math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 8.7s

Alternatives: 24

Speedup: 1.5×

• 50.1% 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: 76.9% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma (fma (* (* 0.08333333333333333 re) im) im re) (* im im) (* 2.0 re))
      (fma (* re re) -0.08333333333333333 0.5))
     (if (<= t_0 1.0)
       (* (sin re) 1.0)
       (* (fma (/ (* (* 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(((0.08333333333333333 * re) * im), im, re), (im * im), (2.0 * re)) * fma((re * re), -0.08333333333333333, 0.5);
	} else if (t_0 <= 1.0) {
		tmp = sin(re) * 1.0;
	} else {
		tmp = fma((((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(Float64(Float64(0.08333333333333333 * re) * im), im, re), Float64(im * im), Float64(2.0 * re)) * fma(Float64(re * re), -0.08333333333333333, 0.5));
	elseif (t_0 <= 1.0)
		tmp = Float64(sin(re) * 1.0);
	else
		tmp = Float64(fma(Float64(Float64(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[(N[(0.08333333333333333 * re), $MachinePrecision] * im), $MachinePrecision] * im + re), $MachinePrecision] * N[(im * im), $MachinePrecision] + N[(2.0 * re), $MachinePrecision]), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[(N[Sin[re], $MachinePrecision] * 1.0), $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 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 re around 0

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

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

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out-- N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-neg-in N/A

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

      7. cancel-sign-sub N/A

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

   5. Applied rewrites 71.8%

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

   6. Taylor expanded in im around 0

      \[\leadsto \left(2 \cdot re + {im}^{2} \cdot \left(im \cdot \left(-1 \cdot \left(im \cdot \left(-1 \cdot \left(-1 \cdot \left(-1 \cdot re + \frac{1}{2} \cdot re\right) + \left(\frac{-1}{2} \cdot re + \frac{1}{6} \cdot re\right)\right) + \left(\frac{-1}{2} \cdot \left(-1 \cdot re + \frac{1}{2} \cdot re\right) + \frac{-1}{6} \cdot re\right)\right)\right) - \left(-1 \cdot \left(-1 \cdot re + \frac{1}{2} \cdot re\right) + \frac{-1}{2} \cdot re\right)\right) - -1 \cdot re\right)\right) \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{12}, \frac{1}{2}\right) \]

   8. Applied rewrites 52.4%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, re, \mathsf{fma}\left(-0.16666666666666666, re, \mathsf{fma}\left(0.3333333333333333, re, -0.5 \cdot re\right)\right)\right), -im, 0 \cdot \left(-0.5 \cdot re\right)\right), im, re\right), im \cdot im, 2 \cdot re\right) \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, -0.08333333333333333, 0.5\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 52.4%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(re, 0.16666666666666666, \mathsf{fma}\left(-0.5, re, 0.25 \cdot re\right)\right), -im, 0 \cdot \left(-0.5 \cdot re\right)\right), im, re\right), im \cdot im, 2 \cdot re\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \]

   11. Taylor expanded in re around 0

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

   13. Applied rewrites 52.4%

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

   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. 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. Taylor expanded in im around 0

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

   9. Applied rewrites 97.5%

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

   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. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt1-in N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 43.0

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

   5. Applied rewrites 43.0%

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

   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 38.8%

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

   10. Applied rewrites 48.5%

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

4. Final simplification 72.4%

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

Alternative 3: 53.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.0001:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(0.08333333333333333 \cdot re\right) \cdot im, im, re\right), im \cdot im, 2 \cdot re\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(im \cdot im\right) \cdot im}{im}, 0.5, 1\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im))) 0.0001)
   (*
    (fma (fma (* (* 0.08333333333333333 re) im) im re) (* im im) (* 2.0 re))
    (fma (* re re) -0.08333333333333333 0.5))
   (* (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))) <= 0.0001) {
		tmp = fma(fma(((0.08333333333333333 * re) * im), im, re), (im * im), (2.0 * re)) * fma((re * re), -0.08333333333333333, 0.5);
	} 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))) <= 0.0001)
		tmp = Float64(fma(fma(Float64(Float64(0.08333333333333333 * re) * im), im, re), Float64(im * im), Float64(2.0 * re)) * fma(Float64(re * re), -0.08333333333333333, 0.5));
	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], 0.0001], N[(N[(N[(N[(N[(0.08333333333333333 * re), $MachinePrecision] * im), $MachinePrecision] * im + re), $MachinePrecision] * N[(im * im), $MachinePrecision] + N[(2.0 * re), $MachinePrecision]), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $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))) < 1.00000000000000005e-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 re around 0

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

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

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out-- N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-neg-in N/A

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

      7. cancel-sign-sub N/A

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

   5. Applied rewrites 68.8%

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

   6. Taylor expanded in im around 0

      \[\leadsto \left(2 \cdot re + {im}^{2} \cdot \left(im \cdot \left(-1 \cdot \left(im \cdot \left(-1 \cdot \left(-1 \cdot \left(-1 \cdot re + \frac{1}{2} \cdot re\right) + \left(\frac{-1}{2} \cdot re + \frac{1}{6} \cdot re\right)\right) + \left(\frac{-1}{2} \cdot \left(-1 \cdot re + \frac{1}{2} \cdot re\right) + \frac{-1}{6} \cdot re\right)\right)\right) - \left(-1 \cdot \left(-1 \cdot re + \frac{1}{2} \cdot re\right) + \frac{-1}{2} \cdot re\right)\right) - -1 \cdot re\right)\right) \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{12}, \frac{1}{2}\right) \]

   8. Applied rewrites 59.0%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, re, \mathsf{fma}\left(-0.16666666666666666, re, \mathsf{fma}\left(0.3333333333333333, re, -0.5 \cdot re\right)\right)\right), -im, 0 \cdot \left(-0.5 \cdot re\right)\right), im, re\right), im \cdot im, 2 \cdot re\right) \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, -0.08333333333333333, 0.5\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 59.0%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(re, 0.16666666666666666, \mathsf{fma}\left(-0.5, re, 0.25 \cdot re\right)\right), -im, 0 \cdot \left(-0.5 \cdot re\right)\right), im, re\right), im \cdot im, 2 \cdot re\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \]

   11. Taylor expanded in re around 0

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

   13. Applied rewrites 59.0%

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

   if 1.00000000000000005e-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. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt1-in N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 62.2

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

   5. Applied rewrites 62.2%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 26.8%

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

   10. Applied rewrites 33.3%

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

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.0001:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(0.08333333333333333 \cdot re\right) \cdot im, im, re\right), im \cdot im, 2 \cdot re\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(im \cdot im\right) \cdot im}{im}, 0.5, 1\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 51.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.0001:\\ \;\;\;\;\left(\mathsf{fma}\left(im, im, 2\right) \cdot re\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(im \cdot im\right) \cdot im}{im}, 0.5, 1\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im))) 0.0001)
   (* (* (fma im im 2.0) re) (fma (* re re) -0.08333333333333333 0.5))
   (* (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))) <= 0.0001) {
		tmp = (fma(im, im, 2.0) * re) * fma((re * re), -0.08333333333333333, 0.5);
	} 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))) <= 0.0001)
		tmp = Float64(Float64(fma(im, im, 2.0) * re) * fma(Float64(re * re), -0.08333333333333333, 0.5));
	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], 0.0001], N[(N[(N[(im * im + 2.0), $MachinePrecision] * re), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $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))) < 1.00000000000000005e-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 re around 0

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

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

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out-- N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-neg-in N/A

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

      7. cancel-sign-sub N/A

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

   5. Applied rewrites 68.8%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 55.3%

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

   if 1.00000000000000005e-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. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt1-in N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 62.2

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

   5. Applied rewrites 62.2%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 26.8%

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

   10. Applied rewrites 33.3%

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

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.0001:\\ \;\;\;\;\left(\mathsf{fma}\left(im, im, 2\right) \cdot re\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(im \cdot im\right) \cdot im}{im}, 0.5, 1\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 49.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.0001:\\ \;\;\;\;\left(\mathsf{fma}\left(im, im, 2\right) \cdot re\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, 0.5, 1\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im))) 0.0001)
   (* (* (fma im im 2.0) re) (fma (* re re) -0.08333333333333333 0.5))
   (* (fma (* 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))) <= 0.0001) {
		tmp = (fma(im, im, 2.0) * re) * fma((re * re), -0.08333333333333333, 0.5);
	} else {
		tmp = fma((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))) <= 0.0001)
		tmp = Float64(Float64(fma(im, im, 2.0) * re) * fma(Float64(re * re), -0.08333333333333333, 0.5));
	else
		tmp = Float64(fma(Float64(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], 0.0001], N[(N[(N[(im * im + 2.0), $MachinePrecision] * re), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(im * 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))) < 1.00000000000000005e-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 re around 0

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

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

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out-- N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-neg-in N/A

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

      7. cancel-sign-sub N/A

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

   5. Applied rewrites 68.8%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 55.3%

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

   if 1.00000000000000005e-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. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt1-in N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 62.2

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

   5. Applied rewrites 62.2%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 26.8%

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

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.0001:\\ \;\;\;\;\left(\mathsf{fma}\left(im, im, 2\right) \cdot re\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, 0.5, 1\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 49.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.0001:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, 0.5, 1\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im))) 0.0001)
   (* (fma im im 2.0) (* (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)) * (exp(-im) + exp(im))) <= 0.0001) {
		tmp = fma(im, im, 2.0) * (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(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) + exp(im))) <= 0.0001)
		tmp = Float64(fma(im, im, 2.0) * Float64(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[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0001], N[(N[(im * im + 2.0), $MachinePrecision] * N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision], N[(N[(N[(im * 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))) < 1.00000000000000005e-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 re around 0

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

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

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out-- N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-neg-in N/A

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

      7. cancel-sign-sub N/A

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

   5. Applied rewrites 68.8%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 55.3%

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

   if 1.00000000000000005e-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. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt1-in N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 62.2

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

   5. Applied rewrites 62.2%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 26.8%

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

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.0001:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, 0.5, 1\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 46.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.05:\\ \;\;\;\;\left(\left(im \cdot re\right) \cdot im\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, 0.5, 1\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im))) -0.05)
   (* (* (* im re) im) (fma (* re re) -0.08333333333333333 0.5))
   (* (fma (* 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))) <= -0.05) {
		tmp = ((im * re) * im) * fma((re * re), -0.08333333333333333, 0.5);
	} else {
		tmp = fma((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))) <= -0.05)
		tmp = Float64(Float64(Float64(im * re) * im) * fma(Float64(re * re), -0.08333333333333333, 0.5));
	else
		tmp = Float64(fma(Float64(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], -0.05], N[(N[(N[(im * re), $MachinePrecision] * im), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(im * 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))) < -0.050000000000000003

   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 re around 0

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

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

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out-- N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-neg-in N/A

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

      7. cancel-sign-sub N/A

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

   5. Applied rewrites 52.1%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 25.0%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0 \cdot \left(-0.5 \cdot re\right), im, re\right) \cdot im, im, 2 \cdot re\right) \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, -0.08333333333333333, 0.5\right) \]

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 25.1%

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

   if -0.050000000000000003 < (*.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. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt1-in N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 76.2

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

   5. Applied rewrites 76.2%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 54.6%

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

4. Final simplification 42.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.05:\\ \;\;\;\;\left(\left(im \cdot re\right) \cdot im\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, 0.5, 1\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 41.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.0001:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, 0.5, 1\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im))) 0.0001)
   (* (* (fma (* re re) -0.08333333333333333 0.5) re) 2.0)
   (* (fma (* 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))) <= 0.0001) {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * re) * 2.0;
	} else {
		tmp = fma((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))) <= 0.0001)
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re) * 2.0);
	else
		tmp = Float64(fma(Float64(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], 0.0001], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(im * 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))) < 1.00000000000000005e-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

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

   5. Applied rewrites 52.8%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 42.2

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

   8. Applied rewrites 42.2%

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

   if 1.00000000000000005e-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. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt1-in N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 62.2

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

   5. Applied rewrites 62.2%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 26.8%

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

4. Final simplification 36.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.0001:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, 0.5, 1\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 37.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.94:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot 0.5\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im))) 0.94)
   (* (* re 0.5) 2.0)
   (* (* (* im im) 0.5) re)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(-im) + exp(im))) <= 0.94) {
		tmp = (re * 0.5) * 2.0;
	} else {
		tmp = ((im * im) * 0.5) * re;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (((0.5d0 * sin(re)) * (exp(-im) + exp(im))) <= 0.94d0) then
        tmp = (re * 0.5d0) * 2.0d0
    else
        tmp = ((im * im) * 0.5d0) * re
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (((0.5 * Math.sin(re)) * (Math.exp(-im) + Math.exp(im))) <= 0.94) {
		tmp = (re * 0.5) * 2.0;
	} else {
		tmp = ((im * im) * 0.5) * re;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if ((0.5 * math.sin(re)) * (math.exp(-im) + math.exp(im))) <= 0.94:
		tmp = (re * 0.5) * 2.0
	else:
		tmp = ((im * im) * 0.5) * re
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) + exp(im))) <= 0.94)
		tmp = Float64(Float64(re * 0.5) * 2.0);
	else
		tmp = Float64(Float64(Float64(im * im) * 0.5) * re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (((0.5 * sin(re)) * (exp(-im) + exp(im))) <= 0.94)
		tmp = (re * 0.5) * 2.0;
	else
		tmp = ((im * im) * 0.5) * re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.94], N[(N[(re * 0.5), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(im * im), $MachinePrecision] * 0.5), $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))) < 0.93999999999999995

   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

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

   5. Applied rewrites 59.6%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 30.9

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

   8. Applied rewrites 30.9%

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

   if 0.93999999999999995 < (*.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. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt1-in N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 44.9

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

   5. Applied rewrites 44.9%

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

   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 37.6%

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

   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 37.6%

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

4. Final simplification 32.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.94:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot 0.5\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 34.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.94:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(im \cdot re\right) \cdot im\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im))) 0.94)
   (* (* re 0.5) 2.0)
   (* (* (* im re) im) 0.5)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(-im) + exp(im))) <= 0.94) {
		tmp = (re * 0.5) * 2.0;
	} else {
		tmp = ((im * re) * im) * 0.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (((0.5d0 * sin(re)) * (exp(-im) + exp(im))) <= 0.94d0) then
        tmp = (re * 0.5d0) * 2.0d0
    else
        tmp = ((im * re) * im) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (((0.5 * Math.sin(re)) * (Math.exp(-im) + Math.exp(im))) <= 0.94) {
		tmp = (re * 0.5) * 2.0;
	} else {
		tmp = ((im * re) * im) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if ((0.5 * math.sin(re)) * (math.exp(-im) + math.exp(im))) <= 0.94:
		tmp = (re * 0.5) * 2.0
	else:
		tmp = ((im * re) * im) * 0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) + exp(im))) <= 0.94)
		tmp = Float64(Float64(re * 0.5) * 2.0);
	else
		tmp = Float64(Float64(Float64(im * re) * im) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (((0.5 * sin(re)) * (exp(-im) + exp(im))) <= 0.94)
		tmp = (re * 0.5) * 2.0;
	else
		tmp = ((im * re) * im) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.94], N[(N[(re * 0.5), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(im * re), $MachinePrecision] * im), $MachinePrecision] * 0.5), $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))) < 0.93999999999999995

   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

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

   5. Applied rewrites 59.6%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 30.9

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

   8. Applied rewrites 30.9%

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

   if 0.93999999999999995 < (*.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. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt1-in N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 44.9

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

   5. Applied rewrites 44.9%

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

   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 37.6%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 31.6%

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

4. Final simplification 31.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.94:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(im \cdot re\right) \cdot im\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 49.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq 10^{-297}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, re, \mathsf{fma}\left(-0.16666666666666666, re, \mathsf{fma}\left(0.3333333333333333, re, -0.5 \cdot re\right)\right)\right), -im, 0\right), im, re\right), im \cdot im, 2 \cdot re\right) \cdot \left(\left(\left(\frac{0.5}{re \cdot re} - 0.08333333333333333\right) \cdot re\right) \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(im \cdot im\right) \cdot im}{im}, 0.5, 1\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) 1e-297)
   (*
    (fma
     (fma
      (fma
       (fma
        0.25
        re
        (fma -0.16666666666666666 re (fma 0.3333333333333333 re (* -0.5 re))))
       (- im)
       0.0)
      im
      re)
     (* im im)
     (* 2.0 re))
    (* (* (- (/ 0.5 (* re re)) 0.08333333333333333) re) 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)) <= 1e-297) {
		tmp = fma(fma(fma(fma(0.25, re, fma(-0.16666666666666666, re, fma(0.3333333333333333, re, (-0.5 * re)))), -im, 0.0), im, re), (im * im), (2.0 * re)) * ((((0.5 / (re * re)) - 0.08333333333333333) * re) * 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(0.5 * sin(re)) <= 1e-297)
		tmp = Float64(fma(fma(fma(fma(0.25, re, fma(-0.16666666666666666, re, fma(0.3333333333333333, re, Float64(-0.5 * re)))), Float64(-im), 0.0), im, re), Float64(im * im), Float64(2.0 * re)) * Float64(Float64(Float64(Float64(0.5 / Float64(re * re)) - 0.08333333333333333) * re) * 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[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], 1e-297], N[(N[(N[(N[(N[(0.25 * re + N[(-0.16666666666666666 * re + N[(0.3333333333333333 * re + N[(-0.5 * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-im) + 0.0), $MachinePrecision] * im + re), $MachinePrecision] * N[(im * im), $MachinePrecision] + N[(2.0 * re), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(0.5 / N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.08333333333333333), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision]), $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 #s(literal 1/2 binary64) (sin.f64 re)) < 1.00000000000000004e-297

   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 re around 0

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

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

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out-- N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-neg-in N/A

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

      7. cancel-sign-sub N/A

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

   5. Applied rewrites 62.5%

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

   6. Taylor expanded in im around 0

      \[\leadsto \left(2 \cdot re + {im}^{2} \cdot \left(im \cdot \left(-1 \cdot \left(im \cdot \left(-1 \cdot \left(-1 \cdot \left(-1 \cdot re + \frac{1}{2} \cdot re\right) + \left(\frac{-1}{2} \cdot re + \frac{1}{6} \cdot re\right)\right) + \left(\frac{-1}{2} \cdot \left(-1 \cdot re + \frac{1}{2} \cdot re\right) + \frac{-1}{6} \cdot re\right)\right)\right) - \left(-1 \cdot \left(-1 \cdot re + \frac{1}{2} \cdot re\right) + \frac{-1}{2} \cdot re\right)\right) - -1 \cdot re\right)\right) \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{12}, \frac{1}{2}\right) \]

   8. Applied rewrites 50.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, re, \mathsf{fma}\left(-0.16666666666666666, re, \mathsf{fma}\left(0.3333333333333333, re, -0.5 \cdot re\right)\right)\right), -im, 0 \cdot \left(-0.5 \cdot re\right)\right), im, re\right), im \cdot im, 2 \cdot re\right) \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, -0.08333333333333333, 0.5\right) \]

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, re, \mathsf{fma}\left(\frac{-1}{6}, re, \mathsf{fma}\left(\frac{1}{3}, re, \frac{-1}{2} \cdot re\right)\right)\right), -im, 0 \cdot \left(\frac{-1}{2} \cdot re\right)\right), im, re\right), im \cdot im, 2 \cdot re\right) \cdot \left({re}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{{re}^{2}} - \frac{1}{12}\right)}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 44.2%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, re, \mathsf{fma}\left(-0.16666666666666666, re, \mathsf{fma}\left(0.3333333333333333, re, -0.5 \cdot re\right)\right)\right), -im, 0 \cdot \left(-0.5 \cdot re\right)\right), im, re\right), im \cdot im, 2 \cdot re\right) \cdot \left(\left(\left(\frac{0.5}{re \cdot re} - 0.08333333333333333\right) \cdot re\right) \cdot \color{blue}{re}\right) \]

   if 1.00000000000000004e-297 < (*.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. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt1-in N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 70.9

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

   5. Applied rewrites 70.9%

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

   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 44.0%

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

   10. Applied rewrites 48.9%

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

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq 10^{-297}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, re, \mathsf{fma}\left(-0.16666666666666666, re, \mathsf{fma}\left(0.3333333333333333, re, -0.5 \cdot re\right)\right)\right), -im, 0\right), im, re\right), im \cdot im, 2 \cdot re\right) \cdot \left(\left(\left(\frac{0.5}{re \cdot re} - 0.08333333333333333\right) \cdot re\right) \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(im \cdot im\right) \cdot im}{im}, 0.5, 1\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 45.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\\ \mathbf{if}\;0.5 \cdot \sin re \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, im, 0.5\right), im, 1\right), im, 1\right), re, \frac{re}{1 + im}\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\left(-im\right) \cdot \mathsf{fma}\left(0.25, re, 0\right), im, re\right), 2 \cdot re\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (fma (* re re) -0.08333333333333333 0.5)))
   (if (<= (* 0.5 (sin re)) 2e-6)
     (*
      (fma
       (fma (fma (fma 0.16666666666666666 im 0.5) im 1.0) im 1.0)
       re
       (/ re (+ 1.0 im)))
      t_0)
     (*
      (fma (* im im) (fma (* (- im) (fma 0.25 re 0.0)) im re) (* 2.0 re))
      t_0))))

C

double code(double re, double im) {
	double t_0 = fma((re * re), -0.08333333333333333, 0.5);
	double tmp;
	if ((0.5 * sin(re)) <= 2e-6) {
		tmp = fma(fma(fma(fma(0.16666666666666666, im, 0.5), im, 1.0), im, 1.0), re, (re / (1.0 + im))) * t_0;
	} else {
		tmp = fma((im * im), fma((-im * fma(0.25, re, 0.0)), im, re), (2.0 * re)) * t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(Float64(re * re), -0.08333333333333333, 0.5)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= 2e-6)
		tmp = Float64(fma(fma(fma(fma(0.16666666666666666, im, 0.5), im, 1.0), im, 1.0), re, Float64(re / Float64(1.0 + im))) * t_0);
	else
		tmp = Float64(fma(Float64(im * im), fma(Float64(Float64(-im) * fma(0.25, re, 0.0)), im, re), Float64(2.0 * re)) * t_0);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]}, If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], 2e-6], N[(N[(N[(N[(N[(0.16666666666666666 * im + 0.5), $MachinePrecision] * im + 1.0), $MachinePrecision] * im + 1.0), $MachinePrecision] * re + N[(re / N[(1.0 + im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(N[(im * im), $MachinePrecision] * N[(N[((-im) * N[(0.25 * re + 0.0), $MachinePrecision]), $MachinePrecision] * im + re), $MachinePrecision] + N[(2.0 * re), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

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

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out-- N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-neg-in N/A

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

      7. cancel-sign-sub N/A

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

   5. Applied rewrites 73.9%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 54.7%

      \[\leadsto \mathsf{fma}\left(e^{im}, re, \frac{re}{1 + im}\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \]

   9. Taylor expanded in im around 0

      \[\leadsto \mathsf{fma}\left(1 + im \cdot \left(1 + im \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot im\right)\right), re, \frac{re}{1 + im}\right) \cdot \mathsf{fma}\left(\color{blue}{re} \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 48.5%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, im, 0.5\right), im, 1\right), im, 1\right), re, \frac{re}{1 + im}\right) \cdot \mathsf{fma}\left(\color{blue}{re} \cdot re, -0.08333333333333333, 0.5\right) \]

   if 1.99999999999999991e-6 < (*.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 re around 0

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

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

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out-- N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-neg-in N/A

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

      7. cancel-sign-sub N/A

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

   5. Applied rewrites 21.1%

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

   6. Taylor expanded in im around 0

      \[\leadsto \left(2 \cdot re + {im}^{2} \cdot \left(im \cdot \left(-1 \cdot \left(im \cdot \left(-1 \cdot \left(-1 \cdot \left(-1 \cdot re + \frac{1}{2} \cdot re\right) + \left(\frac{-1}{2} \cdot re + \frac{1}{6} \cdot re\right)\right) + \left(\frac{-1}{2} \cdot \left(-1 \cdot re + \frac{1}{2} \cdot re\right) + \frac{-1}{6} \cdot re\right)\right)\right) - \left(-1 \cdot \left(-1 \cdot re + \frac{1}{2} \cdot re\right) + \frac{-1}{2} \cdot re\right)\right) - -1 \cdot re\right)\right) \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{12}, \frac{1}{2}\right) \]

   8. Applied rewrites 16.1%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, re, \mathsf{fma}\left(-0.16666666666666666, re, \mathsf{fma}\left(0.3333333333333333, re, -0.5 \cdot re\right)\right)\right), -im, 0 \cdot \left(-0.5 \cdot re\right)\right), im, re\right), im \cdot im, 2 \cdot re\right) \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, -0.08333333333333333, 0.5\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 26.4%

       \[\leadsto \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\left(-im\right) \cdot \mathsf{fma}\left(0.25, re, 0\right), im, re\right), 2 \cdot re\right) \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, -0.08333333333333333, 0.5\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 94.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 5.2:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 7.2 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(e^{im}, re, \frac{re}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, im, 0.5\right), im, 1\right), im, 1\right)}\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889 \cdot \left(im \cdot im\right), im \cdot im, 0.5\right), im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 5.2)
   (*
    (fma
     (fma
      (fma 0.001388888888888889 (* im im) 0.041666666666666664)
      (* im im)
      0.5)
     (* im im)
     1.0)
    (sin re))
   (if (<= im 7.2e+51)
     (*
      (fma
       (exp im)
       re
       (/ re (fma (fma (fma 0.16666666666666666 im 0.5) im 1.0) im 1.0)))
      (fma (* re re) -0.08333333333333333 0.5))
     (*
      (sin re)
      (fma
       (fma (* 0.001388888888888889 (* im im)) (* im im) 0.5)
       (* im im)
       1.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 5.2) {
		tmp = fma(fma(fma(0.001388888888888889, (im * im), 0.041666666666666664), (im * im), 0.5), (im * im), 1.0) * sin(re);
	} else if (im <= 7.2e+51) {
		tmp = fma(exp(im), re, (re / fma(fma(fma(0.16666666666666666, im, 0.5), im, 1.0), im, 1.0))) * fma((re * re), -0.08333333333333333, 0.5);
	} else {
		tmp = sin(re) * fma(fma((0.001388888888888889 * (im * im)), (im * im), 0.5), (im * im), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 5.2)
		tmp = Float64(fma(fma(fma(0.001388888888888889, Float64(im * im), 0.041666666666666664), Float64(im * im), 0.5), Float64(im * im), 1.0) * sin(re));
	elseif (im <= 7.2e+51)
		tmp = Float64(fma(exp(im), re, Float64(re / fma(fma(fma(0.16666666666666666, im, 0.5), im, 1.0), im, 1.0))) * fma(Float64(re * re), -0.08333333333333333, 0.5));
	else
		tmp = Float64(sin(re) * fma(fma(Float64(0.001388888888888889 * Float64(im * im)), Float64(im * im), 0.5), Float64(im * im), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 5.2], 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[Sin[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 7.2e+51], N[(N[(N[Exp[im], $MachinePrecision] * re + N[(re / N[(N[(N[(0.16666666666666666 * im + 0.5), $MachinePrecision] * im + 1.0), $MachinePrecision] * im + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(N[(N[(0.001388888888888889 * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 5.20000000000000018

   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 92.7

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

   7. Applied rewrites 92.7%

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

   if 5.20000000000000018 < im < 7.20000000000000022e51

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

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

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out-- N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-neg-in N/A

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

      7. cancel-sign-sub N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{fma}\left(e^{im}, re, \frac{re}{1 + im \cdot \left(1 + im \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot im\right)\right)}\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 96.9%

      \[\leadsto \mathsf{fma}\left(e^{im}, re, \frac{re}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, im, 0.5\right), im, 1\right), im, 1\right)}\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \]

   if 7.20000000000000022e51 < 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. 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. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 100.0

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

   9. Applied rewrites 100.0%

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

   10. Taylor expanded in im around inf

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

   12. Applied rewrites 100.0%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 5.2:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 7.2 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(e^{im}, re, \frac{re}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, im, 0.5\right), im, 1\right), im, 1\right)}\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889 \cdot \left(im \cdot im\right), im \cdot im, 0.5\right), im \cdot im, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 94.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 6:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 7.2 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(e^{im}, re, \frac{re}{\mathsf{fma}\left(\mathsf{fma}\left(im, 0.5, 1\right), im, 1\right)}\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889 \cdot \left(im \cdot im\right), im \cdot im, 0.5\right), im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 6.0)
   (*
    (fma
     (fma
      (fma 0.001388888888888889 (* im im) 0.041666666666666664)
      (* im im)
      0.5)
     (* im im)
     1.0)
    (sin re))
   (if (<= im 7.2e+51)
     (*
      (fma (exp im) re (/ re (fma (fma im 0.5 1.0) im 1.0)))
      (fma (* re re) -0.08333333333333333 0.5))
     (*
      (sin re)
      (fma
       (fma (* 0.001388888888888889 (* im im)) (* im im) 0.5)
       (* im im)
       1.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 6.0) {
		tmp = fma(fma(fma(0.001388888888888889, (im * im), 0.041666666666666664), (im * im), 0.5), (im * im), 1.0) * sin(re);
	} else if (im <= 7.2e+51) {
		tmp = fma(exp(im), re, (re / fma(fma(im, 0.5, 1.0), im, 1.0))) * fma((re * re), -0.08333333333333333, 0.5);
	} else {
		tmp = sin(re) * fma(fma((0.001388888888888889 * (im * im)), (im * im), 0.5), (im * im), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 6.0)
		tmp = Float64(fma(fma(fma(0.001388888888888889, Float64(im * im), 0.041666666666666664), Float64(im * im), 0.5), Float64(im * im), 1.0) * sin(re));
	elseif (im <= 7.2e+51)
		tmp = Float64(fma(exp(im), re, Float64(re / fma(fma(im, 0.5, 1.0), im, 1.0))) * fma(Float64(re * re), -0.08333333333333333, 0.5));
	else
		tmp = Float64(sin(re) * fma(fma(Float64(0.001388888888888889 * Float64(im * im)), Float64(im * im), 0.5), Float64(im * im), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 6.0], 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[Sin[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 7.2e+51], N[(N[(N[Exp[im], $MachinePrecision] * re + N[(re / N[(N[(im * 0.5 + 1.0), $MachinePrecision] * im + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(N[(N[(0.001388888888888889 * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 6

   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 92.7

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

   7. Applied rewrites 92.7%

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

   if 6 < im < 7.20000000000000022e51

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

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

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out-- N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-neg-in N/A

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

      7. cancel-sign-sub N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 96.6%

      \[\leadsto \mathsf{fma}\left(e^{im}, re, \frac{re}{\mathsf{fma}\left(\mathsf{fma}\left(im, 0.5, 1\right), im, 1\right)}\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \]

   if 7.20000000000000022e51 < 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. 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. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 100.0

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

   9. Applied rewrites 100.0%

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

   10. Taylor expanded in im around inf

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

   12. Applied rewrites 100.0%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 6:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 7.2 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(e^{im}, re, \frac{re}{\mathsf{fma}\left(\mathsf{fma}\left(im, 0.5, 1\right), im, 1\right)}\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889 \cdot \left(im \cdot im\right), im \cdot im, 0.5\right), im \cdot im, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 94.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 6:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 7.2 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(e^{im}, re, \frac{re}{1 + im}\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889 \cdot \left(im \cdot im\right), im \cdot im, 0.5\right), im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 6.0)
   (*
    (fma
     (fma
      (fma 0.001388888888888889 (* im im) 0.041666666666666664)
      (* im im)
      0.5)
     (* im im)
     1.0)
    (sin re))
   (if (<= im 7.2e+51)
     (*
      (fma (exp im) re (/ re (+ 1.0 im)))
      (fma (* re re) -0.08333333333333333 0.5))
     (*
      (sin re)
      (fma
       (fma (* 0.001388888888888889 (* im im)) (* im im) 0.5)
       (* im im)
       1.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 6.0) {
		tmp = fma(fma(fma(0.001388888888888889, (im * im), 0.041666666666666664), (im * im), 0.5), (im * im), 1.0) * sin(re);
	} else if (im <= 7.2e+51) {
		tmp = fma(exp(im), re, (re / (1.0 + im))) * fma((re * re), -0.08333333333333333, 0.5);
	} else {
		tmp = sin(re) * fma(fma((0.001388888888888889 * (im * im)), (im * im), 0.5), (im * im), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 6.0)
		tmp = Float64(fma(fma(fma(0.001388888888888889, Float64(im * im), 0.041666666666666664), Float64(im * im), 0.5), Float64(im * im), 1.0) * sin(re));
	elseif (im <= 7.2e+51)
		tmp = Float64(fma(exp(im), re, Float64(re / Float64(1.0 + im))) * fma(Float64(re * re), -0.08333333333333333, 0.5));
	else
		tmp = Float64(sin(re) * fma(fma(Float64(0.001388888888888889 * Float64(im * im)), Float64(im * im), 0.5), Float64(im * im), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 6.0], 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[Sin[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 7.2e+51], N[(N[(N[Exp[im], $MachinePrecision] * re + N[(re / N[(1.0 + im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(N[(N[(0.001388888888888889 * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 6

   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 92.7

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

   7. Applied rewrites 92.7%

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

   if 6 < im < 7.20000000000000022e51

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

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

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out-- N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-neg-in N/A

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

      7. cancel-sign-sub N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 96.2%

      \[\leadsto \mathsf{fma}\left(e^{im}, re, \frac{re}{1 + im}\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \]

   if 7.20000000000000022e51 < 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. 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. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 100.0

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

   9. Applied rewrites 100.0%

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

   10. Taylor expanded in im around inf

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

   12. Applied rewrites 100.0%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 6:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 7.2 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(e^{im}, re, \frac{re}{1 + im}\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889 \cdot \left(im \cdot im\right), im \cdot im, 0.5\right), im \cdot im, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 92.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 5.2:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 7.2 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(e^{im}, re, \frac{re}{1 + im}\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889 \cdot \left(im \cdot im\right), im \cdot im, 0.5\right), im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 5.2)
   (* (fma (fma 0.041666666666666664 (* im im) 0.5) (* im im) 1.0) (sin re))
   (if (<= im 7.2e+51)
     (*
      (fma (exp im) re (/ re (+ 1.0 im)))
      (fma (* re re) -0.08333333333333333 0.5))
     (*
      (sin re)
      (fma
       (fma (* 0.001388888888888889 (* im im)) (* im im) 0.5)
       (* im im)
       1.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 5.2) {
		tmp = fma(fma(0.041666666666666664, (im * im), 0.5), (im * im), 1.0) * sin(re);
	} else if (im <= 7.2e+51) {
		tmp = fma(exp(im), re, (re / (1.0 + im))) * fma((re * re), -0.08333333333333333, 0.5);
	} else {
		tmp = sin(re) * fma(fma((0.001388888888888889 * (im * im)), (im * im), 0.5), (im * im), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 5.2)
		tmp = Float64(fma(fma(0.041666666666666664, Float64(im * im), 0.5), Float64(im * im), 1.0) * sin(re));
	elseif (im <= 7.2e+51)
		tmp = Float64(fma(exp(im), re, Float64(re / Float64(1.0 + im))) * fma(Float64(re * re), -0.08333333333333333, 0.5));
	else
		tmp = Float64(sin(re) * fma(fma(Float64(0.001388888888888889 * Float64(im * im)), Float64(im * im), 0.5), Float64(im * im), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 5.2], N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sin[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 7.2e+51], N[(N[(N[Exp[im], $MachinePrecision] * re + N[(re / N[(1.0 + im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(N[(N[(0.001388888888888889 * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 5.20000000000000018

   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 86.0

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

   7. Applied rewrites 86.0%

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

   if 5.20000000000000018 < im < 7.20000000000000022e51

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

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

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out-- N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-neg-in N/A

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

      7. cancel-sign-sub N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 96.2%

      \[\leadsto \mathsf{fma}\left(e^{im}, re, \frac{re}{1 + im}\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \]

   if 7.20000000000000022e51 < 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. 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. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 100.0

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

   9. Applied rewrites 100.0%

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

   10. Taylor expanded in im around inf

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

   12. Applied rewrites 100.0%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 5.2:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 7.2 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(e^{im}, re, \frac{re}{1 + im}\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889 \cdot \left(im \cdot im\right), im \cdot im, 0.5\right), im \cdot im, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 92.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 5.2:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;\mathsf{fma}\left(e^{im}, re, \frac{re}{1 + im}\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right), im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 5.2)
   (* (fma (fma 0.041666666666666664 (* im im) 0.5) (* im im) 1.0) (sin re))
   (if (<= im 2.6e+77)
     (*
      (fma (exp im) re (/ re (+ 1.0 im)))
      (fma (* re re) -0.08333333333333333 0.5))
     (* (sin re) (fma (* 0.041666666666666664 (* im im)) (* im im) 1.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 5.2) {
		tmp = fma(fma(0.041666666666666664, (im * im), 0.5), (im * im), 1.0) * sin(re);
	} else if (im <= 2.6e+77) {
		tmp = fma(exp(im), re, (re / (1.0 + im))) * fma((re * re), -0.08333333333333333, 0.5);
	} else {
		tmp = sin(re) * fma((0.041666666666666664 * (im * im)), (im * im), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 5.2)
		tmp = Float64(fma(fma(0.041666666666666664, Float64(im * im), 0.5), Float64(im * im), 1.0) * sin(re));
	elseif (im <= 2.6e+77)
		tmp = Float64(fma(exp(im), re, Float64(re / Float64(1.0 + im))) * fma(Float64(re * re), -0.08333333333333333, 0.5));
	else
		tmp = Float64(sin(re) * fma(Float64(0.041666666666666664 * Float64(im * im)), Float64(im * im), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 5.2], N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sin[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.6e+77], N[(N[(N[Exp[im], $MachinePrecision] * re + N[(re / N[(1.0 + im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 5.20000000000000018

   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 86.0

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

   7. Applied rewrites 86.0%

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

   if 5.20000000000000018 < im < 2.6000000000000002e77

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

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

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out-- N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-neg-in N/A

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

      7. cancel-sign-sub N/A

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

   5. Applied rewrites 94.4%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 92.1%

      \[\leadsto \mathsf{fma}\left(e^{im}, re, \frac{re}{1 + im}\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \]

   if 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. 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. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 100.0

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

   9. Applied rewrites 100.0%

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

   10. Taylor expanded in im around inf

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

   12. Applied rewrites 100.0%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 5.2:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;\mathsf{fma}\left(e^{im}, re, \frac{re}{1 + im}\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right), im \cdot im, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 92.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 6:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;\mathsf{fma}\left(e^{im}, re, \frac{re}{1}\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right), im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 6.0)
   (* (fma (fma 0.041666666666666664 (* im im) 0.5) (* im im) 1.0) (sin re))
   (if (<= im 2.6e+77)
     (* (fma (exp im) re (/ re 1.0)) (fma (* re re) -0.08333333333333333 0.5))
     (* (sin re) (fma (* 0.041666666666666664 (* im im)) (* im im) 1.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 6.0) {
		tmp = fma(fma(0.041666666666666664, (im * im), 0.5), (im * im), 1.0) * sin(re);
	} else if (im <= 2.6e+77) {
		tmp = fma(exp(im), re, (re / 1.0)) * fma((re * re), -0.08333333333333333, 0.5);
	} else {
		tmp = sin(re) * fma((0.041666666666666664 * (im * im)), (im * im), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 6.0)
		tmp = Float64(fma(fma(0.041666666666666664, Float64(im * im), 0.5), Float64(im * im), 1.0) * sin(re));
	elseif (im <= 2.6e+77)
		tmp = Float64(fma(exp(im), re, Float64(re / 1.0)) * fma(Float64(re * re), -0.08333333333333333, 0.5));
	else
		tmp = Float64(sin(re) * fma(Float64(0.041666666666666664 * Float64(im * im)), Float64(im * im), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 6.0], N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sin[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.6e+77], N[(N[(N[Exp[im], $MachinePrecision] * re + N[(re / 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 6

   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 86.0

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

   7. Applied rewrites 86.0%

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

   if 6 < im < 2.6000000000000002e77

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

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

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out-- N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-neg-in N/A

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

      7. cancel-sign-sub N/A

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

   5. Applied rewrites 94.4%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 91.8%

      \[\leadsto \mathsf{fma}\left(e^{im}, re, \frac{re}{1}\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \]

   if 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. 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. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 100.0

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

   9. Applied rewrites 100.0%

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

   10. Taylor expanded in im around inf

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

   12. Applied rewrites 100.0%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 6:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;\mathsf{fma}\left(e^{im}, re, \frac{re}{1}\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right), im \cdot im, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 92.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 8:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;\mathsf{fma}\left(e^{im}, re, \mathsf{fma}\left(-re, im, re\right)\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right), im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 8.0)
   (* (fma (fma 0.041666666666666664 (* im im) 0.5) (* im im) 1.0) (sin re))
   (if (<= im 2.6e+77)
     (*
      (fma (exp im) re (fma (- re) im re))
      (fma (* re re) -0.08333333333333333 0.5))
     (* (sin re) (fma (* 0.041666666666666664 (* im im)) (* im im) 1.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 8.0) {
		tmp = fma(fma(0.041666666666666664, (im * im), 0.5), (im * im), 1.0) * sin(re);
	} else if (im <= 2.6e+77) {
		tmp = fma(exp(im), re, fma(-re, im, re)) * fma((re * re), -0.08333333333333333, 0.5);
	} else {
		tmp = sin(re) * fma((0.041666666666666664 * (im * im)), (im * im), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 8.0)
		tmp = Float64(fma(fma(0.041666666666666664, Float64(im * im), 0.5), Float64(im * im), 1.0) * sin(re));
	elseif (im <= 2.6e+77)
		tmp = Float64(fma(exp(im), re, fma(Float64(-re), im, re)) * fma(Float64(re * re), -0.08333333333333333, 0.5));
	else
		tmp = Float64(sin(re) * fma(Float64(0.041666666666666664 * Float64(im * im)), Float64(im * im), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 8.0], N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sin[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.6e+77], N[(N[(N[Exp[im], $MachinePrecision] * re + N[((-re) * im + re), $MachinePrecision]), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 8

   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 86.0

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

   7. Applied rewrites 86.0%

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

   if 8 < im < 2.6000000000000002e77

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

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

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out-- N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-neg-in N/A

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

      7. cancel-sign-sub N/A

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

   5. Applied rewrites 94.4%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 85.9%

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

   if 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. 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. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 100.0

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

   9. Applied rewrites 100.0%

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

   10. Taylor expanded in im around inf

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

   12. Applied rewrites 100.0%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 8:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;\mathsf{fma}\left(e^{im}, re, \mathsf{fma}\left(-re, im, re\right)\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right), im \cdot im, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 90.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 21000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 8 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, re, \mathsf{fma}\left(-0.16666666666666666, re, \mathsf{fma}\left(0.3333333333333333, re, -0.5 \cdot re\right)\right)\right), -im, 0\right), im, re\right), im \cdot im, 2 \cdot re\right) \cdot \left(\left(\left(\frac{0.5}{re \cdot re} - 0.08333333333333333\right) \cdot re\right) \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right), im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 21000.0)
   (* (fma (fma 0.041666666666666664 (* im im) 0.5) (* im im) 1.0) (sin re))
   (if (<= im 8e+75)
     (*
      (fma
       (fma
        (fma
         (fma
          0.25
          re
          (fma
           -0.16666666666666666
           re
           (fma 0.3333333333333333 re (* -0.5 re))))
         (- im)
         0.0)
        im
        re)
       (* im im)
       (* 2.0 re))
      (* (* (- (/ 0.5 (* re re)) 0.08333333333333333) re) re))
     (* (sin re) (fma (* 0.041666666666666664 (* im im)) (* im im) 1.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 21000.0) {
		tmp = fma(fma(0.041666666666666664, (im * im), 0.5), (im * im), 1.0) * sin(re);
	} else if (im <= 8e+75) {
		tmp = fma(fma(fma(fma(0.25, re, fma(-0.16666666666666666, re, fma(0.3333333333333333, re, (-0.5 * re)))), -im, 0.0), im, re), (im * im), (2.0 * re)) * ((((0.5 / (re * re)) - 0.08333333333333333) * re) * re);
	} else {
		tmp = sin(re) * fma((0.041666666666666664 * (im * im)), (im * im), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 21000.0)
		tmp = Float64(fma(fma(0.041666666666666664, Float64(im * im), 0.5), Float64(im * im), 1.0) * sin(re));
	elseif (im <= 8e+75)
		tmp = Float64(fma(fma(fma(fma(0.25, re, fma(-0.16666666666666666, re, fma(0.3333333333333333, re, Float64(-0.5 * re)))), Float64(-im), 0.0), im, re), Float64(im * im), Float64(2.0 * re)) * Float64(Float64(Float64(Float64(0.5 / Float64(re * re)) - 0.08333333333333333) * re) * re));
	else
		tmp = Float64(sin(re) * fma(Float64(0.041666666666666664 * Float64(im * im)), Float64(im * im), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 21000.0], N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sin[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 8e+75], N[(N[(N[(N[(N[(0.25 * re + N[(-0.16666666666666666 * re + N[(0.3333333333333333 * re + N[(-0.5 * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-im) + 0.0), $MachinePrecision] * im + re), $MachinePrecision] * N[(im * im), $MachinePrecision] + N[(2.0 * re), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(0.5 / N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.08333333333333333), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 21000

   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 84.7

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

   7. Applied rewrites 84.7%

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

   if 21000 < im < 7.99999999999999941e75

   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 re around 0

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

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

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out-- N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-neg-in N/A

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

      7. cancel-sign-sub N/A

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

   5. Applied rewrites 93.3%

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

   6. Taylor expanded in im around 0

      \[\leadsto \left(2 \cdot re + {im}^{2} \cdot \left(im \cdot \left(-1 \cdot \left(im \cdot \left(-1 \cdot \left(-1 \cdot \left(-1 \cdot re + \frac{1}{2} \cdot re\right) + \left(\frac{-1}{2} \cdot re + \frac{1}{6} \cdot re\right)\right) + \left(\frac{-1}{2} \cdot \left(-1 \cdot re + \frac{1}{2} \cdot re\right) + \frac{-1}{6} \cdot re\right)\right)\right) - \left(-1 \cdot \left(-1 \cdot re + \frac{1}{2} \cdot re\right) + \frac{-1}{2} \cdot re\right)\right) - -1 \cdot re\right)\right) \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{12}, \frac{1}{2}\right) \]

   8. Applied rewrites 16.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, re, \mathsf{fma}\left(-0.16666666666666666, re, \mathsf{fma}\left(0.3333333333333333, re, -0.5 \cdot re\right)\right)\right), -im, 0 \cdot \left(-0.5 \cdot re\right)\right), im, re\right), im \cdot im, 2 \cdot re\right) \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, -0.08333333333333333, 0.5\right) \]

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, re, \mathsf{fma}\left(\frac{-1}{6}, re, \mathsf{fma}\left(\frac{1}{3}, re, \frac{-1}{2} \cdot re\right)\right)\right), -im, 0 \cdot \left(\frac{-1}{2} \cdot re\right)\right), im, re\right), im \cdot im, 2 \cdot re\right) \cdot \left({re}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{{re}^{2}} - \frac{1}{12}\right)}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 42.2%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, re, \mathsf{fma}\left(-0.16666666666666666, re, \mathsf{fma}\left(0.3333333333333333, re, -0.5 \cdot re\right)\right)\right), -im, 0 \cdot \left(-0.5 \cdot re\right)\right), im, re\right), im \cdot im, 2 \cdot re\right) \cdot \left(\left(\left(\frac{0.5}{re \cdot re} - 0.08333333333333333\right) \cdot re\right) \cdot \color{blue}{re}\right) \]

   if 7.99999999999999941e75 < 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. 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. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 100.0

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

   9. Applied rewrites 100.0%

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

   10. Taylor expanded in im around inf

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

   12. Applied rewrites 100.0%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 21000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 8 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, re, \mathsf{fma}\left(-0.16666666666666666, re, \mathsf{fma}\left(0.3333333333333333, re, -0.5 \cdot re\right)\right)\right), -im, 0\right), im, re\right), im \cdot im, 2 \cdot re\right) \cdot \left(\left(\left(\frac{0.5}{re \cdot re} - 0.08333333333333333\right) \cdot re\right) \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right), im \cdot im, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 84.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 14.5:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 8 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, re, \mathsf{fma}\left(-0.16666666666666666, re, \mathsf{fma}\left(0.3333333333333333, re, -0.5 \cdot re\right)\right)\right), -im, 0\right), im, re\right), im \cdot im, 2 \cdot re\right) \cdot \left(\left(\left(\frac{0.5}{re \cdot re} - 0.08333333333333333\right) \cdot re\right) \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right), im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 14.5)
   (* (fma (* 0.5 im) im 1.0) (sin re))
   (if (<= im 8e+75)
     (*
      (fma
       (fma
        (fma
         (fma
          0.25
          re
          (fma
           -0.16666666666666666
           re
           (fma 0.3333333333333333 re (* -0.5 re))))
         (- im)
         0.0)
        im
        re)
       (* im im)
       (* 2.0 re))
      (* (* (- (/ 0.5 (* re re)) 0.08333333333333333) re) re))
     (* (sin re) (fma (* 0.041666666666666664 (* im im)) (* im im) 1.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 14.5) {
		tmp = fma((0.5 * im), im, 1.0) * sin(re);
	} else if (im <= 8e+75) {
		tmp = fma(fma(fma(fma(0.25, re, fma(-0.16666666666666666, re, fma(0.3333333333333333, re, (-0.5 * re)))), -im, 0.0), im, re), (im * im), (2.0 * re)) * ((((0.5 / (re * re)) - 0.08333333333333333) * re) * re);
	} else {
		tmp = sin(re) * fma((0.041666666666666664 * (im * im)), (im * im), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 14.5)
		tmp = Float64(fma(Float64(0.5 * im), im, 1.0) * sin(re));
	elseif (im <= 8e+75)
		tmp = Float64(fma(fma(fma(fma(0.25, re, fma(-0.16666666666666666, re, fma(0.3333333333333333, re, Float64(-0.5 * re)))), Float64(-im), 0.0), im, re), Float64(im * im), Float64(2.0 * re)) * Float64(Float64(Float64(Float64(0.5 / Float64(re * re)) - 0.08333333333333333) * re) * re));
	else
		tmp = Float64(sin(re) * fma(Float64(0.041666666666666664 * Float64(im * im)), Float64(im * im), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 14.5], N[(N[(N[(0.5 * im), $MachinePrecision] * im + 1.0), $MachinePrecision] * N[Sin[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 8e+75], N[(N[(N[(N[(N[(0.25 * re + N[(-0.16666666666666666 * re + N[(0.3333333333333333 * re + N[(-0.5 * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-im) + 0.0), $MachinePrecision] * im + re), $MachinePrecision] * N[(im * im), $MachinePrecision] + N[(2.0 * re), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(0.5 / N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.08333333333333333), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 14.5

   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. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt1-in N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 74.4

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

   5. Applied rewrites 74.4%

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

   if 14.5 < im < 7.99999999999999941e75

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

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

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out-- N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-neg-in N/A

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

      7. cancel-sign-sub N/A

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

   5. Applied rewrites 94.4%

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

   6. Taylor expanded in im around 0

      \[\leadsto \left(2 \cdot re + {im}^{2} \cdot \left(im \cdot \left(-1 \cdot \left(im \cdot \left(-1 \cdot \left(-1 \cdot \left(-1 \cdot re + \frac{1}{2} \cdot re\right) + \left(\frac{-1}{2} \cdot re + \frac{1}{6} \cdot re\right)\right) + \left(\frac{-1}{2} \cdot \left(-1 \cdot re + \frac{1}{2} \cdot re\right) + \frac{-1}{6} \cdot re\right)\right)\right) - \left(-1 \cdot \left(-1 \cdot re + \frac{1}{2} \cdot re\right) + \frac{-1}{2} \cdot re\right)\right) - -1 \cdot re\right)\right) \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{12}, \frac{1}{2}\right) \]

   8. Applied rewrites 14.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, re, \mathsf{fma}\left(-0.16666666666666666, re, \mathsf{fma}\left(0.3333333333333333, re, -0.5 \cdot re\right)\right)\right), -im, 0 \cdot \left(-0.5 \cdot re\right)\right), im, re\right), im \cdot im, 2 \cdot re\right) \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, -0.08333333333333333, 0.5\right) \]

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, re, \mathsf{fma}\left(\frac{-1}{6}, re, \mathsf{fma}\left(\frac{1}{3}, re, \frac{-1}{2} \cdot re\right)\right)\right), -im, 0 \cdot \left(\frac{-1}{2} \cdot re\right)\right), im, re\right), im \cdot im, 2 \cdot re\right) \cdot \left({re}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{{re}^{2}} - \frac{1}{12}\right)}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 36.3%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, re, \mathsf{fma}\left(-0.16666666666666666, re, \mathsf{fma}\left(0.3333333333333333, re, -0.5 \cdot re\right)\right)\right), -im, 0 \cdot \left(-0.5 \cdot re\right)\right), im, re\right), im \cdot im, 2 \cdot re\right) \cdot \left(\left(\left(\frac{0.5}{re \cdot re} - 0.08333333333333333\right) \cdot re\right) \cdot \color{blue}{re}\right) \]

   if 7.99999999999999941e75 < 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. 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. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 100.0

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

   9. Applied rewrites 100.0%

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

   10. Taylor expanded in im around inf

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

   12. Applied rewrites 100.0%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 14.5:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 8 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, re, \mathsf{fma}\left(-0.16666666666666666, re, \mathsf{fma}\left(0.3333333333333333, re, -0.5 \cdot re\right)\right)\right), -im, 0\right), im, re\right), im \cdot im, 2 \cdot re\right) \cdot \left(\left(\left(\frac{0.5}{re \cdot re} - 0.08333333333333333\right) \cdot re\right) \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right), im \cdot im, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 83.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \sin re\\ \mathbf{if}\;im \leq 14.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im \leq 1.75 \cdot 10^{+124}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, re, \mathsf{fma}\left(-0.16666666666666666, re, \mathsf{fma}\left(0.3333333333333333, re, -0.5 \cdot re\right)\right)\right), -im, 0\right), im, re\right), im \cdot im, 2 \cdot re\right) \cdot \left(\left(\left(\frac{0.5}{re \cdot re} - 0.08333333333333333\right) \cdot re\right) \cdot re\right)\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(im \cdot im\right) \cdot im}{im}, 0.5, 1\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (fma (* 0.5 im) im 1.0) (sin re))))
   (if (<= im 14.5)
     t_0
     (if (<= im 1.75e+124)
       (*
        (fma
         (fma
          (fma
           (fma
            0.25
            re
            (fma
             -0.16666666666666666
             re
             (fma 0.3333333333333333 re (* -0.5 re))))
           (- im)
           0.0)
          im
          re)
         (* im im)
         (* 2.0 re))
        (* (* (- (/ 0.5 (* re re)) 0.08333333333333333) re) re))
       (if (<= im 1.9e+154)
         (* (fma (/ (* (* im im) im) im) 0.5 1.0) re)
         t_0)))))

C

double code(double re, double im) {
	double t_0 = fma((0.5 * im), im, 1.0) * sin(re);
	double tmp;
	if (im <= 14.5) {
		tmp = t_0;
	} else if (im <= 1.75e+124) {
		tmp = fma(fma(fma(fma(0.25, re, fma(-0.16666666666666666, re, fma(0.3333333333333333, re, (-0.5 * re)))), -im, 0.0), im, re), (im * im), (2.0 * re)) * ((((0.5 / (re * re)) - 0.08333333333333333) * re) * re);
	} else if (im <= 1.9e+154) {
		tmp = fma((((im * im) * im) / im), 0.5, 1.0) * re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(fma(Float64(0.5 * im), im, 1.0) * sin(re))
	tmp = 0.0
	if (im <= 14.5)
		tmp = t_0;
	elseif (im <= 1.75e+124)
		tmp = Float64(fma(fma(fma(fma(0.25, re, fma(-0.16666666666666666, re, fma(0.3333333333333333, re, Float64(-0.5 * re)))), Float64(-im), 0.0), im, re), Float64(im * im), Float64(2.0 * re)) * Float64(Float64(Float64(Float64(0.5 / Float64(re * re)) - 0.08333333333333333) * re) * re));
	elseif (im <= 1.9e+154)
		tmp = Float64(fma(Float64(Float64(Float64(im * im) * im) / im), 0.5, 1.0) * re);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[(0.5 * im), $MachinePrecision] * im + 1.0), $MachinePrecision] * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 14.5], t$95$0, If[LessEqual[im, 1.75e+124], N[(N[(N[(N[(N[(0.25 * re + N[(-0.16666666666666666 * re + N[(0.3333333333333333 * re + N[(-0.5 * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-im) + 0.0), $MachinePrecision] * im + re), $MachinePrecision] * N[(im * im), $MachinePrecision] + N[(2.0 * re), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(0.5 / N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.08333333333333333), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.9e+154], N[(N[(N[(N[(N[(im * im), $MachinePrecision] * im), $MachinePrecision] / im), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * re), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < 14.5 or 1.8999999999999999e154 < 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. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt1-in N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 78.7

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

   5. Applied rewrites 78.7%

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

   if 14.5 < im < 1.7500000000000001e124

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

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

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out-- N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-neg-in N/A

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

      7. cancel-sign-sub N/A

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

   5. Applied rewrites 90.9%

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

   6. Taylor expanded in im around 0

      \[\leadsto \left(2 \cdot re + {im}^{2} \cdot \left(im \cdot \left(-1 \cdot \left(im \cdot \left(-1 \cdot \left(-1 \cdot \left(-1 \cdot re + \frac{1}{2} \cdot re\right) + \left(\frac{-1}{2} \cdot re + \frac{1}{6} \cdot re\right)\right) + \left(\frac{-1}{2} \cdot \left(-1 \cdot re + \frac{1}{2} \cdot re\right) + \frac{-1}{6} \cdot re\right)\right)\right) - \left(-1 \cdot \left(-1 \cdot re + \frac{1}{2} \cdot re\right) + \frac{-1}{2} \cdot re\right)\right) - -1 \cdot re\right)\right) \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{12}, \frac{1}{2}\right) \]

   8. Applied rewrites 25.7%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, re, \mathsf{fma}\left(-0.16666666666666666, re, \mathsf{fma}\left(0.3333333333333333, re, -0.5 \cdot re\right)\right)\right), -im, 0 \cdot \left(-0.5 \cdot re\right)\right), im, re\right), im \cdot im, 2 \cdot re\right) \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, -0.08333333333333333, 0.5\right) \]

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, re, \mathsf{fma}\left(\frac{-1}{6}, re, \mathsf{fma}\left(\frac{1}{3}, re, \frac{-1}{2} \cdot re\right)\right)\right), -im, 0 \cdot \left(\frac{-1}{2} \cdot re\right)\right), im, re\right), im \cdot im, 2 \cdot re\right) \cdot \left({re}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{{re}^{2}} - \frac{1}{12}\right)}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 43.3%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, re, \mathsf{fma}\left(-0.16666666666666666, re, \mathsf{fma}\left(0.3333333333333333, re, -0.5 \cdot re\right)\right)\right), -im, 0 \cdot \left(-0.5 \cdot re\right)\right), im, re\right), im \cdot im, 2 \cdot re\right) \cdot \left(\left(\left(\frac{0.5}{re \cdot re} - 0.08333333333333333\right) \cdot re\right) \cdot \color{blue}{re}\right) \]

   if 1.7500000000000001e124 < im < 1.8999999999999999e154

   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. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt1-in N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 7.8

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

   5. Applied rewrites 7.8%

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

   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 51.5%

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

   10. Applied rewrites 83.3%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 14.5:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 1.75 \cdot 10^{+124}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, re, \mathsf{fma}\left(-0.16666666666666666, re, \mathsf{fma}\left(0.3333333333333333, re, -0.5 \cdot re\right)\right)\right), -im, 0\right), im, re\right), im \cdot im, 2 \cdot re\right) \cdot \left(\left(\left(\frac{0.5}{re \cdot re} - 0.08333333333333333\right) \cdot re\right) \cdot re\right)\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(im \cdot im\right) \cdot im}{im}, 0.5, 1\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \sin re\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 48.7% 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. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt1-in N/A

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

   5. lower-*.f64 N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sin.f64 70.7

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

5. Applied rewrites 70.7%

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

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 45.7%

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

Alternative 24: 26.3% accurate, 28.8× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

5. Applied rewrites 46.8%

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

6. Taylor expanded in re around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 24.1

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

8. Applied rewrites 24.1%

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

Reproduce

herbie shell --seed 2024358 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))