im \cdot im}, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{im \cdot im}, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), \color{blue}{im \cdot im}, 1\right) \cdot \sin re \]

      14. lower-*.f64 76.8

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), \color{blue}{im \cdot im}, 1\right) \cdot \sin re \]

   7. Applied rewrites 76.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right)} \cdot \sin re \]

   8. Taylor expanded in re around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \color{blue}{\left(re \cdot \left(1 + {re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \left(re \cdot \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \color{blue}{\left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right) + re \cdot 1\right)} \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \left(\color{blue}{\left(re \cdot {re}^{2}\right) \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)} + re \cdot 1\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \left(\left(re \cdot \color{blue}{\left(re \cdot re\right)}\right) \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) + re \cdot 1\right) \]

      5. cube-unmult N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \left(\color{blue}{{re}^{3}} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) + re \cdot 1\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \left({re}^{\color{blue}{\left(2 + 1\right)}} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) + re \cdot 1\right) \]

      7. pow-plus N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \left(\color{blue}{\left({re}^{2} \cdot re\right)} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) + re \cdot 1\right) \]

      8. *-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \left(\left({re}^{2} \cdot re\right) \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) + \color{blue}{re}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \color{blue}{\mathsf{fma}\left({re}^{2} \cdot re, \frac{1}{120} \cdot {re}^{2} - \frac{1}{6}, re\right)} \]

      10. pow-plus N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left(\color{blue}{{re}^{\left(2 + 1\right)}}, \frac{1}{120} \cdot {re}^{2} - \frac{1}{6}, re\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left(\color{blue}{{re}^{\left(2 + 1\right)}}, \frac{1}{120} \cdot {re}^{2} - \frac{1}{6}, re\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left({re}^{\color{blue}{3}}, \frac{1}{120} \cdot {re}^{2} - \frac{1}{6}, re\right) \]

      13. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left({re}^{3}, \color{blue}{\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}}, re\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left({re}^{3}, \color{blue}{\frac{1}{120} \cdot {re}^{2}} - \frac{1}{6}, re\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left({re}^{3}, \frac{1}{120} \cdot \color{blue}{\left(re \cdot re\right)} - \frac{1}{6}, re\right) \]

      16. lower-*.f64 62.7

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left({re}^{3}, 0.008333333333333333 \cdot \color{blue}{\left(re \cdot re\right)} - 0.16666666666666666, re\right) \]

   10. Applied rewrites 62.7%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \color{blue}{\mathsf{fma}\left({re}^{3}, 0.008333333333333333 \cdot \left(re \cdot re\right) - 0.16666666666666666, re\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(im, \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right), {re}^{4} \cdot \left(im \cdot \mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re \cdot re, 0.004166666666666667\right)\right)\right) \cdot re\right) \cdot im\\ \mathbf{elif}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left({re}^{3}, 0.008333333333333333 \cdot \left(re \cdot re\right) - 0.16666666666666666, re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left(re \cdot re, re \cdot -0.16666666666666666, re\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{\left(im \cdot im\right) \cdot \left(im \cdot im\right)}{im}}{im}, 0.5, 1\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma
       (fma
        (fma 0.001388888888888889 (* im im) 0.041666666666666664)
        (* im im)
        0.5)
       (* im im)
       1.0)
      (fma (* re re) (* re -0.16666666666666666) re))
     (if (<= t_0 1.0)
       (* (fma (* 0.5 im) im 1.0) (sin re))
       (* (fma (/ (/ (* (* im im) (* im im)) im) im) 0.5 1.0) re)))))

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(fma(fma(0.001388888888888889, (im * im), 0.041666666666666664), (im * im), 0.5), (im * im), 1.0) * fma((re * re), (re * -0.16666666666666666), re);
	} else if (t_0 <= 1.0) {
		tmp = fma((0.5 * im), im, 1.0) * sin(re);
	} else {
		tmp = fma(((((im * im) * (im * im)) / im) / im), 0.5, 1.0) * re;
	}
	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(fma(fma(fma(0.001388888888888889, Float64(im * im), 0.041666666666666664), Float64(im * im), 0.5), Float64(im * im), 1.0) * fma(Float64(re * re), Float64(re * -0.16666666666666666), re));
	elseif (t_0 <= 1.0)
		tmp = Float64(fma(Float64(0.5 * im), im, 1.0) * sin(re));
	else
		tmp = Float64(fma(Float64(Float64(Float64(Float64(im * im) * Float64(im * im)) / im) / im), 0.5, 1.0) * re);
	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[(0.001388888888888889 * N[(im * im), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * N[(re * -0.16666666666666666), $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[(N[(N[(0.5 * im), $MachinePrecision] * im + 1.0), $MachinePrecision] * N[Sin[re], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] / im), $MachinePrecision] / im), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * re), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(\frac{1}{2} \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)}\right) \]

      6. +-commutative N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(e^{im} + e^{0 - im}\right)}\right) \]

      7. lift-exp.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(\color{blue}{e^{im}} + e^{0 - im}\right)\right) \]

      8. lift-exp.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + \color{blue}{e^{0 - im}}\right)\right) \]

      9. lift--.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + e^{\color{blue}{0 - im}}\right)\right) \]

      10. sub0-neg N/A

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

      11. cosh-undef N/A

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

      12. associate-*r* N/A

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

      13. metadata-eval N/A

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

      14. cosh-0 N/A

          \[\leadsto \sin re \cdot \left(\color{blue}{\cosh 0} \cdot \cosh im\right) \]

      15. *-commutative N/A

          \[\leadsto \color{blue}{\left(\cosh 0 \cdot \cosh im\right) \cdot \sin re} \]

      16. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\cosh 0 \cdot \cosh im\right) \cdot \sin re} \]

      17. cosh-0 N/A

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

      18. exp-0 N/A

          \[\leadsto \left(\color{blue}{e^{0}} \cdot \cosh im\right) \cdot \sin re \]

      19. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(e^{0} \cdot \cosh im\right)} \cdot \sin re \]

      20. exp-0 N/A

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

      21. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \cdot \sin re \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + 1\right)} \cdot \sin re \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) \cdot {im}^{2}} + 1\right) \cdot \sin re \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right), {im}^{2}, 1\right)} \cdot \sin re \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) + \frac{1}{2}}, {im}^{2}, 1\right) \cdot \sin re \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) \cdot {im}^{2}} + \frac{1}{2}, {im}^{2}, 1\right) \cdot \sin re \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}, {im}^{2}, \frac{1}{2}\right)}, {im}^{2}, 1\right) \cdot \sin re \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{720} \cdot {im}^{2} + \frac{1}{24}}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{720}, {im}^{2}, \frac{1}{24}\right)}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{im \cdot im}, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{im \cdot im}, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), \color{blue}{im \cdot im}, 1\right) \cdot \sin re \]

      14. lower-*.f64 79.5

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), \color{blue}{im \cdot im}, 1\right) \cdot \sin re \]

   7. Applied rewrites 79.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right)} \cdot \sin re \]

   8. Taylor expanded in re around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \left(re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2} + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \color{blue}{\left(re \cdot \left(\frac{-1}{6} \cdot {re}^{2}\right) + re \cdot 1\right)} \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \left(re \cdot \color{blue}{\left({re}^{2} \cdot \frac{-1}{6}\right)} + re \cdot 1\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \left(\color{blue}{\left(re \cdot {re}^{2}\right) \cdot \frac{-1}{6}} + re \cdot 1\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \left(\left(re \cdot \color{blue}{\left(re \cdot re\right)}\right) \cdot \frac{-1}{6} + re \cdot 1\right) \]

      6. cube-unmult N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \left(\color{blue}{{re}^{3}} \cdot \frac{-1}{6} + re \cdot 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \left({re}^{\color{blue}{\left(2 + 1\right)}} \cdot \frac{-1}{6} + re \cdot 1\right) \]

      8. pow-plus N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \left(\color{blue}{\left({re}^{2} \cdot re\right)} \cdot \frac{-1}{6} + re \cdot 1\right) \]

      9. *-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \left(\left({re}^{2} \cdot re\right) \cdot \frac{-1}{6} + \color{blue}{re}\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \color{blue}{\mathsf{fma}\left({re}^{2} \cdot re, \frac{-1}{6}, re\right)} \]

      11. pow-plus N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left(\color{blue}{{re}^{\left(2 + 1\right)}}, \frac{-1}{6}, re\right) \]

      12. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left(\color{blue}{{re}^{\left(2 + 1\right)}}, \frac{-1}{6}, re\right) \]

      13. metadata-eval 55.9

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left({re}^{\color{blue}{3}}, -0.16666666666666666, re\right) \]

   10. Applied rewrites 55.9%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \color{blue}{\mathsf{fma}\left({re}^{3}, -0.16666666666666666, re\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 55.9%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{re \cdot -0.16666666666666666}, re\right) \]

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 99.8

          \[\leadsto \mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \color{blue}{\sin re} \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \sin re} \]

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 49.3

          \[\leadsto \mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \color{blue}{\sin re} \]

   5. Applied rewrites 49.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \sin re} \]

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 43.8%

      \[\leadsto \mathsf{fma}\left(im \cdot im, 0.5, 1\right) \cdot \color{blue}{re} \]
   9. Step-by-step derivation

   10. Applied rewrites 49.5%

       \[\leadsto \mathsf{fma}\left(\frac{\left(-im \cdot im\right) \cdot \left(-im\right)}{im}, 0.5, 1\right) \cdot re \]
   11. Step-by-step derivation

   12. Applied rewrites 50.9%

       \[\leadsto \mathsf{fma}\left(\frac{\frac{\left(\left(-im\right) \cdot im\right) \cdot \left(\left(-im\right) \cdot im\right)}{im}}{im}, 0.5, 1\right) \cdot re \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left(re \cdot re, re \cdot -0.16666666666666666, re\right)\\ \mathbf{elif}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 1:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{\left(im \cdot im\right) \cdot \left(im \cdot im\right)}{im}}{im}, 0.5, 1\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left(re \cdot re, re \cdot -0.16666666666666666, re\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;1 \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{\left(im \cdot im\right) \cdot \left(im \cdot im\right)}{im}}{im}, 0.5, 1\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma
       (fma
        (fma 0.001388888888888889 (* im im) 0.041666666666666664)
        (* im im)
        0.5)
       (* im im)
       1.0)
      (fma (* re re) (* re -0.16666666666666666) re))
     (if (<= t_0 1.0)
       (* 1.0 (sin re))
       (* (fma (/ (/ (* (* im im) (* im im)) im) im) 0.5 1.0) re)))))

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(fma(fma(0.001388888888888889, (im * im), 0.041666666666666664), (im * im), 0.5), (im * im), 1.0) * fma((re * re), (re * -0.16666666666666666), re);
	} else if (t_0 <= 1.0) {
		tmp = 1.0 * sin(re);
	} else {
		tmp = fma(((((im * im) * (im * im)) / im) / im), 0.5, 1.0) * re;
	}
	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(fma(fma(fma(0.001388888888888889, Float64(im * im), 0.041666666666666664), Float64(im * im), 0.5), Float64(im * im), 1.0) * fma(Float64(re * re), Float64(re * -0.16666666666666666), re));
	elseif (t_0 <= 1.0)
		tmp = Float64(1.0 * sin(re));
	else
		tmp = Float64(fma(Float64(Float64(Float64(Float64(im * im) * Float64(im * im)) / im) / im), 0.5, 1.0) * re);
	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[(0.001388888888888889 * N[(im * im), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * N[(re * -0.16666666666666666), $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[(1.0 * N[Sin[re], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] / im), $MachinePrecision] / im), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * re), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(\frac{1}{2} \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)}\right) \]

      6. +-commutative N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(e^{im} + e^{0 - im}\right)}\right) \]

      7. lift-exp.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(\color{blue}{e^{im}} + e^{0 - im}\right)\right) \]

      8. lift-exp.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + \color{blue}{e^{0 - im}}\right)\right) \]

      9. lift--.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + e^{\color{blue}{0 - im}}\right)\right) \]

      10. sub0-neg N/A

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

      11. cosh-undef N/A

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

      12. associate-*r* N/A

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

      13. metadata-eval N/A

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

      14. cosh-0 N/A

          \[\leadsto \sin re \cdot \left(\color{blue}{\cosh 0} \cdot \cosh im\right) \]

      15. *-commutative N/A

          \[\leadsto \color{blue}{\left(\cosh 0 \cdot \cosh im\right) \cdot \sin re} \]

      16. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\cosh 0 \cdot \cosh im\right) \cdot \sin re} \]

      17. cosh-0 N/A

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

      18. exp-0 N/A

          \[\leadsto \left(\color{blue}{e^{0}} \cdot \cosh im\right) \cdot \sin re \]

      19. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(e^{0} \cdot \cosh im\right)} \cdot \sin re \]

      20. exp-0 N/A

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

      21. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \cdot \sin re \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + 1\right)} \cdot \sin re \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) \cdot {im}^{2}} + 1\right) \cdot \sin re \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right), {im}^{2}, 1\right)} \cdot \sin re \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) + \frac{1}{2}}, {im}^{2}, 1\right) \cdot \sin re \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) \cdot {im}^{2}} + \frac{1}{2}, {im}^{2}, 1\right) \cdot \sin re \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}, {im}^{2}, \frac{1}{2}\right)}, {im}^{2}, 1\right) \cdot \sin re \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{720} \cdot {im}^{2} + \frac{1}{24}}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{720}, {im}^{2}, \frac{1}{24}\right)}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{im \cdot im}, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{im \cdot im}, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), \color{blue}{im \cdot im}, 1\right) \cdot \sin re \]

      14. lower-*.f64 79.5

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), \color{blue}{im \cdot im}, 1\right) \cdot \sin re \]

   7. Applied rewrites 79.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right)} \cdot \sin re \]

   8. Taylor expanded in re around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \left(re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2} + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \color{blue}{\left(re \cdot \left(\frac{-1}{6} \cdot {re}^{2}\right) + re \cdot 1\right)} \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \left(re \cdot \color{blue}{\left({re}^{2} \cdot \frac{-1}{6}\right)} + re \cdot 1\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \left(\color{blue}{\left(re \cdot {re}^{2}\right) \cdot \frac{-1}{6}} + re \cdot 1\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \left(\left(re \cdot \color{blue}{\left(re \cdot re\right)}\right) \cdot \frac{-1}{6} + re \cdot 1\right) \]

      6. cube-unmult N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \left(\color{blue}{{re}^{3}} \cdot \frac{-1}{6} + re \cdot 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \left({re}^{\color{blue}{\left(2 + 1\right)}} \cdot \frac{-1}{6} + re \cdot 1\right) \]

      8. pow-plus N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \left(\color{blue}{\left({re}^{2} \cdot re\right)} \cdot \frac{-1}{6} + re \cdot 1\right) \]

      9. *-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \left(\left({re}^{2} \cdot re\right) \cdot \frac{-1}{6} + \color{blue}{re}\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \color{blue}{\mathsf{fma}\left({re}^{2} \cdot re, \frac{-1}{6}, re\right)} \]

      11. pow-plus N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left(\color{blue}{{re}^{\left(2 + 1\right)}}, \frac{-1}{6}, re\right) \]

      12. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left(\color{blue}{{re}^{\left(2 + 1\right)}}, \frac{-1}{6}, re\right) \]

      13. metadata-eval 55.9

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left({re}^{\color{blue}{3}}, -0.16666666666666666, re\right) \]

   10. Applied rewrites 55.9%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \color{blue}{\mathsf{fma}\left({re}^{3}, -0.16666666666666666, re\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 55.9%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{re \cdot -0.16666666666666666}, re\right) \]

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 99.8

          \[\leadsto \mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \color{blue}{\sin re} \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \sin re} \]

   6. Taylor expanded in im around 0

      \[\leadsto 1 \cdot \sin \color{blue}{re} \]
   7. Step-by-step derivation

   8. Applied rewrites 99.6%

      \[\leadsto 1 \cdot \sin \color{blue}{re} \]

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 49.3

          \[\leadsto \mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \color{blue}{\sin re} \]

   5. Applied rewrites 49.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \sin re} \]

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 43.8%

      \[\leadsto \mathsf{fma}\left(im \cdot im, 0.5, 1\right) \cdot \color{blue}{re} \]
   9. Step-by-step derivation

   10. Applied rewrites 49.5%

       \[\leadsto \mathsf{fma}\left(\frac{\left(-im \cdot im\right) \cdot \left(-im\right)}{im}, 0.5, 1\right) \cdot re \]
   11. Step-by-step derivation

   12. Applied rewrites 50.9%

       \[\leadsto \mathsf{fma}\left(\frac{\frac{\left(\left(-im\right) \cdot im\right) \cdot \left(\left(-im\right) \cdot im\right)}{im}}{im}, 0.5, 1\right) \cdot re \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left(re \cdot re, re \cdot -0.16666666666666666, re\right)\\ \mathbf{elif}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 1:\\ \;\;\;\;1 \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{\left(im \cdot im\right) \cdot \left(im \cdot im\right)}{im}}{im}, 0.5, 1\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(im, \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right), {re}^{4} \cdot \left(im \cdot \mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re \cdot re, 0.004166666666666667\right)\right)\right) \cdot re\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right) \cdot im, im, 1\right) \cdot \sin re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im))) (- INFINITY))
   (*
    (*
     (fma
      im
      (fma (* re re) -0.08333333333333333 0.5)
      (*
       (pow re 4.0)
       (* im (fma -9.92063492063492e-5 (* re re) 0.004166666666666667))))
     re)
    im)
   (*
    (fma
     (*
      (fma
       (fma 0.001388888888888889 (* im im) 0.041666666666666664)
       (* im im)
       0.5)
      im)
     im
     1.0)
    (sin re))))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(-im) + exp(im))) <= -((double) INFINITY)) {
		tmp = (fma(im, fma((re * re), -0.08333333333333333, 0.5), (pow(re, 4.0) * (im * fma(-9.92063492063492e-5, (re * re), 0.004166666666666667)))) * re) * im;
	} else {
		tmp = fma((fma(fma(0.001388888888888889, (im * im), 0.041666666666666664), (im * im), 0.5) * im), im, 1.0) * 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(fma(im, fma(Float64(re * re), -0.08333333333333333, 0.5), Float64((re ^ 4.0) * Float64(im * fma(-9.92063492063492e-5, Float64(re * re), 0.004166666666666667)))) * re) * im);
	else
		tmp = Float64(fma(Float64(fma(fma(0.001388888888888889, Float64(im * im), 0.041666666666666664), Float64(im * im), 0.5) * im), im, 1.0) * sin(re));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(N[(N[(im * N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] + N[(N[Power[re, 4.0], $MachinePrecision] * N[(im * N[(-9.92063492063492e-5 * N[(re * re), $MachinePrecision] + 0.004166666666666667), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(N[(N[(0.001388888888888889 * N[(im * im), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * im), $MachinePrecision] * im + 1.0), $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 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -inf.0

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 45.0

          \[\leadsto \mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \color{blue}{\sin re} \]

   5. Applied rewrites 45.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \sin re} \]

   6. Taylor expanded in im around inf

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

   8. Applied rewrites 36.7%

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

   9. Taylor expanded in re around 0

      \[\leadsto \left(re \cdot \left(\frac{1}{2} \cdot im + {re}^{2} \cdot \left(\frac{-1}{12} \cdot im + {re}^{2} \cdot \left(\frac{-1}{10080} \cdot \left(im \cdot {re}^{2}\right) + \frac{1}{240} \cdot im\right)\right)\right)\right) \cdot im \]
   10. Step-by-step derivation

   11. Applied rewrites 32.0%

       \[\leadsto \left(\mathsf{fma}\left(im, \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right), {re}^{4} \cdot \left(im \cdot \mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re \cdot re, 0.004166666666666667\right)\right)\right) \cdot re\right) \cdot im \]

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(\frac{1}{2} \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)}\right) \]

      6. +-commutative N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(e^{im} + e^{0 - im}\right)}\right) \]

      7. lift-exp.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(\color{blue}{e^{im}} + e^{0 - im}\right)\right) \]

      8. lift-exp.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + \color{blue}{e^{0 - im}}\right)\right) \]

      9. lift--.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + e^{\color{blue}{0 - im}}\right)\right) \]

      10. sub0-neg N/A

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

      11. cosh-undef N/A

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

      12. associate-*r* N/A

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

      13. metadata-eval N/A

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

      14. cosh-0 N/A

          \[\leadsto \sin re \cdot \left(\color{blue}{\cosh 0} \cdot \cosh im\right) \]

      15. *-commutative N/A

          \[\leadsto \color{blue}{\left(\cosh 0 \cdot \cosh im\right) \cdot \sin re} \]

      16. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\cosh 0 \cdot \cosh im\right) \cdot \sin re} \]

      17. cosh-0 N/A

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

      18. exp-0 N/A

          \[\leadsto \left(\color{blue}{e^{0}} \cdot \cosh im\right) \cdot \sin re \]

      19. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(e^{0} \cdot \cosh im\right)} \cdot \sin re \]

      20. exp-0 N/A

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

      21. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \cdot \sin re \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + 1\right)} \cdot \sin re \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) \cdot {im}^{2}} + 1\right) \cdot \sin re \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right), {im}^{2}, 1\right)} \cdot \sin re \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) + \frac{1}{2}}, {im}^{2}, 1\right) \cdot \sin re \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) \cdot {im}^{2}} + \frac{1}{2}, {im}^{2}, 1\right) \cdot \sin re \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}, {im}^{2}, \frac{1}{2}\right)}, {im}^{2}, 1\right) \cdot \sin re \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{720} \cdot {im}^{2} + \frac{1}{24}}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{720}, {im}^{2}, \frac{1}{24}\right)}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{im \cdot im}, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{im \cdot im}, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), \color{blue}{im \cdot im}, 1\right) \cdot \sin re \]

      14. lower-*.f64 92.3

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), \color{blue}{im \cdot im}, 1\right) \cdot \sin re \]

   7. Applied rewrites 92.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right)} \cdot \sin re \]
   8. Step-by-step derivation

   9. Applied rewrites 92.3%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right) \cdot im, \color{blue}{im}, 1\right) \cdot \sin re \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(im, \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right), {re}^{4} \cdot \left(im \cdot \mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re \cdot re, 0.004166666666666667\right)\right)\right) \cdot re\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right) \cdot im, im, 1\right) \cdot \sin re\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right)\\ \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(t\_0, im \cdot im, 1\right) \cdot \mathsf{fma}\left(re \cdot re, re \cdot -0.16666666666666666, re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0 \cdot im, im, 1\right) \cdot \sin re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (fma
          (fma 0.001388888888888889 (* im im) 0.041666666666666664)
          (* im im)
          0.5)))
   (if (<= (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im))) (- INFINITY))
     (* (fma t_0 (* im im) 1.0) (fma (* re re) (* re -0.16666666666666666) re))
     (* (fma (* t_0 im) im 1.0) (sin re)))))

C

double code(double re, double im) {
	double t_0 = fma(fma(0.001388888888888889, (im * im), 0.041666666666666664), (im * im), 0.5);
	double tmp;
	if (((0.5 * sin(re)) * (exp(-im) + exp(im))) <= -((double) INFINITY)) {
		tmp = fma(t_0, (im * im), 1.0) * fma((re * re), (re * -0.16666666666666666), re);
	} else {
		tmp = fma((t_0 * im), im, 1.0) * sin(re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(fma(0.001388888888888889, Float64(im * im), 0.041666666666666664), Float64(im * im), 0.5)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) + exp(im))) <= Float64(-Inf))
		tmp = Float64(fma(t_0, Float64(im * im), 1.0) * fma(Float64(re * re), Float64(re * -0.16666666666666666), re));
	else
		tmp = Float64(fma(Float64(t_0 * im), im, 1.0) * sin(re));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.001388888888888889 * N[(im * im), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision]}, 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[(t$95$0 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * N[(re * -0.16666666666666666), $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$0 * im), $MachinePrecision] * im + 1.0), $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 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -inf.0

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(\frac{1}{2} \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)}\right) \]

      6. +-commutative N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(e^{im} + e^{0 - im}\right)}\right) \]

      7. lift-exp.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(\color{blue}{e^{im}} + e^{0 - im}\right)\right) \]

      8. lift-exp.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + \color{blue}{e^{0 - im}}\right)\right) \]

      9. lift--.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + e^{\color{blue}{0 - im}}\right)\right) \]

      10. sub0-neg N/A

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

      11. cosh-undef N/A

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

      12. associate-*r* N/A

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

      13. metadata-eval N/A

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

      14. cosh-0 N/A

          \[\leadsto \sin re \cdot \left(\color{blue}{\cosh 0} \cdot \cosh im\right) \]

      15. *-commutative N/A

          \[\leadsto \color{blue}{\left(\cosh 0 \cdot \cosh im\right) \cdot \sin re} \]

      16. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\cosh 0 \cdot \cosh im\right) \cdot \sin re} \]

      17. cosh-0 N/A

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

      18. exp-0 N/A

          \[\leadsto \left(\color{blue}{e^{0}} \cdot \cosh im\right) \cdot \sin re \]

      19. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(e^{0} \cdot \cosh im\right)} \cdot \sin re \]

      20. exp-0 N/A

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

      21. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \cdot \sin re \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + 1\right)} \cdot \sin re \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) \cdot {im}^{2}} + 1\right) \cdot \sin re \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right), {im}^{2}, 1\right)} \cdot \sin re \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) + \frac{1}{2}}, {im}^{2}, 1\right) \cdot \sin re \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) \cdot {im}^{2}} + \frac{1}{2}, {im}^{2}, 1\right) \cdot \sin re \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}, {im}^{2}, \frac{1}{2}\right)}, {im}^{2}, 1\right) \cdot \sin re \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{720} \cdot {im}^{2} + \frac{1}{24}}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{720}, {im}^{2}, \frac{1}{24}\right)}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{im \cdot im}, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{im \cdot im}, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), \color{blue}{im \cdot im}, 1\right) \cdot \sin re \]

      14. lower-*.f64 79.5

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), \color{blue}{im \cdot im}, 1\right) \cdot \sin re \]

   7. Applied rewrites 79.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right)} \cdot \sin re \]

   8. Taylor expanded in re around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \left(re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2} + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \color{blue}{\left(re \cdot \left(\frac{-1}{6} \cdot {re}^{2}\right) + re \cdot 1\right)} \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \left(re \cdot \color{blue}{\left({re}^{2} \cdot \frac{-1}{6}\right)} + re \cdot 1\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \left(\color{blue}{\left(re \cdot {re}^{2}\right) \cdot \frac{-1}{6}} + re \cdot 1\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \left(\left(re \cdot \color{blue}{\left(re \cdot re\right)}\right) \cdot \frac{-1}{6} + re \cdot 1\right) \]

      6. cube-unmult N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \left(\color{blue}{{re}^{3}} \cdot \frac{-1}{6} + re \cdot 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \left({re}^{\color{blue}{\left(2 + 1\right)}} \cdot \frac{-1}{6} + re \cdot 1\right) \]

      8. pow-plus N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \left(\color{blue}{\left({re}^{2} \cdot re\right)} \cdot \frac{-1}{6} + re \cdot 1\right) \]

      9. *-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \left(\left({re}^{2} \cdot re\right) \cdot \frac{-1}{6} + \color{blue}{re}\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \color{blue}{\mathsf{fma}\left({re}^{2} \cdot re, \frac{-1}{6}, re\right)} \]

      11. pow-plus N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left(\color{blue}{{re}^{\left(2 + 1\right)}}, \frac{-1}{6}, re\right) \]

      12. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left(\color{blue}{{re}^{\left(2 + 1\right)}}, \frac{-1}{6}, re\right) \]

      13. metadata-eval 55.9

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left({re}^{\color{blue}{3}}, -0.16666666666666666, re\right) \]

   10. Applied rewrites 55.9%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \color{blue}{\mathsf{fma}\left({re}^{3}, -0.16666666666666666, re\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 55.9%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{re \cdot -0.16666666666666666}, re\right) \]

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(\frac{1}{2} \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)}\right) \]

      6. +-commutative N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(e^{im} + e^{0 - im}\right)}\right) \]

      7. lift-exp.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(\color{blue}{e^{im}} + e^{0 - im}\right)\right) \]

      8. lift-exp.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + \color{blue}{e^{0 - im}}\right)\right) \]

      9. lift--.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + e^{\color{blue}{0 - im}}\right)\right) \]

      10. sub0-neg N/A

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

      11. cosh-undef N/A

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

      12. associate-*r* N/A

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

      13. metadata-eval N/A

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

      14. cosh-0 N/A

          \[\leadsto \sin re \cdot \left(\color{blue}{\cosh 0} \cdot \cosh im\right) \]

      15. *-commutative N/A

          \[\leadsto \color{blue}{\left(\cosh 0 \cdot \cosh im\right) \cdot \sin re} \]

      16. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\cosh 0 \cdot \cosh im\right) \cdot \sin re} \]

      17. cosh-0 N/A

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

      18. exp-0 N/A

          \[\leadsto \left(\color{blue}{e^{0}} \cdot \cosh im\right) \cdot \sin re \]

      19. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(e^{0} \cdot \cosh im\right)} \cdot \sin re \]

      20. exp-0 N/A

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

      21. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \cdot \sin re \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + 1\right)} \cdot \sin re \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) \cdot {im}^{2}} + 1\right) \cdot \sin re \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right), {im}^{2}, 1\right)} \cdot \sin re \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) + \frac{1}{2}}, {im}^{2}, 1\right) \cdot \sin re \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) \cdot {im}^{2}} + \frac{1}{2}, {im}^{2}, 1\right) \cdot \sin re \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}, {im}^{2}, \frac{1}{2}\right)}, {im}^{2}, 1\right) \cdot \sin re \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{720} \cdot {im}^{2} + \frac{1}{24}}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{720}, {im}^{2}, \frac{1}{24}\right)}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{im \cdot im}, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{im \cdot im}, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), \color{blue}{im \cdot im}, 1\right) \cdot \sin re \]

      14. lower-*.f64 92.3

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), \color{blue}{im \cdot im}, 1\right) \cdot \sin re \]

   7. Applied rewrites 92.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right)} \cdot \sin re \]
   8. Step-by-step derivation

   9. Applied rewrites 92.3%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right) \cdot im, \color{blue}{im}, 1\right) \cdot \sin re \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left(re \cdot re, re \cdot -0.16666666666666666, re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right) \cdot im, im, 1\right) \cdot \sin re\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left(re \cdot re, re \cdot -0.16666666666666666, re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889 \cdot \left(im \cdot im\right), im \cdot im, 0.5\right) \cdot im, im, 1\right) \cdot \sin re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im))) (- INFINITY))
   (*
    (fma
     (fma
      (fma 0.001388888888888889 (* im im) 0.041666666666666664)
      (* im im)
      0.5)
     (* im im)
     1.0)
    (fma (* re re) (* re -0.16666666666666666) re))
   (*
    (fma (* (fma (* 0.001388888888888889 (* im im)) (* im im) 0.5) im) im 1.0)
    (sin re))))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(-im) + exp(im))) <= -((double) INFINITY)) {
		tmp = fma(fma(fma(0.001388888888888889, (im * im), 0.041666666666666664), (im * im), 0.5), (im * im), 1.0) * fma((re * re), (re * -0.16666666666666666), re);
	} else {
		tmp = fma((fma((0.001388888888888889 * (im * im)), (im * im), 0.5) * im), im, 1.0) * 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(fma(fma(fma(0.001388888888888889, Float64(im * im), 0.041666666666666664), Float64(im * im), 0.5), Float64(im * im), 1.0) * fma(Float64(re * re), Float64(re * -0.16666666666666666), re));
	else
		tmp = Float64(fma(Float64(fma(Float64(0.001388888888888889 * Float64(im * im)), Float64(im * im), 0.5) * im), im, 1.0) * sin(re));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(N[(N[(N[(0.001388888888888889 * N[(im * im), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * N[(re * -0.16666666666666666), $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.001388888888888889 * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * im), $MachinePrecision] * im + 1.0), $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 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -inf.0

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(\frac{1}{2} \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)}\right) \]

      6. +-commutative N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(e^{im} + e^{0 - im}\right)}\right) \]

      7. lift-exp.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(\color{blue}{e^{im}} + e^{0 - im}\right)\right) \]

      8. lift-exp.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + \color{blue}{e^{0 - im}}\right)\right) \]

      9. lift--.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + e^{\color{blue}{0 - im}}\right)\right) \]

      10. sub0-neg N/A

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

      11. cosh-undef N/A

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

      12. associate-*r* N/A

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

      13. metadata-eval N/A

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

      14. cosh-0 N/A

          \[\leadsto \sin re \cdot \left(\color{blue}{\cosh 0} \cdot \cosh im\right) \]

      15. *-commutative N/A

          \[\leadsto \color{blue}{\left(\cosh 0 \cdot \cosh im\right) \cdot \sin re} \]

      16. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\cosh 0 \cdot \cosh im\right) \cdot \sin re} \]

      17. cosh-0 N/A

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

      18. exp-0 N/A

          \[\leadsto \left(\color{blue}{e^{0}} \cdot \cosh im\right) \cdot \sin re \]

      19. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(e^{0} \cdot \cosh im\right)} \cdot \sin re \]

      20. exp-0 N/A

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

      21. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \cdot \sin re \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + 1\right)} \cdot \sin re \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) \cdot {im}^{2}} + 1\right) \cdot \sin re \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right), {im}^{2}, 1\right)} \cdot \sin re \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) + \frac{1}{2}}, {im}^{2}, 1\right) \cdot \sin re \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) \cdot {im}^{2}} + \frac{1}{2}, {im}^{2}, 1\right) \cdot \sin re \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}, {im}^{2}, \frac{1}{2}\right)}, {im}^{2}, 1\right) \cdot \sin re \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{720} \cdot {im}^{2} + \frac{1}{24}}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{720}, {im}^{2}, \frac{1}{24}\right)}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{im \cdot im}, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{im \cdot im}, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), \color{blue}{im \cdot im}, 1\right) \cdot \sin re \]

      14. lower-*.f64 79.5

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), \color{blue}{im \cdot im}, 1\right) \cdot \sin re \]

   7. Applied rewrites 79.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right)} \cdot \sin re \]

   8. Taylor expanded in re around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \left(re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2} + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \color{blue}{\left(re \cdot \left(\frac{-1}{6} \cdot {re}^{2}\right) + re \cdot 1\right)} \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \left(re \cdot \color{blue}{\left({re}^{2} \cdot \frac{-1}{6}\right)} + re \cdot 1\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \left(\color{blue}{\left(re \cdot {re}^{2}\right) \cdot \frac{-1}{6}} + re \cdot 1\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \left(\left(re \cdot \color{blue}{\left(re \cdot re\right)}\right) \cdot \frac{-1}{6} + re \cdot 1\right) \]

      6. cube-unmult N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \left(\color{blue}{{re}^{3}} \cdot \frac{-1}{6} + re \cdot 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \left({re}^{\color{blue}{\left(2 + 1\right)}} \cdot \frac{-1}{6} + re \cdot 1\right) \]

      8. pow-plus N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \left(\color{blue}{\left({re}^{2} \cdot re\right)} \cdot \frac{-1}{6} + re \cdot 1\right) \]

      9. *-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \left(\left({re}^{2} \cdot re\right) \cdot \frac{-1}{6} + \color{blue}{re}\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \color{blue}{\mathsf{fma}\left({re}^{2} \cdot re, \frac{-1}{6}, re\right)} \]

      11. pow-plus N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left(\color{blue}{{re}^{\left(2 + 1\right)}}, \frac{-1}{6}, re\right) \]

      12. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left(\color{blue}{{re}^{\left(2 + 1\right)}}, \frac{-1}{6}, re\right) \]

      13. metadata-eval 55.9

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left({re}^{\color{blue}{3}}, -0.16666666666666666, re\right) \]

   10. Applied rewrites 55.9%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \color{blue}{\mathsf{fma}\left({re}^{3}, -0.16666666666666666, re\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 55.9%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{re \cdot -0.16666666666666666}, re\right) \]

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(\frac{1}{2} \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)}\right) \]

      6. +-commutative N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(e^{im} + e^{0 - im}\right)}\right) \]

      7. lift-exp.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(\color{blue}{e^{im}} + e^{0 - im}\right)\right) \]

      8. lift-exp.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + \color{blue}{e^{0 - im}}\right)\right) \]

      9. lift--.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + e^{\color{blue}{0 - im}}\right)\right) \]

      10. sub0-neg N/A

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

      11. cosh-undef N/A

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

      12. associate-*r* N/A

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

      13. metadata-eval N/A

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

      14. cosh-0 N/A

          \[\leadsto \sin re \cdot \left(\color{blue}{\cosh 0} \cdot \cosh im\right) \]

      15. *-commutative N/A

          \[\leadsto \color{blue}{\left(\cosh 0 \cdot \cosh im\right) \cdot \sin re} \]

      16. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\cosh 0 \cdot \cosh im\right) \cdot \sin re} \]

      17. cosh-0 N/A

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

      18. exp-0 N/A

          \[\leadsto \left(\color{blue}{e^{0}} \cdot \cosh im\right) \cdot \sin re \]

      19. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(e^{0} \cdot \cosh im\right)} \cdot \sin re \]

      20. exp-0 N/A

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

      21. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \cdot \sin re \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + 1\right)} \cdot \sin re \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) \cdot {im}^{2}} + 1\right) \cdot \sin re \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right), {im}^{2}, 1\right)} \cdot \sin re \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) + \frac{1}{2}}, {im}^{2}, 1\right) \cdot \sin re \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) \cdot {im}^{2}} + \frac{1}{2}, {im}^{2}, 1\right) \cdot \sin re \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}, {im}^{2}, \frac{1}{2}\right)}, {im}^{2}, 1\right) \cdot \sin re \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{720} \cdot {im}^{2} + \frac{1}{24}}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{720}, {im}^{2}, \frac{1}{24}\right)}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{im \cdot im}, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{im \cdot im}, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), \color{blue}{im \cdot im}, 1\right) \cdot \sin re \]

      14. lower-*.f64 92.3

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), \color{blue}{im \cdot im}, 1\right) \cdot \sin re \]

   7. Applied rewrites 92.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right)} \cdot \sin re \]
   8. Step-by-step derivation

   9. Applied rewrites 92.3%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right) \cdot im, \color{blue}{im}, 1\right) \cdot \sin re \]

   10. Taylor expanded in im around inf

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720} \cdot {im}^{2}, im \cdot im, \frac{1}{2}\right) \cdot im, im, 1\right) \cdot \sin re \]
   11. Step-by-step derivation

   12. Applied rewrites 92.2%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889 \cdot \left(im \cdot im\right), im \cdot im, 0.5\right) \cdot im, im, 1\right) \cdot \sin re \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left(re \cdot re, re \cdot -0.16666666666666666, re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889 \cdot \left(im \cdot im\right), im \cdot im, 0.5\right) \cdot im, im, 1\right) \cdot \sin re\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 50.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 4.00000000000000019e-4

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 80.7

          \[\leadsto \mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \color{blue}{\sin re} \]

   5. Applied rewrites 80.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \sin re} \]

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 55.0%

      \[\leadsto \left(\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, 0.5, 1\right)\right) \cdot \color{blue}{re} \]

   if 4.00000000000000019e-4 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 64.7

          \[\leadsto \mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \color{blue}{\sin re} \]

   5. Applied rewrites 64.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \sin re} \]

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 31.5%

      \[\leadsto \mathsf{fma}\left(im \cdot im, 0.5, 1\right) \cdot \color{blue}{re} \]
   9. Step-by-step derivation

   10. Applied rewrites 35.5%

       \[\leadsto \mathsf{fma}\left(\frac{\left(-im \cdot im\right) \cdot \left(-im\right)}{im}, 0.5, 1\right) \cdot re \]
   11. Step-by-step derivation

   12. Applied rewrites 35.5%

       \[\leadsto \mathsf{fma}\left(\frac{\left(im \cdot im\right) \cdot im}{im}, 0.5, 1\right) \cdot re \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.0004:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, 0.5, 1\right)\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(im \cdot im\right) \cdot im}{im}, 0.5, 1\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 55.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(im \cdot im, 0.5, 1\right)\\ t_1 := 0.5 \cdot \sin re\\ \mathbf{if}\;t\_1 \leq -0.01:\\ \;\;\;\;\left(\left(im \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re\right) \cdot im\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{\left(im \cdot im\right) \cdot \left(im \cdot im\right)}{im}}{im}, 0.5, 1\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, 0.008333333333333333, -0.16666666666666666\right) \cdot t\_0, re \cdot re, t\_0\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (fma (* im im) 0.5 1.0)) (t_1 (* 0.5 (sin re))))
   (if (<= t_1 -0.01)
     (* (* (* im (fma (* re re) -0.08333333333333333 0.5)) re) im)
     (if (<= t_1 4e-7)
       (* (fma (/ (/ (* (* im im) (* im im)) im) im) 0.5 1.0) re)
       (*
        (fma
         (* (fma (* re re) 0.008333333333333333 -0.16666666666666666) t_0)
         (* re re)
         t_0)
        re)))))

C

double code(double re, double im) {
	double t_0 = fma((im * im), 0.5, 1.0);
	double t_1 = 0.5 * sin(re);
	double tmp;
	if (t_1 <= -0.01) {
		tmp = ((im * fma((re * re), -0.08333333333333333, 0.5)) * re) * im;
	} else if (t_1 <= 4e-7) {
		tmp = fma(((((im * im) * (im * im)) / im) / im), 0.5, 1.0) * re;
	} else {
		tmp = fma((fma((re * re), 0.008333333333333333, -0.16666666666666666) * t_0), (re * re), t_0) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(Float64(im * im), 0.5, 1.0)
	t_1 = Float64(0.5 * sin(re))
	tmp = 0.0
	if (t_1 <= -0.01)
		tmp = Float64(Float64(Float64(im * fma(Float64(re * re), -0.08333333333333333, 0.5)) * re) * im);
	elseif (t_1 <= 4e-7)
		tmp = Float64(fma(Float64(Float64(Float64(Float64(im * im) * Float64(im * im)) / im) / im), 0.5, 1.0) * re);
	else
		tmp = Float64(fma(Float64(fma(Float64(re * re), 0.008333333333333333, -0.16666666666666666) * t_0), Float64(re * re), t_0) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(im * im), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.01], N[(N[(N[(im * N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision] * im), $MachinePrecision], If[LessEqual[t$95$1, 4e-7], N[(N[(N[(N[(N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] / im), $MachinePrecision] / im), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(N[(N[(re * re), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(re * re), $MachinePrecision] + t$95$0), $MachinePrecision] * re), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 78.5

          \[\leadsto \mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \color{blue}{\sin re} \]

   5. Applied rewrites 78.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \sin re} \]

   6. Taylor expanded in im around inf

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

   8. Applied rewrites 22.6%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 18.0%

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

   if -0.0100000000000000002 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < 3.9999999999999998e-7

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 73.4

          \[\leadsto \mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \color{blue}{\sin re} \]

   5. Applied rewrites 73.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \sin re} \]

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 72.7%

      \[\leadsto \mathsf{fma}\left(im \cdot im, 0.5, 1\right) \cdot \color{blue}{re} \]
   9. Step-by-step derivation

   10. Applied rewrites 79.4%

       \[\leadsto \mathsf{fma}\left(\frac{\left(-im \cdot im\right) \cdot \left(-im\right)}{im}, 0.5, 1\right) \cdot re \]
   11. Step-by-step derivation

   12. Applied rewrites 81.7%

       \[\leadsto \mathsf{fma}\left(\frac{\frac{\left(\left(-im\right) \cdot im\right) \cdot \left(\left(-im\right) \cdot im\right)}{im}}{im}, 0.5, 1\right) \cdot re \]

   if 3.9999999999999998e-7 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 73.5

          \[\leadsto \mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \color{blue}{\sin re} \]

   5. Applied rewrites 73.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \sin re} \]

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 26.5%

      \[\leadsto \mathsf{fma}\left(im \cdot 0.5, im, \mathsf{fma}\left(\left(\mathsf{fma}\left(im \cdot im, 0.5, 1\right) \cdot re\right) \cdot re, \mathsf{fma}\left(0.008333333333333333, re \cdot re, -0.16666666666666666\right), 1\right)\right) \cdot \color{blue}{re} \]
   9. Step-by-step derivation

   10. Applied rewrites 26.5%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, 0.008333333333333333, -0.16666666666666666\right) \cdot \mathsf{fma}\left(im \cdot im, 0.5, 1\right), re \cdot re, \mathsf{fma}\left(im \cdot im, 0.5, 1\right)\right) \cdot re \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.01:\\ \;\;\;\;\left(\left(im \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re\right) \cdot im\\ \mathbf{elif}\;0.5 \cdot \sin re \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{\left(im \cdot im\right) \cdot \left(im \cdot im\right)}{im}}{im}, 0.5, 1\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, 0.008333333333333333, -0.16666666666666666\right) \cdot \mathsf{fma}\left(im \cdot im, 0.5, 1\right), re \cdot re, \mathsf{fma}\left(im \cdot im, 0.5, 1\right)\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 55.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ \mathbf{if}\;t\_0 \leq -0.01:\\ \;\;\;\;\left(\left(im \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re\right) \cdot im\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{\left(im \cdot im\right) \cdot \left(im \cdot im\right)}{im}}{im}, 0.5, 1\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot 0.5, im, \mathsf{fma}\left(\left(\mathsf{fma}\left(im \cdot im, 0.5, 1\right) \cdot re\right) \cdot re, \mathsf{fma}\left(0.008333333333333333, re \cdot re, -0.16666666666666666\right), 1\right)\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re))))
   (if (<= t_0 -0.01)
     (* (* (* im (fma (* re re) -0.08333333333333333 0.5)) re) im)
     (if (<= t_0 4e-7)
       (* (fma (/ (/ (* (* im im) (* im im)) im) im) 0.5 1.0) re)
       (*
        (fma
         (* im 0.5)
         im
         (fma
          (* (* (fma (* im im) 0.5 1.0) re) re)
          (fma 0.008333333333333333 (* re re) -0.16666666666666666)
          1.0))
        re)))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sin(re);
	double tmp;
	if (t_0 <= -0.01) {
		tmp = ((im * fma((re * re), -0.08333333333333333, 0.5)) * re) * im;
	} else if (t_0 <= 4e-7) {
		tmp = fma(((((im * im) * (im * im)) / im) / im), 0.5, 1.0) * re;
	} else {
		tmp = fma((im * 0.5), im, fma(((fma((im * im), 0.5, 1.0) * re) * re), fma(0.008333333333333333, (re * re), -0.16666666666666666), 1.0)) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * sin(re))
	tmp = 0.0
	if (t_0 <= -0.01)
		tmp = Float64(Float64(Float64(im * fma(Float64(re * re), -0.08333333333333333, 0.5)) * re) * im);
	elseif (t_0 <= 4e-7)
		tmp = Float64(fma(Float64(Float64(Float64(Float64(im * im) * Float64(im * im)) / im) / im), 0.5, 1.0) * re);
	else
		tmp = Float64(fma(Float64(im * 0.5), im, fma(Float64(Float64(fma(Float64(im * im), 0.5, 1.0) * re) * re), fma(0.008333333333333333, Float64(re * re), -0.16666666666666666), 1.0)) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.01], N[(N[(N[(im * N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision] * im), $MachinePrecision], If[LessEqual[t$95$0, 4e-7], N[(N[(N[(N[(N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] / im), $MachinePrecision] / im), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(im * 0.5), $MachinePrecision] * im + N[(N[(N[(N[(N[(im * im), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision] * N[(0.008333333333333333 * N[(re * re), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 78.5

          \[\leadsto \mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \color{blue}{\sin re} \]

   5. Applied rewrites 78.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \sin re} \]

   6. Taylor expanded in im around inf

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

   8. Applied rewrites 22.6%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 18.0%

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

   if -0.0100000000000000002 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < 3.9999999999999998e-7

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 73.4

          \[\leadsto \mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \color{blue}{\sin re} \]

   5. Applied rewrites 73.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \sin re} \]

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 72.7%

      \[\leadsto \mathsf{fma}\left(im \cdot im, 0.5, 1\right) \cdot \color{blue}{re} \]
   9. Step-by-step derivation

   10. Applied rewrites 79.4%

       \[\leadsto \mathsf{fma}\left(\frac{\left(-im \cdot im\right) \cdot \left(-im\right)}{im}, 0.5, 1\right) \cdot re \]
   11. Step-by-step derivation

   12. Applied rewrites 81.7%

       \[\leadsto \mathsf{fma}\left(\frac{\frac{\left(\left(-im\right) \cdot im\right) \cdot \left(\left(-im\right) \cdot im\right)}{im}}{im}, 0.5, 1\right) \cdot re \]

   if 3.9999999999999998e-7 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 73.5

          \[\leadsto \mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \color{blue}{\sin re} \]

   5. Applied rewrites 73.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \sin re} \]

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 26.5%

      \[\leadsto \mathsf{fma}\left(im \cdot 0.5, im, \mathsf{fma}\left(\left(\mathsf{fma}\left(im \cdot im, 0.5, 1\right) \cdot re\right) \cdot re, \mathsf{fma}\left(0.008333333333333333, re \cdot re, -0.16666666666666666\right), 1\right)\right) \cdot \color{blue}{re} \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.01:\\ \;\;\;\;\left(\left(im \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re\right) \cdot im\\ \mathbf{elif}\;0.5 \cdot \sin re \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{\left(im \cdot im\right) \cdot \left(im \cdot im\right)}{im}}{im}, 0.5, 1\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot 0.5, im, \mathsf{fma}\left(\left(\mathsf{fma}\left(im \cdot im, 0.5, 1\right) \cdot re\right) \cdot re, \mathsf{fma}\left(0.008333333333333333, re \cdot re, -0.16666666666666666\right), 1\right)\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 55.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ \mathbf{if}\;t\_0 \leq -0.01:\\ \;\;\;\;\left(\left(im \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re\right) \cdot im\\ \mathbf{elif}\;t\_0 \leq 0.15:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{\left(im \cdot im\right) \cdot \left(im \cdot im\right)}{im}}{im}, 0.5, 1\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(re \cdot re\right) - 0.16666666666666666, re \cdot re, 1\right) \cdot 0.5\right) \cdot im\right) \cdot im\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re))))
   (if (<= t_0 -0.01)
     (* (* (* im (fma (* re re) -0.08333333333333333 0.5)) re) im)
     (if (<= t_0 0.15)
       (* (fma (/ (/ (* (* im im) (* im im)) im) im) 0.5 1.0) re)
       (*
        (*
         (*
          (*
           (fma
            (- (* 0.008333333333333333 (* re re)) 0.16666666666666666)
            (* re re)
            1.0)
           0.5)
          im)
         im)
        re)))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sin(re);
	double tmp;
	if (t_0 <= -0.01) {
		tmp = ((im * fma((re * re), -0.08333333333333333, 0.5)) * re) * im;
	} else if (t_0 <= 0.15) {
		tmp = fma(((((im * im) * (im * im)) / im) / im), 0.5, 1.0) * re;
	} else {
		tmp = (((fma(((0.008333333333333333 * (re * re)) - 0.16666666666666666), (re * re), 1.0) * 0.5) * im) * im) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * sin(re))
	tmp = 0.0
	if (t_0 <= -0.01)
		tmp = Float64(Float64(Float64(im * fma(Float64(re * re), -0.08333333333333333, 0.5)) * re) * im);
	elseif (t_0 <= 0.15)
		tmp = Float64(fma(Float64(Float64(Float64(Float64(im * im) * Float64(im * im)) / im) / im), 0.5, 1.0) * re);
	else
		tmp = Float64(Float64(Float64(Float64(fma(Float64(Float64(0.008333333333333333 * Float64(re * re)) - 0.16666666666666666), Float64(re * re), 1.0) * 0.5) * im) * im) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.01], N[(N[(N[(im * N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision] * im), $MachinePrecision], If[LessEqual[t$95$0, 0.15], N[(N[(N[(N[(N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] / im), $MachinePrecision] / im), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(0.008333333333333333 * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * re), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 78.5

          \[\leadsto \mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \color{blue}{\sin re} \]

   5. Applied rewrites 78.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \sin re} \]

   6. Taylor expanded in im around inf

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

   8. Applied rewrites 22.6%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 18.0%

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

   if -0.0100000000000000002 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < 0.149999999999999994

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 73.7

          \[\leadsto \mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \color{blue}{\sin re} \]

   5. Applied rewrites 73.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \sin re} \]

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 66.3%

      \[\leadsto \mathsf{fma}\left(im \cdot im, 0.5, 1\right) \cdot \color{blue}{re} \]
   9. Step-by-step derivation

   10. Applied rewrites 72.3%

       \[\leadsto \mathsf{fma}\left(\frac{\left(-im \cdot im\right) \cdot \left(-im\right)}{im}, 0.5, 1\right) \cdot re \]
   11. Step-by-step derivation

   12. Applied rewrites 74.4%

       \[\leadsto \mathsf{fma}\left(\frac{\frac{\left(\left(-im\right) \cdot im\right) \cdot \left(\left(-im\right) \cdot im\right)}{im}}{im}, 0.5, 1\right) \cdot re \]

   if 0.149999999999999994 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 72.6

          \[\leadsto \mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \color{blue}{\sin re} \]

   5. Applied rewrites 72.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \sin re} \]

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 29.8%

      \[\leadsto \mathsf{fma}\left(im \cdot 0.5, im, \mathsf{fma}\left(\left(\mathsf{fma}\left(im \cdot im, 0.5, 1\right) \cdot re\right) \cdot re, \mathsf{fma}\left(0.008333333333333333, re \cdot re, -0.16666666666666666\right), 1\right)\right) \cdot \color{blue}{re} \]

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 29.9%

       \[\leadsto \left(\left(\left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(re \cdot re\right) - 0.16666666666666666, re \cdot re, 1\right) \cdot 0.5\right) \cdot im\right) \cdot im\right) \cdot re \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.01:\\ \;\;\;\;\left(\left(im \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re\right) \cdot im\\ \mathbf{elif}\;0.5 \cdot \sin re \leq 0.15:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{\left(im \cdot im\right) \cdot \left(im \cdot im\right)}{im}}{im}, 0.5, 1\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(re \cdot re\right) - 0.16666666666666666, re \cdot re, 1\right) \cdot 0.5\right) \cdot im\right) \cdot im\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 53.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ \mathbf{if}\;t\_0 \leq -0.01:\\ \;\;\;\;\left(\left(im \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re\right) \cdot im\\ \mathbf{elif}\;t\_0 \leq 0.15:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(im \cdot im\right) \cdot im}{im}, 0.5, 1\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(re \cdot re\right) - 0.16666666666666666, re \cdot re, 1\right) \cdot 0.5\right) \cdot im\right) \cdot im\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re))))
   (if (<= t_0 -0.01)
     (* (* (* im (fma (* re re) -0.08333333333333333 0.5)) re) im)
     (if (<= t_0 0.15)
       (* (fma (/ (* (* im im) im) im) 0.5 1.0) re)
       (*
        (*
         (*
          (*
           (fma
            (- (* 0.008333333333333333 (* re re)) 0.16666666666666666)
            (* re re)
            1.0)
           0.5)
          im)
         im)
        re)))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sin(re);
	double tmp;
	if (t_0 <= -0.01) {
		tmp = ((im * fma((re * re), -0.08333333333333333, 0.5)) * re) * im;
	} else if (t_0 <= 0.15) {
		tmp = fma((((im * im) * im) / im), 0.5, 1.0) * re;
	} else {
		tmp = (((fma(((0.008333333333333333 * (re * re)) - 0.16666666666666666), (re * re), 1.0) * 0.5) * im) * im) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * sin(re))
	tmp = 0.0
	if (t_0 <= -0.01)
		tmp = Float64(Float64(Float64(im * fma(Float64(re * re), -0.08333333333333333, 0.5)) * re) * im);
	elseif (t_0 <= 0.15)
		tmp = Float64(fma(Float64(Float64(Float64(im * im) * im) / im), 0.5, 1.0) * re);
	else
		tmp = Float64(Float64(Float64(Float64(fma(Float64(Float64(0.008333333333333333 * Float64(re * re)) - 0.16666666666666666), Float64(re * re), 1.0) * 0.5) * im) * im) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.01], N[(N[(N[(im * N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision] * im), $MachinePrecision], If[LessEqual[t$95$0, 0.15], N[(N[(N[(N[(N[(im * im), $MachinePrecision] * im), $MachinePrecision] / im), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(0.008333333333333333 * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * re), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 78.5

          \[\leadsto \mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \color{blue}{\sin re} \]

   5. Applied rewrites 78.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \sin re} \]

   6. Taylor expanded in im around inf

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

   8. Applied rewrites 22.6%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 18.0%

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

   if -0.0100000000000000002 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < 0.149999999999999994

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 73.7

          \[\leadsto \mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \color{blue}{\sin re} \]

   5. Applied rewrites 73.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \sin re} \]

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 66.3%

      \[\leadsto \mathsf{fma}\left(im \cdot im, 0.5, 1\right) \cdot \color{blue}{re} \]
   9. Step-by-step derivation

   10. Applied rewrites 72.3%

       \[\leadsto \mathsf{fma}\left(\frac{\left(-im \cdot im\right) \cdot \left(-im\right)}{im}, 0.5, 1\right) \cdot re \]
   11. Step-by-step derivation

   12. Applied rewrites 72.3%

       \[\leadsto \mathsf{fma}\left(\frac{\left(im \cdot im\right) \cdot im}{im}, 0.5, 1\right) \cdot re \]

   if 0.149999999999999994 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 72.6

          \[\leadsto \mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \color{blue}{\sin re} \]

   5. Applied rewrites 72.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \sin re} \]

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 29.8%

      \[\leadsto \mathsf{fma}\left(im \cdot 0.5, im, \mathsf{fma}\left(\left(\mathsf{fma}\left(im \cdot im, 0.5, 1\right) \cdot re\right) \cdot re, \mathsf{fma}\left(0.008333333333333333, re \cdot re, -0.16666666666666666\right), 1\right)\right) \cdot \color{blue}{re} \]

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 29.9%

       \[\leadsto \left(\left(\left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(re \cdot re\right) - 0.16666666666666666, re \cdot re, 1\right) \cdot 0.5\right) \cdot im\right) \cdot im\right) \cdot re \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 57.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(im \cdot im, 0.5, 1\right)\\ \mathbf{if}\;0.5 \cdot \sin re \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left(re \cdot re, re \cdot -0.16666666666666666, re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, 0.008333333333333333, -0.16666666666666666\right) \cdot t\_0, re \cdot re, t\_0\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (fma (* im im) 0.5 1.0)))
   (if (<= (* 0.5 (sin re)) 4e-7)
     (*
      (fma
       (fma
        (fma 0.001388888888888889 (* im im) 0.041666666666666664)
        (* im im)
        0.5)
       (* im im)
       1.0)
      (fma (* re re) (* re -0.16666666666666666) re))
     (*
      (fma
       (* (fma (* re re) 0.008333333333333333 -0.16666666666666666) t_0)
       (* re re)
       t_0)
      re))))

C

double code(double re, double im) {
	double t_0 = fma((im * im), 0.5, 1.0);
	double tmp;
	if ((0.5 * sin(re)) <= 4e-7) {
		tmp = fma(fma(fma(0.001388888888888889, (im * im), 0.041666666666666664), (im * im), 0.5), (im * im), 1.0) * fma((re * re), (re * -0.16666666666666666), re);
	} else {
		tmp = fma((fma((re * re), 0.008333333333333333, -0.16666666666666666) * t_0), (re * re), t_0) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(Float64(im * im), 0.5, 1.0)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= 4e-7)
		tmp = Float64(fma(fma(fma(0.001388888888888889, Float64(im * im), 0.041666666666666664), Float64(im * im), 0.5), Float64(im * im), 1.0) * fma(Float64(re * re), Float64(re * -0.16666666666666666), re));
	else
		tmp = Float64(fma(Float64(fma(Float64(re * re), 0.008333333333333333, -0.16666666666666666) * t_0), Float64(re * re), t_0) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(im * im), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]}, If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], 4e-7], N[(N[(N[(N[(0.001388888888888889 * N[(im * im), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * N[(re * -0.16666666666666666), $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(re * re), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(re * re), $MachinePrecision] + t$95$0), $MachinePrecision] * re), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < 3.9999999999999998e-7

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(\frac{1}{2} \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)}\right) \]

      6. +-commutative N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(e^{im} + e^{0 - im}\right)}\right) \]

      7. lift-exp.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(\color{blue}{e^{im}} + e^{0 - im}\right)\right) \]

      8. lift-exp.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + \color{blue}{e^{0 - im}}\right)\right) \]

      9. lift--.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + e^{\color{blue}{0 - im}}\right)\right) \]

      10. sub0-neg N/A

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

      11. cosh-undef N/A

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

      12. associate-*r* N/A

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

      13. metadata-eval N/A

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

      14. cosh-0 N/A

          \[\leadsto \sin re \cdot \left(\color{blue}{\cosh 0} \cdot \cosh im\right) \]

      15. *-commutative N/A

          \[\leadsto \color{blue}{\left(\cosh 0 \cdot \cosh im\right) \cdot \sin re} \]

      16. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\cosh 0 \cdot \cosh im\right) \cdot \sin re} \]

      17. cosh-0 N/A

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

      18. exp-0 N/A

          \[\leadsto \left(\color{blue}{e^{0}} \cdot \cosh im\right) \cdot \sin re \]

      19. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(e^{0} \cdot \cosh im\right)} \cdot \sin re \]

      20. exp-0 N/A

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

      21. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \cdot \sin re \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + 1\right)} \cdot \sin re \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) \cdot {im}^{2}} + 1\right) \cdot \sin re \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right), {im}^{2}, 1\right)} \cdot \sin re \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) + \frac{1}{2}}, {im}^{2}, 1\right) \cdot \sin re \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) \cdot {im}^{2}} + \frac{1}{2}, {im}^{2}, 1\right) \cdot \sin re \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}, {im}^{2}, \frac{1}{2}\right)}, {im}^{2}, 1\right) \cdot \sin re \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{720} \cdot {im}^{2} + \frac{1}{24}}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{720}, {im}^{2}, \frac{1}{24}\right)}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{im \cdot im}, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{im \cdot im}, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), \color{blue}{im \cdot im}, 1\right) \cdot \sin re \]

      14. lower-*.f64 90.2

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), \color{blue}{im \cdot im}, 1\right) \cdot \sin re \]

   7. Applied rewrites 90.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right)} \cdot \sin re \]

   8. Taylor expanded in re around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \left(re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2} + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \color{blue}{\left(re \cdot \left(\frac{-1}{6} \cdot {re}^{2}\right) + re \cdot 1\right)} \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \left(re \cdot \color{blue}{\left({re}^{2} \cdot \frac{-1}{6}\right)} + re \cdot 1\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \left(\color{blue}{\left(re \cdot {re}^{2}\right) \cdot \frac{-1}{6}} + re \cdot 1\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \left(\left(re \cdot \color{blue}{\left(re \cdot re\right)}\right) \cdot \frac{-1}{6} + re \cdot 1\right) \]

      6. cube-unmult N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \left(\color{blue}{{re}^{3}} \cdot \frac{-1}{6} + re \cdot 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \left({re}^{\color{blue}{\left(2 + 1\right)}} \cdot \frac{-1}{6} + re \cdot 1\right) \]

      8. pow-plus N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \left(\color{blue}{\left({re}^{2} \cdot re\right)} \cdot \frac{-1}{6} + re \cdot 1\right) \]

      9. *-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \left(\left({re}^{2} \cdot re\right) \cdot \frac{-1}{6} + \color{blue}{re}\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \color{blue}{\mathsf{fma}\left({re}^{2} \cdot re, \frac{-1}{6}, re\right)} \]

      11. pow-plus N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left(\color{blue}{{re}^{\left(2 + 1\right)}}, \frac{-1}{6}, re\right) \]

      12. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left(\color{blue}{{re}^{\left(2 + 1\right)}}, \frac{-1}{6}, re\right) \]

      13. metadata-eval 63.4

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left({re}^{\color{blue}{3}}, -0.16666666666666666, re\right) \]

   10. Applied rewrites 63.4%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \color{blue}{\mathsf{fma}\left({re}^{3}, -0.16666666666666666, re\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 63.4%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{re \cdot -0.16666666666666666}, re\right) \]

   if 3.9999999999999998e-7 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 73.5

          \[\leadsto \mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \color{blue}{\sin re} \]

   5. Applied rewrites 73.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \sin re} \]

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 26.5%

      \[\leadsto \mathsf{fma}\left(im \cdot 0.5, im, \mathsf{fma}\left(\left(\mathsf{fma}\left(im \cdot im, 0.5, 1\right) \cdot re\right) \cdot re, \mathsf{fma}\left(0.008333333333333333, re \cdot re, -0.16666666666666666\right), 1\right)\right) \cdot \color{blue}{re} \]
   9. Step-by-step derivation

   10. Applied rewrites 26.5%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, 0.008333333333333333, -0.16666666666666666\right) \cdot \mathsf{fma}\left(im \cdot im, 0.5, 1\right), re \cdot re, \mathsf{fma}\left(im \cdot im, 0.5, 1\right)\right) \cdot re \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 48.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(im \cdot im, 0.5, 1\right)\\ \mathbf{if}\;0.5 \cdot \sin re \leq 0.0002:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot t\_0\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (fma (* im im) 0.5 1.0)))
   (if (<= (* 0.5 (sin re)) 0.0002)
     (* (* (fma -0.16666666666666666 (* re re) 1.0) t_0) re)
     (* t_0 re))))

C

double code(double re, double im) {
	double t_0 = fma((im * im), 0.5, 1.0);
	double tmp;
	if ((0.5 * sin(re)) <= 0.0002) {
		tmp = (fma(-0.16666666666666666, (re * re), 1.0) * t_0) * re;
	} else {
		tmp = t_0 * re;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(Float64(im * im), 0.5, 1.0)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= 0.0002)
		tmp = Float64(Float64(fma(-0.16666666666666666, Float64(re * re), 1.0) * t_0) * re);
	else
		tmp = Float64(t_0 * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(im * im), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]}, If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], 0.0002], N[(N[(N[(-0.16666666666666666 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision] * re), $MachinePrecision], N[(t$95$0 * re), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < 2.0000000000000001e-4

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 75.2

          \[\leadsto \mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \color{blue}{\sin re} \]

   5. Applied rewrites 75.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \sin re} \]

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 53.9%

      \[\leadsto \left(\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, 0.5, 1\right)\right) \cdot \color{blue}{re} \]

   if 2.0000000000000001e-4 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 73.5

          \[\leadsto \mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \color{blue}{\sin re} \]

   5. Applied rewrites 73.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \sin re} \]

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 21.8%

      \[\leadsto \mathsf{fma}\left(im \cdot im, 0.5, 1\right) \cdot \color{blue}{re} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 90.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3500000000 \lor \neg \left(im \leq 2.6 \cdot 10^{+77}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left(re \cdot re, re \cdot -0.16666666666666666, re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 3500000000.0) (not (<= im 2.6e+77)))
   (* (fma (fma 0.041666666666666664 (* im im) 0.5) (* im im) 1.0) (sin re))
   (*
    (fma
     (fma
      (fma 0.001388888888888889 (* im im) 0.041666666666666664)
      (* im im)
      0.5)
     (* im im)
     1.0)
    (fma (* re re) (* re -0.16666666666666666) re))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 3500000000.0) || !(im <= 2.6e+77)) {
		tmp = fma(fma(0.041666666666666664, (im * im), 0.5), (im * im), 1.0) * sin(re);
	} else {
		tmp = fma(fma(fma(0.001388888888888889, (im * im), 0.041666666666666664), (im * im), 0.5), (im * im), 1.0) * fma((re * re), (re * -0.16666666666666666), re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 3500000000.0) || !(im <= 2.6e+77))
		tmp = Float64(fma(fma(0.041666666666666664, Float64(im * im), 0.5), Float64(im * im), 1.0) * sin(re));
	else
		tmp = Float64(fma(fma(fma(0.001388888888888889, Float64(im * im), 0.041666666666666664), Float64(im * im), 0.5), Float64(im * im), 1.0) * fma(Float64(re * re), Float64(re * -0.16666666666666666), re));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 3500000000.0], N[Not[LessEqual[im, 2.6e+77]], $MachinePrecision]], N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sin[re], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.001388888888888889 * N[(im * im), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * N[(re * -0.16666666666666666), $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 3.5e9 or 2.6000000000000002e77 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(\frac{1}{2} \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)}\right) \]

      6. +-commutative N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(e^{im} + e^{0 - im}\right)}\right) \]

      7. lift-exp.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(\color{blue}{e^{im}} + e^{0 - im}\right)\right) \]

      8. lift-exp.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + \color{blue}{e^{0 - im}}\right)\right) \]

      9. lift--.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + e^{\color{blue}{0 - im}}\right)\right) \]

      10. sub0-neg N/A

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

      11. cosh-undef N/A

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

      12. associate-*r* N/A

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

      13. metadata-eval N/A

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

      14. cosh-0 N/A

          \[\leadsto \sin re \cdot \left(\color{blue}{\cosh 0} \cdot \cosh im\right) \]

      15. *-commutative N/A

          \[\leadsto \color{blue}{\left(\cosh 0 \cdot \cosh im\right) \cdot \sin re} \]

      16. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\cosh 0 \cdot \cosh im\right) \cdot \sin re} \]

      17. cosh-0 N/A

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

      18. exp-0 N/A

          \[\leadsto \left(\color{blue}{e^{0}} \cdot \cosh im\right) \cdot \sin re \]

      19. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(e^{0} \cdot \cosh im\right)} \cdot \sin re \]

      20. exp-0 N/A

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

      21. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \cdot \sin re \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) + 1\right)} \cdot \sin re \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot {im}^{2}} + 1\right) \cdot \sin re \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}, {im}^{2}, 1\right)} \cdot \sin re \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}}, {im}^{2}, 1\right) \cdot \sin re \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {im}^{2}, \frac{1}{2}\right)}, {im}^{2}, 1\right) \cdot \sin re \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right), \color{blue}{im \cdot im}, 1\right) \cdot \sin re \]

      9. lower-*.f64 88.9

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), \color{blue}{im \cdot im}, 1\right) \cdot \sin re \]

   7. Applied rewrites 88.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right)} \cdot \sin re \]

   if 3.5e9 < im < 2.6000000000000002e77

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(\frac{1}{2} \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)}\right) \]

      6. +-commutative N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(e^{im} + e^{0 - im}\right)}\right) \]

      7. lift-exp.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(\color{blue}{e^{im}} + e^{0 - im}\right)\right) \]

      8. lift-exp.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + \color{blue}{e^{0 - im}}\right)\right) \]

      9. lift--.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + e^{\color{blue}{0 - im}}\right)\right) \]

      10. sub0-neg N/A

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

      11. cosh-undef N/A

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

      12. associate-*r* N/A

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

      13. metadata-eval N/A

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

      14. cosh-0 N/A

          \[\leadsto \sin re \cdot \left(\color{blue}{\cosh 0} \cdot \cosh im\right) \]

      15. *-commutative N/A

          \[\leadsto \color{blue}{\left(\cosh 0 \cdot \cosh im\right) \cdot \sin re} \]

      16. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\cosh 0 \cdot \cosh im\right) \cdot \sin re} \]

      17. cosh-0 N/A

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

      18. exp-0 N/A

          \[\leadsto \left(\color{blue}{e^{0}} \cdot \cosh im\right) \cdot \sin re \]

      19. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(e^{0} \cdot \cosh im\right)} \cdot \sin re \]

      20. exp-0 N/A

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

      21. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \cdot \sin re \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + 1\right)} \cdot \sin re \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) \cdot {im}^{2}} + 1\right) \cdot \sin re \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right), {im}^{2}, 1\right)} \cdot \sin re \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) + \frac{1}{2}}, {im}^{2}, 1\right) \cdot \sin re \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) \cdot {im}^{2}} + \frac{1}{2}, {im}^{2}, 1\right) \cdot \sin re \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}, {im}^{2}, \frac{1}{2}\right)}, {im}^{2}, 1\right) \cdot \sin re \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{720} \cdot {im}^{2} + \frac{1}{24}}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{720}, {im}^{2}, \frac{1}{24}\right)}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{im \cdot im}, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{im \cdot im}, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), \color{blue}{im \cdot im}, 1\right) \cdot \sin re \]

      14. lower-*.f64 30.0

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), \color{blue}{im \cdot im}, 1\right) \cdot \sin re \]

   7. Applied rewrites 30.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right)} \cdot \sin re \]

   8. Taylor expanded in re around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \left(re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2} + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \color{blue}{\left(re \cdot \left(\frac{-1}{6} \cdot {re}^{2}\right) + re \cdot 1\right)} \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \left(re \cdot \color{blue}{\left({re}^{2} \cdot \frac{-1}{6}\right)} + re \cdot 1\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \left(\color{blue}{\left(re \cdot {re}^{2}\right) \cdot \frac{-1}{6}} + re \cdot 1\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \left(\left(re \cdot \color{blue}{\left(re \cdot re\right)}\right) \cdot \frac{-1}{6} + re \cdot 1\right) \]

      6. cube-unmult N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \left(\color{blue}{{re}^{3}} \cdot \frac{-1}{6} + re \cdot 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \left({re}^{\color{blue}{\left(2 + 1\right)}} \cdot \frac{-1}{6} + re \cdot 1\right) \]

      8. pow-plus N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \left(\color{blue}{\left({re}^{2} \cdot re\right)} \cdot \frac{-1}{6} + re \cdot 1\right) \]

      9. *-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \left(\left({re}^{2} \cdot re\right) \cdot \frac{-1}{6} + \color{blue}{re}\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \color{blue}{\mathsf{fma}\left({re}^{2} \cdot re, \frac{-1}{6}, re\right)} \]

      11. pow-plus N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left(\color{blue}{{re}^{\left(2 + 1\right)}}, \frac{-1}{6}, re\right) \]

      12. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left(\color{blue}{{re}^{\left(2 + 1\right)}}, \frac{-1}{6}, re\right) \]

      13. metadata-eval 41.6

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left({re}^{\color{blue}{3}}, -0.16666666666666666, re\right) \]

   10. Applied rewrites 41.6%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \color{blue}{\mathsf{fma}\left({re}^{3}, -0.16666666666666666, re\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 41.6%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{re \cdot -0.16666666666666666}, re\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3500000000 \lor \neg \left(im \leq 2.6 \cdot 10^{+77}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left(re \cdot re, re \cdot -0.16666666666666666, re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 48.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.01:\\ \;\;\;\;\left(\left(im \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, 0.5, 1\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.01)
   (* (* (* im (fma (* re re) -0.08333333333333333 0.5)) re) im)
   (* (fma (* im im) 0.5 1.0) re)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.01) {
		tmp = ((im * fma((re * re), -0.08333333333333333, 0.5)) * re) * im;
	} else {
		tmp = fma((im * im), 0.5, 1.0) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.01)
		tmp = Float64(Float64(Float64(im * fma(Float64(re * re), -0.08333333333333333, 0.5)) * re) * im);
	else
		tmp = Float64(fma(Float64(im * im), 0.5, 1.0) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.01], N[(N[(N[(im * N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(im * im), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 78.5

          \[\leadsto \mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \color{blue}{\sin re} \]

   5. Applied rewrites 78.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \sin re} \]

   6. Taylor expanded in im around inf

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

   8. Applied rewrites 22.6%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 18.0%

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

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 73.4

          \[\leadsto \mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \color{blue}{\sin re} \]

   5. Applied rewrites 73.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \sin re} \]

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 56.2%

      \[\leadsto \mathsf{fma}\left(im \cdot im, 0.5, 1\right) \cdot \color{blue}{re} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 48.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.01:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, 0.5, 1\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.01)
   (* (* (* im im) (fma (* re re) -0.08333333333333333 0.5)) re)
   (* (fma (* im im) 0.5 1.0) re)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.01) {
		tmp = ((im * im) * fma((re * re), -0.08333333333333333, 0.5)) * re;
	} else {
		tmp = fma((im * im), 0.5, 1.0) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.01)
		tmp = Float64(Float64(Float64(im * im) * fma(Float64(re * re), -0.08333333333333333, 0.5)) * re);
	else
		tmp = Float64(fma(Float64(im * im), 0.5, 1.0) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.01], N[(N[(N[(im * im), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(im * im), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 78.5

          \[\leadsto \mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \color{blue}{\sin re} \]

   5. Applied rewrites 78.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \sin re} \]

   6. Taylor expanded in im around inf

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

   8. Applied rewrites 22.6%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 17.6%

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

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 73.4

          \[\leadsto \mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \color{blue}{\sin re} \]

   5. Applied rewrites 73.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \sin re} \]

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 56.2%

      \[\leadsto \mathsf{fma}\left(im \cdot im, 0.5, 1\right) \cdot \color{blue}{re} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 48.2% accurate, 18.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(im \cdot im, 0.5, 1\right) \cdot re \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (fma (* im im) 0.5 1.0) re))

C

double code(double re, double im) {
	return fma((im * im), 0.5, 1.0) * re;
}

Julia

function code(re, im)
	return Float64(fma(Float64(im * im), 0.5, 1.0) * re)
end

Wolfram

code[re_, im_] := N[(N[(N[(im * im), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * re), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in im around 0

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

   1. associate-*r* N/A

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

   2. distribute-rgt1-in N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sin.f64 74.8

       \[\leadsto \mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \color{blue}{\sin re} \]

5. Applied rewrites 74.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \sin re} \]

6. Taylor expanded in re around 0

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

8. Applied rewrites 46.5%

   \[\leadsto \mathsf{fma}\left(im \cdot im, 0.5, 1\right) \cdot \color{blue}{re} \]
9. Add Preprocessing

Alternative 19: 25.1% accurate, 19.8× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	return ((im * im) * 0.5) * 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 = ((im * im) * 0.5d0) * re
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in im around 0

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

   1. associate-*r* N/A

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

   2. distribute-rgt1-in N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sin.f64 74.8

       \[\leadsto \mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \color{blue}{\sin re} \]

5. Applied rewrites 74.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \sin re} \]

6. Taylor expanded in re around 0

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

8. Applied rewrites 46.5%

   \[\leadsto \mathsf{fma}\left(im \cdot im, 0.5, 1\right) \cdot \color{blue}{re} \]

9. Taylor expanded in im around inf

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

11. Applied rewrites 22.8%

    \[\leadsto \left(\left(im \cdot im\right) \cdot 0.5\right) \cdot re \]
12. Add Preprocessing

Alternative 20: 19.2% accurate, 19.8× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in im around 0

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

   1. associate-*r* N/A

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

   2. distribute-rgt1-in N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sin.f64 74.8

       \[\leadsto \mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \color{blue}{\sin re} \]

5. Applied rewrites 74.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \sin re} \]

6. Taylor expanded in im around inf

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

8. Applied rewrites 20.0%

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

9. Taylor expanded in re around 0

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

11. Applied rewrites 17.8%

    \[\leadsto \left(\left(0.5 \cdot re\right) \cdot im\right) \cdot im \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024363 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))