math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 4.5s

Alternatives: 19

Speedup: 1.5×

• 49.5% 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

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

FPCore

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

C

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

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 = ((2.0d0 * cosh(im)) * sin(re)) * 0.5d0
end function

Java

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

Python

def code(re, im):
	return ((2.0 * math.cosh(im)) * math.sin(re)) * 0.5

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. 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. lift-sin.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-exp.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. distribute-rgt-in N/A

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

   9. sub0-neg N/A

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

   10. distribute-rgt-in N/A

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

   11. +-commutative N/A

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

   12. associate-*r* N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

3. Applied rewrites 100.0%

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

Alternative 2: 84.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re \cdot re, 0.004166666666666667\right) \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \sin re\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(2 \cdot \cosh im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (*
      (*
       (fma
        (-
         (*
          (fma -9.92063492063492e-5 (* re re) 0.004166666666666667)
          (* re re))
         0.08333333333333333)
        (* re re)
        0.5)
       re)
      (fma (fma (* im im) 0.08333333333333333 1.0) (* im im) 2.0))
     (if (<= t_0 1.0)
       (*
        (*
         (fma
          (fma
           (fma 0.002777777777777778 (* im im) 0.08333333333333333)
           (* im im)
           1.0)
          (* im im)
          2.0)
         (sin re))
        0.5)
       (* (* re 0.5) (* 2.0 (cosh im)))))))

C

double code(double re, double im) {
	double t_0 = (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma(((fma(-9.92063492063492e-5, (re * re), 0.004166666666666667) * (re * re)) - 0.08333333333333333), (re * re), 0.5) * re) * fma(fma((im * im), 0.08333333333333333, 1.0), (im * im), 2.0);
	} else if (t_0 <= 1.0) {
		tmp = (fma(fma(fma(0.002777777777777778, (im * im), 0.08333333333333333), (im * im), 1.0), (im * im), 2.0) * sin(re)) * 0.5;
	} else {
		tmp = (re * 0.5) * (2.0 * cosh(im));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(fma(Float64(Float64(fma(-9.92063492063492e-5, Float64(re * re), 0.004166666666666667) * Float64(re * re)) - 0.08333333333333333), Float64(re * re), 0.5) * re) * fma(fma(Float64(im * im), 0.08333333333333333, 1.0), Float64(im * im), 2.0));
	elseif (t_0 <= 1.0)
		tmp = Float64(Float64(fma(fma(fma(0.002777777777777778, Float64(im * im), 0.08333333333333333), Float64(im * im), 1.0), Float64(im * im), 2.0) * sin(re)) * 0.5);
	else
		tmp = Float64(Float64(re * 0.5) * Float64(2.0 * cosh(im)));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(N[(-9.92063492063492e-5 * N[(re * re), $MachinePrecision] + 0.004166666666666667), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.08333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * 0.08333333333333333 + 1.0), $MachinePrecision] * N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[(N[(N[(N[(N[(0.002777777777777778 * N[(im * im), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision] * N[Sin[re], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(re * 0.5), $MachinePrecision] * N[(2.0 * N[Cosh[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(1 + \frac{1}{12} \cdot {im}^{2}\right) \cdot {im}^{2} + 2\right) \]

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 76.4

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot \color{blue}{im}, 2\right) \]

   4. Applied rewrites 76.4%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right)} \]

   5. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{12}, 1\right), im \cdot im, 2\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right) \cdot \color{blue}{re}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{12}, 1\right), im \cdot im, 2\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right) \cdot \color{blue}{re}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{12}, 1\right), im \cdot im, 2\right) \]

   7. Applied rewrites 62.7%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re \cdot re, 0.004166666666666667\right) \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right) \]

   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. 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. lift-sin.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. distribute-rgt-in N/A

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

      9. sub0-neg N/A

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

      10. distribute-rgt-in N/A

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

      11. +-commutative N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in im around 0

      \[\leadsto \left(\color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \cdot \sin re\right) \cdot \frac{1}{2} \]
   5. Step-by-step derivation

      1. cosh-undef-rev N/A

         \[\leadsto \left(\left(\color{blue}{2} + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      2. sub0-neg N/A

         \[\leadsto \left(\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{2} + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      4. +-commutative N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + \color{blue}{2}\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) \cdot {im}^{2} + 2\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right), \color{blue}{{im}^{2}}, 2\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      7. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left({im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) + 1, {\color{blue}{im}}^{2}, 2\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      8. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) \cdot {im}^{2} + 1, {im}^{2}, 2\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}, {im}^{2}, 1\right), {\color{blue}{im}}^{2}, 2\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      10. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360} \cdot {im}^{2} + \frac{1}{12}, {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      11. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, {im}^{2}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      12. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      14. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), {im}^{2}, 2\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      15. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), {im}^{2}, 2\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      16. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot \color{blue}{im}, 2\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      17. lift-*.f64 99.3

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot \color{blue}{im}, 2\right) \cdot \sin re\right) \cdot 0.5 \]

   6. Applied rewrites 99.3%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right)} \cdot \sin re\right) \cdot 0.5 \]

   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. Taylor expanded in re around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. cosh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cosh.f64 74.3

         \[\leadsto \left(re \cdot 0.5\right) \cdot \left(2 \cdot \cosh im\right) \]

   4. Applied rewrites 74.3%

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

Alternative 3: 84.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ t_1 := t\_0 \cdot \left(e^{0 - im} + e^{im}\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re \cdot re, 0.004166666666666667\right) \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right)\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(2 \cdot \cosh im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re))) (t_1 (* t_0 (+ (exp (- 0.0 im)) (exp im)))))
   (if (<= t_1 (- INFINITY))
     (*
      (*
       (fma
        (-
         (*
          (fma -9.92063492063492e-5 (* re re) 0.004166666666666667)
          (* re re))
         0.08333333333333333)
        (* re re)
        0.5)
       re)
      (fma (fma (* im im) 0.08333333333333333 1.0) (* im im) 2.0))
     (if (<= t_1 1.0)
       (*
        t_0
        (fma
         (fma
          (fma 0.002777777777777778 (* im im) 0.08333333333333333)
          (* im im)
          1.0)
         (* im im)
         2.0))
       (* (* re 0.5) (* 2.0 (cosh im)))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sin(re);
	double t_1 = t_0 * (exp((0.0 - im)) + exp(im));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (fma(((fma(-9.92063492063492e-5, (re * re), 0.004166666666666667) * (re * re)) - 0.08333333333333333), (re * re), 0.5) * re) * fma(fma((im * im), 0.08333333333333333, 1.0), (im * im), 2.0);
	} else if (t_1 <= 1.0) {
		tmp = t_0 * fma(fma(fma(0.002777777777777778, (im * im), 0.08333333333333333), (im * im), 1.0), (im * im), 2.0);
	} else {
		tmp = (re * 0.5) * (2.0 * cosh(im));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * sin(re))
	t_1 = Float64(t_0 * Float64(exp(Float64(0.0 - im)) + exp(im)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(fma(Float64(Float64(fma(-9.92063492063492e-5, Float64(re * re), 0.004166666666666667) * Float64(re * re)) - 0.08333333333333333), Float64(re * re), 0.5) * re) * fma(fma(Float64(im * im), 0.08333333333333333, 1.0), Float64(im * im), 2.0));
	elseif (t_1 <= 1.0)
		tmp = Float64(t_0 * fma(fma(fma(0.002777777777777778, Float64(im * im), 0.08333333333333333), Float64(im * im), 1.0), Float64(im * im), 2.0));
	else
		tmp = Float64(Float64(re * 0.5) * Float64(2.0 * cosh(im)));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(N[(N[(-9.92063492063492e-5 * N[(re * re), $MachinePrecision] + 0.004166666666666667), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.08333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * 0.08333333333333333 + 1.0), $MachinePrecision] * N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.0], N[(t$95$0 * N[(N[(N[(0.002777777777777778 * N[(im * im), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(re * 0.5), $MachinePrecision] * N[(2.0 * N[Cosh[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(1 + \frac{1}{12} \cdot {im}^{2}\right) \cdot {im}^{2} + 2\right) \]

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 76.4

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot \color{blue}{im}, 2\right) \]

   4. Applied rewrites 76.4%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right)} \]

   5. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{12}, 1\right), im \cdot im, 2\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right) \cdot \color{blue}{re}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{12}, 1\right), im \cdot im, 2\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right) \cdot \color{blue}{re}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{12}, 1\right), im \cdot im, 2\right) \]

   7. Applied rewrites 62.7%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re \cdot re, 0.004166666666666667\right) \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right) \]

   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. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + \color{blue}{2}\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) \cdot {im}^{2} + 2\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right), \color{blue}{{im}^{2}}, 2\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left({im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) + 1, {\color{blue}{im}}^{2}, 2\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) \cdot {im}^{2} + 1, {im}^{2}, 2\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}, {im}^{2}, 1\right), {\color{blue}{im}}^{2}, 2\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360} \cdot {im}^{2} + \frac{1}{12}, {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, {im}^{2}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      11. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), {im}^{2}, 2\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), {im}^{2}, 2\right) \]

      13. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot \color{blue}{im}, 2\right) \]

      14. lower-*.f64 99.3

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot \color{blue}{im}, 2\right) \]

   4. Applied rewrites 99.3%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right)} \]

   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. Taylor expanded in re around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. cosh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cosh.f64 74.3

         \[\leadsto \left(re \cdot 0.5\right) \cdot \left(2 \cdot \cosh im\right) \]

   4. Applied rewrites 74.3%

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

Alternative 4: 84.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ t_1 := t\_0 \cdot \left(e^{0 - im} + e^{im}\right)\\ t_2 := \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re \cdot re, 0.004166666666666667\right) \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \cdot t\_2\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;t\_0 \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(2 \cdot \cosh im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re)))
        (t_1 (* t_0 (+ (exp (- 0.0 im)) (exp im))))
        (t_2 (fma (fma (* im im) 0.08333333333333333 1.0) (* im im) 2.0)))
   (if (<= t_1 (- INFINITY))
     (*
      (*
       (fma
        (-
         (*
          (fma -9.92063492063492e-5 (* re re) 0.004166666666666667)
          (* re re))
         0.08333333333333333)
        (* re re)
        0.5)
       re)
      t_2)
     (if (<= t_1 1.0) (* t_0 t_2) (* (* re 0.5) (* 2.0 (cosh im)))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sin(re);
	double t_1 = t_0 * (exp((0.0 - im)) + exp(im));
	double t_2 = fma(fma((im * im), 0.08333333333333333, 1.0), (im * im), 2.0);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (fma(((fma(-9.92063492063492e-5, (re * re), 0.004166666666666667) * (re * re)) - 0.08333333333333333), (re * re), 0.5) * re) * t_2;
	} else if (t_1 <= 1.0) {
		tmp = t_0 * t_2;
	} else {
		tmp = (re * 0.5) * (2.0 * cosh(im));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * sin(re))
	t_1 = Float64(t_0 * Float64(exp(Float64(0.0 - im)) + exp(im)))
	t_2 = fma(fma(Float64(im * im), 0.08333333333333333, 1.0), Float64(im * im), 2.0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(fma(Float64(Float64(fma(-9.92063492063492e-5, Float64(re * re), 0.004166666666666667) * Float64(re * re)) - 0.08333333333333333), Float64(re * re), 0.5) * re) * t_2);
	elseif (t_1 <= 1.0)
		tmp = Float64(t_0 * t_2);
	else
		tmp = Float64(Float64(re * 0.5) * Float64(2.0 * cosh(im)));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(im * im), $MachinePrecision] * 0.08333333333333333 + 1.0), $MachinePrecision] * N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(N[(N[(-9.92063492063492e-5 * N[(re * re), $MachinePrecision] + 0.004166666666666667), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.08333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$1, 1.0], N[(t$95$0 * t$95$2), $MachinePrecision], N[(N[(re * 0.5), $MachinePrecision] * N[(2.0 * N[Cosh[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(1 + \frac{1}{12} \cdot {im}^{2}\right) \cdot {im}^{2} + 2\right) \]

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 76.4

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot \color{blue}{im}, 2\right) \]

   4. Applied rewrites 76.4%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right)} \]

   5. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{12}, 1\right), im \cdot im, 2\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right) \cdot \color{blue}{re}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{12}, 1\right), im \cdot im, 2\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right) \cdot \color{blue}{re}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{12}, 1\right), im \cdot im, 2\right) \]

   7. Applied rewrites 62.7%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re \cdot re, 0.004166666666666667\right) \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right) \]

   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. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(1 + \frac{1}{12} \cdot {im}^{2}\right) \cdot {im}^{2} + 2\right) \]

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 99.2

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot \color{blue}{im}, 2\right) \]

   4. Applied rewrites 99.2%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right)} \]

   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. Taylor expanded in re around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. cosh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cosh.f64 74.3

         \[\leadsto \left(re \cdot 0.5\right) \cdot \left(2 \cdot \cosh im\right) \]

   4. Applied rewrites 74.3%

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

Alternative 5: 84.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re \cdot re, 0.004166666666666667\right) \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\mathsf{fma}\left(0.5, im \cdot im, 1\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(2 \cdot \cosh im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (*
      (*
       (fma
        (-
         (*
          (fma -9.92063492063492e-5 (* re re) 0.004166666666666667)
          (* re re))
         0.08333333333333333)
        (* re re)
        0.5)
       re)
      (fma (fma (* im im) 0.08333333333333333 1.0) (* im im) 2.0))
     (if (<= t_0 1.0)
       (* (fma 0.5 (* im im) 1.0) (sin re))
       (* (* re 0.5) (* 2.0 (cosh im)))))))

C

double code(double re, double im) {
	double t_0 = (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma(((fma(-9.92063492063492e-5, (re * re), 0.004166666666666667) * (re * re)) - 0.08333333333333333), (re * re), 0.5) * re) * fma(fma((im * im), 0.08333333333333333, 1.0), (im * im), 2.0);
	} else if (t_0 <= 1.0) {
		tmp = fma(0.5, (im * im), 1.0) * sin(re);
	} else {
		tmp = (re * 0.5) * (2.0 * cosh(im));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(fma(Float64(Float64(fma(-9.92063492063492e-5, Float64(re * re), 0.004166666666666667) * Float64(re * re)) - 0.08333333333333333), Float64(re * re), 0.5) * re) * fma(fma(Float64(im * im), 0.08333333333333333, 1.0), Float64(im * im), 2.0));
	elseif (t_0 <= 1.0)
		tmp = Float64(fma(0.5, Float64(im * im), 1.0) * sin(re));
	else
		tmp = Float64(Float64(re * 0.5) * Float64(2.0 * cosh(im)));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(N[(-9.92063492063492e-5 * N[(re * re), $MachinePrecision] + 0.004166666666666667), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.08333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * 0.08333333333333333 + 1.0), $MachinePrecision] * N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[(N[(0.5 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sin[re], $MachinePrecision]), $MachinePrecision], N[(N[(re * 0.5), $MachinePrecision] * N[(2.0 * N[Cosh[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(1 + \frac{1}{12} \cdot {im}^{2}\right) \cdot {im}^{2} + 2\right) \]

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 76.4

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot \color{blue}{im}, 2\right) \]

   4. Applied rewrites 76.4%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right)} \]

   5. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{12}, 1\right), im \cdot im, 2\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right) \cdot \color{blue}{re}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{12}, 1\right), im \cdot im, 2\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right) \cdot \color{blue}{re}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{12}, 1\right), im \cdot im, 2\right) \]

   7. Applied rewrites 62.7%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re \cdot re, 0.004166666666666667\right) \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right) \]

   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. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sin.f64 99.0

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

   4. Applied rewrites 99.0%

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

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

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. cosh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cosh.f64 74.3

         \[\leadsto \left(re \cdot 0.5\right) \cdot \left(2 \cdot \cosh im\right) \]

   4. Applied rewrites 74.3%

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

Alternative 6: 83.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re \cdot re, 0.004166666666666667\right) \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(2 \cdot \cosh im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (*
      (*
       (fma
        (-
         (*
          (fma -9.92063492063492e-5 (* re re) 0.004166666666666667)
          (* re re))
         0.08333333333333333)
        (* re re)
        0.5)
       re)
      (fma (fma (* im im) 0.08333333333333333 1.0) (* im im) 2.0))
     (if (<= t_0 1.0) (sin re) (* (* re 0.5) (* 2.0 (cosh im)))))))

C

double code(double re, double im) {
	double t_0 = (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma(((fma(-9.92063492063492e-5, (re * re), 0.004166666666666667) * (re * re)) - 0.08333333333333333), (re * re), 0.5) * re) * fma(fma((im * im), 0.08333333333333333, 1.0), (im * im), 2.0);
	} else if (t_0 <= 1.0) {
		tmp = sin(re);
	} else {
		tmp = (re * 0.5) * (2.0 * cosh(im));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(fma(Float64(Float64(fma(-9.92063492063492e-5, Float64(re * re), 0.004166666666666667) * Float64(re * re)) - 0.08333333333333333), Float64(re * re), 0.5) * re) * fma(fma(Float64(im * im), 0.08333333333333333, 1.0), Float64(im * im), 2.0));
	elseif (t_0 <= 1.0)
		tmp = sin(re);
	else
		tmp = Float64(Float64(re * 0.5) * Float64(2.0 * cosh(im)));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(N[(-9.92063492063492e-5 * N[(re * re), $MachinePrecision] + 0.004166666666666667), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.08333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * 0.08333333333333333 + 1.0), $MachinePrecision] * N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[Sin[re], $MachinePrecision], N[(N[(re * 0.5), $MachinePrecision] * N[(2.0 * N[Cosh[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(1 + \frac{1}{12} \cdot {im}^{2}\right) \cdot {im}^{2} + 2\right) \]

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 76.4

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot \color{blue}{im}, 2\right) \]

   4. Applied rewrites 76.4%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right)} \]

   5. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{12}, 1\right), im \cdot im, 2\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right) \cdot \color{blue}{re}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{12}, 1\right), im \cdot im, 2\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right) \cdot \color{blue}{re}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{12}, 1\right), im \cdot im, 2\right) \]

   7. Applied rewrites 62.7%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re \cdot re, 0.004166666666666667\right) \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right) \]

   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. Taylor expanded in im around 0

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

      1. lift-sin.f64 98.5

         \[\leadsto \sin re \]

   4. Applied rewrites 98.5%

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

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

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. cosh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cosh.f64 74.3

         \[\leadsto \left(re \cdot 0.5\right) \cdot \left(2 \cdot \cosh im\right) \]

   4. Applied rewrites 74.3%

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

Alternative 7: 81.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re \cdot re, 0.004166666666666667\right) \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.008333333333333333 - 0.16666666666666666, re \cdot re, 1\right) \cdot re\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (*
      (*
       (fma
        (-
         (*
          (fma -9.92063492063492e-5 (* re re) 0.004166666666666667)
          (* re re))
         0.08333333333333333)
        (* re re)
        0.5)
       re)
      (fma (fma (* im im) 0.08333333333333333 1.0) (* im im) 2.0))
     (if (<= t_0 1.0)
       (sin re)
       (*
        (*
         (fma
          (fma
           (fma 0.002777777777777778 (* im im) 0.08333333333333333)
           (* im im)
           1.0)
          (* im im)
          2.0)
         (*
          (fma
           (- (* (* re re) 0.008333333333333333) 0.16666666666666666)
           (* re re)
           1.0)
          re))
        0.5)))))

C

double code(double re, double im) {
	double t_0 = (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma(((fma(-9.92063492063492e-5, (re * re), 0.004166666666666667) * (re * re)) - 0.08333333333333333), (re * re), 0.5) * re) * fma(fma((im * im), 0.08333333333333333, 1.0), (im * im), 2.0);
	} else if (t_0 <= 1.0) {
		tmp = sin(re);
	} else {
		tmp = (fma(fma(fma(0.002777777777777778, (im * im), 0.08333333333333333), (im * im), 1.0), (im * im), 2.0) * (fma((((re * re) * 0.008333333333333333) - 0.16666666666666666), (re * re), 1.0) * re)) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(fma(Float64(Float64(fma(-9.92063492063492e-5, Float64(re * re), 0.004166666666666667) * Float64(re * re)) - 0.08333333333333333), Float64(re * re), 0.5) * re) * fma(fma(Float64(im * im), 0.08333333333333333, 1.0), Float64(im * im), 2.0));
	elseif (t_0 <= 1.0)
		tmp = sin(re);
	else
		tmp = Float64(Float64(fma(fma(fma(0.002777777777777778, Float64(im * im), 0.08333333333333333), Float64(im * im), 1.0), Float64(im * im), 2.0) * Float64(fma(Float64(Float64(Float64(re * re) * 0.008333333333333333) - 0.16666666666666666), Float64(re * re), 1.0) * re)) * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(N[(-9.92063492063492e-5 * N[(re * re), $MachinePrecision] + 0.004166666666666667), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.08333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * 0.08333333333333333 + 1.0), $MachinePrecision] * N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[Sin[re], $MachinePrecision], N[(N[(N[(N[(N[(0.002777777777777778 * N[(im * im), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[(N[(N[(N[(re * re), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision] * 0.5), $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. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(1 + \frac{1}{12} \cdot {im}^{2}\right) \cdot {im}^{2} + 2\right) \]

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 76.4

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot \color{blue}{im}, 2\right) \]

   4. Applied rewrites 76.4%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right)} \]

   5. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{12}, 1\right), im \cdot im, 2\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right) \cdot \color{blue}{re}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{12}, 1\right), im \cdot im, 2\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right) \cdot \color{blue}{re}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{12}, 1\right), im \cdot im, 2\right) \]

   7. Applied rewrites 62.7%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re \cdot re, 0.004166666666666667\right) \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right) \]

   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. Taylor expanded in im around 0

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

      1. lift-sin.f64 98.5

         \[\leadsto \sin re \]

   4. Applied rewrites 98.5%

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

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. 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. lift-sin.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. distribute-rgt-in N/A

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

      9. sub0-neg N/A

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

      10. distribute-rgt-in N/A

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

      11. +-commutative N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in im around 0

      \[\leadsto \left(\color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \cdot \sin re\right) \cdot \frac{1}{2} \]
   5. Step-by-step derivation

      1. cosh-undef-rev N/A

         \[\leadsto \left(\left(\color{blue}{2} + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      2. sub0-neg N/A

         \[\leadsto \left(\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{2} + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      4. +-commutative N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + \color{blue}{2}\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) \cdot {im}^{2} + 2\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right), \color{blue}{{im}^{2}}, 2\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      7. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left({im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) + 1, {\color{blue}{im}}^{2}, 2\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      8. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) \cdot {im}^{2} + 1, {im}^{2}, 2\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}, {im}^{2}, 1\right), {\color{blue}{im}}^{2}, 2\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      10. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360} \cdot {im}^{2} + \frac{1}{12}, {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      11. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, {im}^{2}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      12. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      14. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), {im}^{2}, 2\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      15. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), {im}^{2}, 2\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      16. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot \color{blue}{im}, 2\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      17. lift-*.f64 84.4

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot \color{blue}{im}, 2\right) \cdot \sin re\right) \cdot 0.5 \]

   6. Applied rewrites 84.4%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right)} \cdot \sin re\right) \cdot 0.5 \]

   7. Taylor expanded in re around 0

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \color{blue}{\left(re \cdot \left(1 + {re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right)\right)}\right) \cdot \frac{1}{2} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left(\left(1 + {re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{re}\right)\right) \cdot \frac{1}{2} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left(\left(1 + {re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{re}\right)\right) \cdot \frac{1}{2} \]

      3. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left(\left({re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) + 1\right) \cdot re\right)\right) \cdot \frac{1}{2} \]

      4. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left(\left(\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot {re}^{2} + 1\right) \cdot re\right)\right) \cdot \frac{1}{2} \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left(\mathsf{fma}\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}, {re}^{2}, 1\right) \cdot re\right)\right) \cdot \frac{1}{2} \]

      6. lower--.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left(\mathsf{fma}\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}, {re}^{2}, 1\right) \cdot re\right)\right) \cdot \frac{1}{2} \]

      7. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left(\mathsf{fma}\left({re}^{2} \cdot \frac{1}{120} - \frac{1}{6}, {re}^{2}, 1\right) \cdot re\right)\right) \cdot \frac{1}{2} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left(\mathsf{fma}\left({re}^{2} \cdot \frac{1}{120} - \frac{1}{6}, {re}^{2}, 1\right) \cdot re\right)\right) \cdot \frac{1}{2} \]

      9. pow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{1}{120} - \frac{1}{6}, {re}^{2}, 1\right) \cdot re\right)\right) \cdot \frac{1}{2} \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{1}{120} - \frac{1}{6}, {re}^{2}, 1\right) \cdot re\right)\right) \cdot \frac{1}{2} \]

      11. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{1}{120} - \frac{1}{6}, re \cdot re, 1\right) \cdot re\right)\right) \cdot \frac{1}{2} \]

      12. lift-*.f64 66.1

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.008333333333333333 - 0.16666666666666666, re \cdot re, 1\right) \cdot re\right)\right) \cdot 0.5 \]

   9. Applied rewrites 66.1%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.008333333333333333 - 0.16666666666666666, re \cdot re, 1\right) \cdot re\right)}\right) \cdot 0.5 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 57.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq -0.05:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re \cdot re, 0.004166666666666667\right) \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.008333333333333333 - 0.16666666666666666, re \cdot re, 1\right) \cdot re\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))) -0.05)
   (*
    (*
     (fma
      (-
       (* (fma -9.92063492063492e-5 (* re re) 0.004166666666666667) (* re re))
       0.08333333333333333)
      (* re re)
      0.5)
     re)
    (fma (fma (* im im) 0.08333333333333333 1.0) (* im im) 2.0))
   (*
    (*
     (fma
      (fma
       (fma 0.002777777777777778 (* im im) 0.08333333333333333)
       (* im im)
       1.0)
      (* im im)
      2.0)
     (*
      (fma
       (- (* (* re re) 0.008333333333333333) 0.16666666666666666)
       (* re re)
       1.0)
      re))
    0.5)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp((0.0 - im)) + exp(im))) <= -0.05) {
		tmp = (fma(((fma(-9.92063492063492e-5, (re * re), 0.004166666666666667) * (re * re)) - 0.08333333333333333), (re * re), 0.5) * re) * fma(fma((im * im), 0.08333333333333333, 1.0), (im * im), 2.0);
	} else {
		tmp = (fma(fma(fma(0.002777777777777778, (im * im), 0.08333333333333333), (im * im), 1.0), (im * im), 2.0) * (fma((((re * re) * 0.008333333333333333) - 0.16666666666666666), (re * re), 1.0) * re)) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im))) <= -0.05)
		tmp = Float64(Float64(fma(Float64(Float64(fma(-9.92063492063492e-5, Float64(re * re), 0.004166666666666667) * Float64(re * re)) - 0.08333333333333333), Float64(re * re), 0.5) * re) * fma(fma(Float64(im * im), 0.08333333333333333, 1.0), Float64(im * im), 2.0));
	else
		tmp = Float64(Float64(fma(fma(fma(0.002777777777777778, Float64(im * im), 0.08333333333333333), Float64(im * im), 1.0), Float64(im * im), 2.0) * Float64(fma(Float64(Float64(Float64(re * re) * 0.008333333333333333) - 0.16666666666666666), Float64(re * re), 1.0) * re)) * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.05], N[(N[(N[(N[(N[(N[(-9.92063492063492e-5 * N[(re * re), $MachinePrecision] + 0.004166666666666667), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.08333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * 0.08333333333333333 + 1.0), $MachinePrecision] * N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.002777777777777778 * N[(im * im), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[(N[(N[(N[(re * re), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * re), $MachinePrecision]), $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))) < -0.050000000000000003

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(1 + \frac{1}{12} \cdot {im}^{2}\right) \cdot {im}^{2} + 2\right) \]

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 84.3

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot \color{blue}{im}, 2\right) \]

   4. Applied rewrites 84.3%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right)} \]

   5. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{12}, 1\right), im \cdot im, 2\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right) \cdot \color{blue}{re}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{12}, 1\right), im \cdot im, 2\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right) \cdot \color{blue}{re}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{12}, 1\right), im \cdot im, 2\right) \]

   7. Applied rewrites 41.7%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re \cdot re, 0.004166666666666667\right) \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right) \]

   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. 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. lift-sin.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. distribute-rgt-in N/A

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

      9. sub0-neg N/A

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

      10. distribute-rgt-in N/A

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

      11. +-commutative N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in im around 0

      \[\leadsto \left(\color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \cdot \sin re\right) \cdot \frac{1}{2} \]
   5. Step-by-step derivation

      1. cosh-undef-rev N/A

         \[\leadsto \left(\left(\color{blue}{2} + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      2. sub0-neg N/A

         \[\leadsto \left(\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{2} + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      4. +-commutative N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + \color{blue}{2}\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) \cdot {im}^{2} + 2\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right), \color{blue}{{im}^{2}}, 2\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      7. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left({im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) + 1, {\color{blue}{im}}^{2}, 2\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      8. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) \cdot {im}^{2} + 1, {im}^{2}, 2\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}, {im}^{2}, 1\right), {\color{blue}{im}}^{2}, 2\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      10. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360} \cdot {im}^{2} + \frac{1}{12}, {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      11. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, {im}^{2}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      12. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      14. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), {im}^{2}, 2\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      15. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), {im}^{2}, 2\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      16. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot \color{blue}{im}, 2\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      17. lift-*.f64 93.3

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot \color{blue}{im}, 2\right) \cdot \sin re\right) \cdot 0.5 \]

   6. Applied rewrites 93.3%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right)} \cdot \sin re\right) \cdot 0.5 \]

   7. Taylor expanded in re around 0

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \color{blue}{\left(re \cdot \left(1 + {re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right)\right)}\right) \cdot \frac{1}{2} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left(\left(1 + {re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{re}\right)\right) \cdot \frac{1}{2} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left(\left(1 + {re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{re}\right)\right) \cdot \frac{1}{2} \]

      3. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left(\left({re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) + 1\right) \cdot re\right)\right) \cdot \frac{1}{2} \]

      4. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left(\left(\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot {re}^{2} + 1\right) \cdot re\right)\right) \cdot \frac{1}{2} \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left(\mathsf{fma}\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}, {re}^{2}, 1\right) \cdot re\right)\right) \cdot \frac{1}{2} \]

      6. lower--.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left(\mathsf{fma}\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}, {re}^{2}, 1\right) \cdot re\right)\right) \cdot \frac{1}{2} \]

      7. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left(\mathsf{fma}\left({re}^{2} \cdot \frac{1}{120} - \frac{1}{6}, {re}^{2}, 1\right) \cdot re\right)\right) \cdot \frac{1}{2} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left(\mathsf{fma}\left({re}^{2} \cdot \frac{1}{120} - \frac{1}{6}, {re}^{2}, 1\right) \cdot re\right)\right) \cdot \frac{1}{2} \]

      9. pow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{1}{120} - \frac{1}{6}, {re}^{2}, 1\right) \cdot re\right)\right) \cdot \frac{1}{2} \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{1}{120} - \frac{1}{6}, {re}^{2}, 1\right) \cdot re\right)\right) \cdot \frac{1}{2} \]

      11. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{1}{120} - \frac{1}{6}, re \cdot re, 1\right) \cdot re\right)\right) \cdot \frac{1}{2} \]

      12. lift-*.f64 66.2

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.008333333333333333 - 0.16666666666666666, re \cdot re, 1\right) \cdot re\right)\right) \cdot 0.5 \]

   9. Applied rewrites 66.2%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.008333333333333333 - 0.16666666666666666, re \cdot re, 1\right) \cdot re\right)}\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 52.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(0.5, im \cdot im, 1\right) \cdot \left(\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot im, im, 0.5\right) \cdot im, im, 1\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))) 5e-8)
   (* (fma 0.5 (* im im) 1.0) (* (fma -0.16666666666666666 (* re re) 1.0) re))
   (* (fma (* (fma (* 0.041666666666666664 im) im 0.5) im) im 1.0) re)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp((0.0 - im)) + exp(im))) <= 5e-8) {
		tmp = fma(0.5, (im * im), 1.0) * (fma(-0.16666666666666666, (re * re), 1.0) * re);
	} else {
		tmp = fma((fma((0.041666666666666664 * im), im, 0.5) * im), im, 1.0) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im))) <= 5e-8)
		tmp = Float64(fma(0.5, Float64(im * im), 1.0) * Float64(fma(-0.16666666666666666, Float64(re * re), 1.0) * re));
	else
		tmp = Float64(fma(Float64(fma(Float64(0.041666666666666664 * im), im, 0.5) * im), im, 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[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-8], N[(N[(0.5 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(-0.16666666666666666 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.041666666666666664 * im), $MachinePrecision] * im + 0.5), $MachinePrecision] * im), $MachinePrecision] * im + 1.0), $MachinePrecision] * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 4.9999999999999998e-8

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sin.f64 81.2

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

   4. Applied rewrites 81.2%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 59.1

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

   7. Applied rewrites 59.1%

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

   if 4.9999999999999998e-8 < (*.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. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-sin.f64 80.2

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

   4. Applied rewrites 80.2%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 41.2

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

   7. Applied rewrites 41.2%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 41.2

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

   9. Applied rewrites 41.2%

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

       1. lift-*.f64 N/A

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

       2. lift-fma.f64 N/A

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

       3. pow2 N/A

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

       4. *-commutative N/A

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

       5. pow2 N/A

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

       6. associate-*r* N/A

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

       7. lower-fma.f64 N/A

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

       8. lower-*.f64 41.2

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

   11. Applied rewrites 41.2%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot im, im, 0.5\right) \cdot im, im, 1\right) \cdot re \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 44.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot im, im, 0.5\right) \cdot im, im, 1\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))) -0.05)
   (* (fma -0.16666666666666666 (* re re) 1.0) re)
   (* (fma (* (fma (* 0.041666666666666664 im) im 0.5) im) im 1.0) re)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp((0.0 - im)) + exp(im))) <= -0.05) {
		tmp = fma(-0.16666666666666666, (re * re), 1.0) * re;
	} else {
		tmp = fma((fma((0.041666666666666664 * im), im, 0.5) * im), im, 1.0) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im))) <= -0.05)
		tmp = Float64(fma(-0.16666666666666666, Float64(re * re), 1.0) * re);
	else
		tmp = Float64(fma(Float64(fma(Float64(0.041666666666666664 * im), im, 0.5) * im), im, 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[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.05], N[(N[(-0.16666666666666666 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(N[(N[(0.041666666666666664 * im), $MachinePrecision] * im + 0.5), $MachinePrecision] * im), $MachinePrecision] * im + 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. Taylor expanded in im around 0

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

      1. lift-sin.f64 36.1

         \[\leadsto \sin re \]

   4. Applied rewrites 36.1%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 13.3

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

   7. Applied rewrites 13.3%

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

   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. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-sin.f64 87.6

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

   4. Applied rewrites 87.6%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 63.4

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

   7. Applied rewrites 63.4%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 63.4

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

   9. Applied rewrites 63.4%

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

       1. lift-*.f64 N/A

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

       2. lift-fma.f64 N/A

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

       3. pow2 N/A

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

       4. *-commutative N/A

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

       5. pow2 N/A

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

       6. associate-*r* N/A

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

       7. lower-fma.f64 N/A

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

       8. lower-*.f64 63.4

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

   11. Applied rewrites 63.4%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot im, im, 0.5\right) \cdot im, im, 1\right) \cdot re \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 44.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(im \cdot im\right) \cdot 0.041666666666666664, im \cdot im, 1\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))) 5e-8)
   (* (fma -0.16666666666666666 (* re re) 1.0) re)
   (* (fma (* (* im im) 0.041666666666666664) (* im im) 1.0) re)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp((0.0 - im)) + exp(im))) <= 5e-8) {
		tmp = fma(-0.16666666666666666, (re * re), 1.0) * re;
	} else {
		tmp = fma(((im * im) * 0.041666666666666664), (im * im), 1.0) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im))) <= 5e-8)
		tmp = Float64(fma(-0.16666666666666666, Float64(re * re), 1.0) * re);
	else
		tmp = Float64(fma(Float64(Float64(im * im) * 0.041666666666666664), Float64(im * im), 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[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-8], N[(N[(-0.16666666666666666 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 4.9999999999999998e-8

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

      1. lift-sin.f64 61.1

         \[\leadsto \sin re \]

   4. Applied rewrites 61.1%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 47.0

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

   7. Applied rewrites 47.0%

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

   if 4.9999999999999998e-8 < (*.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. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-sin.f64 80.2

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

   4. Applied rewrites 80.2%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 41.2

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

   7. Applied rewrites 41.2%

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

   8. Taylor expanded in im around inf

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

      1. pow2 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift-*.f64 41.2

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

   10. Applied rewrites 41.2%

       \[\leadsto \mathsf{fma}\left(\left(im \cdot im\right) \cdot 0.041666666666666664, im \cdot im, 1\right) \cdot re \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 40.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(im \cdot im\right) \cdot re, 0.5, re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))) -0.05)
   (* (fma -0.16666666666666666 (* re re) 1.0) re)
   (fma (* (* im im) re) 0.5 re)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp((0.0 - im)) + exp(im))) <= -0.05) {
		tmp = fma(-0.16666666666666666, (re * re), 1.0) * re;
	} else {
		tmp = fma(((im * im) * re), 0.5, re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im))) <= -0.05)
		tmp = Float64(fma(-0.16666666666666666, Float64(re * re), 1.0) * re);
	else
		tmp = fma(Float64(Float64(im * im) * re), 0.5, re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.05], N[(N[(-0.16666666666666666 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(im * im), $MachinePrecision] * re), $MachinePrecision] * 0.5 + 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. Taylor expanded in im around 0

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

      1. lift-sin.f64 36.1

         \[\leadsto \sin re \]

   4. Applied rewrites 36.1%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 13.3

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

   7. Applied rewrites 13.3%

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

   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. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-sin.f64 87.6

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

   4. Applied rewrites 87.6%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 63.4

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

   7. Applied rewrites 63.4%

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

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 57.1

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

   10. Applied rewrites 57.1%

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

Alternative 13: 58.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left(\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot re\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.004166666666666667, re \cdot re, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) 2e-8)
   (*
    (*
     (fma
      (fma
       (fma 0.002777777777777778 (* im im) 0.08333333333333333)
       (* im im)
       1.0)
      (* im im)
      2.0)
     (* (fma -0.16666666666666666 (* re re) 1.0) re))
    0.5)
   (*
    (* (fma (* (* re re) 0.004166666666666667) (* re re) 0.5) re)
    (fma (fma (* im im) 0.08333333333333333 1.0) (* im im) 2.0))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= 2e-8) {
		tmp = (fma(fma(fma(0.002777777777777778, (im * im), 0.08333333333333333), (im * im), 1.0), (im * im), 2.0) * (fma(-0.16666666666666666, (re * re), 1.0) * re)) * 0.5;
	} else {
		tmp = (fma(((re * re) * 0.004166666666666667), (re * re), 0.5) * re) * fma(fma((im * im), 0.08333333333333333, 1.0), (im * im), 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= 2e-8)
		tmp = Float64(Float64(fma(fma(fma(0.002777777777777778, Float64(im * im), 0.08333333333333333), Float64(im * im), 1.0), Float64(im * im), 2.0) * Float64(fma(-0.16666666666666666, Float64(re * re), 1.0) * re)) * 0.5);
	else
		tmp = Float64(Float64(fma(Float64(Float64(re * re) * 0.004166666666666667), Float64(re * re), 0.5) * re) * fma(fma(Float64(im * im), 0.08333333333333333, 1.0), Float64(im * im), 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], 2e-8], N[(N[(N[(N[(N[(0.002777777777777778 * N[(im * im), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[(-0.16666666666666666 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(N[(re * re), $MachinePrecision] * 0.004166666666666667), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * 0.08333333333333333 + 1.0), $MachinePrecision] * N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < 2e-8

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. 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. lift-sin.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. distribute-rgt-in N/A

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

      9. sub0-neg N/A

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

      10. distribute-rgt-in N/A

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

      11. +-commutative N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in im around 0

      \[\leadsto \left(\color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \cdot \sin re\right) \cdot \frac{1}{2} \]
   5. Step-by-step derivation

      1. cosh-undef-rev N/A

         \[\leadsto \left(\left(\color{blue}{2} + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      2. sub0-neg N/A

         \[\leadsto \left(\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{2} + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      4. +-commutative N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + \color{blue}{2}\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) \cdot {im}^{2} + 2\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right), \color{blue}{{im}^{2}}, 2\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      7. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left({im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) + 1, {\color{blue}{im}}^{2}, 2\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      8. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) \cdot {im}^{2} + 1, {im}^{2}, 2\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}, {im}^{2}, 1\right), {\color{blue}{im}}^{2}, 2\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      10. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360} \cdot {im}^{2} + \frac{1}{12}, {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      11. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, {im}^{2}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      12. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      14. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), {im}^{2}, 2\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      15. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), {im}^{2}, 2\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      16. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot \color{blue}{im}, 2\right) \cdot \sin re\right) \cdot \frac{1}{2} \]

      17. lift-*.f64 91.9

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot \color{blue}{im}, 2\right) \cdot \sin re\right) \cdot 0.5 \]

   6. Applied rewrites 91.9%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right)} \cdot \sin re\right) \cdot 0.5 \]

   7. Taylor expanded in re around 0

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)}\right) \cdot \frac{1}{2} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left(\left(1 + \frac{-1}{6} \cdot {re}^{2}\right) \cdot \color{blue}{re}\right)\right) \cdot \frac{1}{2} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left(\left(1 + \frac{-1}{6} \cdot {re}^{2}\right) \cdot \color{blue}{re}\right)\right) \cdot \frac{1}{2} \]

      3. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left(\left(\frac{-1}{6} \cdot {re}^{2} + 1\right) \cdot re\right)\right) \cdot \frac{1}{2} \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{6}, {re}^{2}, 1\right) \cdot re\right)\right) \cdot \frac{1}{2} \]

      5. pow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{6}, re \cdot re, 1\right) \cdot re\right)\right) \cdot \frac{1}{2} \]

      6. lift-*.f64 69.0

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \left(\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot re\right)\right) \cdot 0.5 \]

   9. Applied rewrites 69.0%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \color{blue}{\left(\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot re\right)}\right) \cdot 0.5 \]

   if 2e-8 < (*.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. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(1 + \frac{1}{12} \cdot {im}^{2}\right) \cdot {im}^{2} + 2\right) \]

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 88.2

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot \color{blue}{im}, 2\right) \]

   4. Applied rewrites 88.2%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right)} \]

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 26.4

          \[\leadsto \left(\mathsf{fma}\left(0.004166666666666667 \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right) \]

   7. Applied rewrites 26.4%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.004166666666666667 \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right) \]

   8. Taylor expanded in re around inf

      \[\leadsto \left(\mathsf{fma}\left(\frac{1}{240} \cdot {re}^{2}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{12}, 1\right), im \cdot im, 2\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 26.1

         \[\leadsto \left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.004166666666666667, re \cdot re, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right) \]

   10. Applied rewrites 26.1%

       \[\leadsto \left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.004166666666666667, re \cdot re, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 57.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.01:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot 0.08333333333333333, im \cdot im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot re\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.01)
   (*
    (* (fma (* re re) -0.08333333333333333 0.5) re)
    (fma (* (* im im) 0.08333333333333333) (* im im) 2.0))
   (*
    (*
     (fma
      (fma
       (fma 0.002777777777777778 (* im im) 0.08333333333333333)
       (* im im)
       1.0)
      (* im im)
      2.0)
     re)
    0.5)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.01) {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * re) * fma(((im * im) * 0.08333333333333333), (im * im), 2.0);
	} else {
		tmp = (fma(fma(fma(0.002777777777777778, (im * im), 0.08333333333333333), (im * im), 1.0), (im * im), 2.0) * re) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.01)
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re) * fma(Float64(Float64(im * im) * 0.08333333333333333), Float64(im * im), 2.0));
	else
		tmp = Float64(Float64(fma(fma(fma(0.002777777777777778, Float64(im * im), 0.08333333333333333), Float64(im * im), 1.0), Float64(im * im), 2.0) * re) * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.01], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * 0.08333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.002777777777777778 * N[(im * im), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision] * re), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(1 + \frac{1}{12} \cdot {im}^{2}\right) \cdot {im}^{2} + 2\right) \]

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 89.4

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot \color{blue}{im}, 2\right) \]

   4. Applied rewrites 89.4%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right)} \]

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 25.7

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right) \]

   7. Applied rewrites 25.7%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right) \]

   8. Taylor expanded in im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 25.7

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

   10. Applied rewrites 25.7%

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

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. 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. lift-sin.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. distribute-rgt-in N/A

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

      9. sub0-neg N/A

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

      10. distribute-rgt-in N/A

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

      11. +-commutative N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in re around 0

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

   6. Applied rewrites 74.7%

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

   7. Taylor expanded in im around 0

      \[\leadsto \left(\color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \cdot re\right) \cdot \frac{1}{2} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + \color{blue}{2}\right) \cdot re\right) \cdot \frac{1}{2} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) \cdot {im}^{2} + 2\right) \cdot re\right) \cdot \frac{1}{2} \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right), \color{blue}{{im}^{2}}, 2\right) \cdot re\right) \cdot \frac{1}{2} \]

      4. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left({im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) + 1, {\color{blue}{im}}^{2}, 2\right) \cdot re\right) \cdot \frac{1}{2} \]

      5. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) \cdot {im}^{2} + 1, {im}^{2}, 2\right) \cdot re\right) \cdot \frac{1}{2} \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}, {im}^{2}, 1\right), {\color{blue}{im}}^{2}, 2\right) \cdot re\right) \cdot \frac{1}{2} \]

      7. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360} \cdot {im}^{2} + \frac{1}{12}, {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot re\right) \cdot \frac{1}{2} \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, {im}^{2}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot re\right) \cdot \frac{1}{2} \]

      9. pow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot re\right) \cdot \frac{1}{2} \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot re\right) \cdot \frac{1}{2} \]

      11. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), {im}^{2}, 2\right) \cdot re\right) \cdot \frac{1}{2} \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), {im}^{2}, 2\right) \cdot re\right) \cdot \frac{1}{2} \]

      13. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot \color{blue}{im}, 2\right) \cdot re\right) \cdot \frac{1}{2} \]

      14. lift-*.f64 68.5

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot \color{blue}{im}, 2\right) \cdot re\right) \cdot 0.5 \]

   9. Applied rewrites 68.5%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right)} \cdot re\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 55.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq 5 \cdot 10^{-34}:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.004166666666666667 \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) 5e-34)
   (*
    (* (fma (* re re) -0.08333333333333333 0.5) re)
    (fma (fma (* im im) 0.08333333333333333 1.0) (* im im) 2.0))
   (*
    (*
     (fma
      (- (* 0.004166666666666667 (* re re)) 0.08333333333333333)
      (* re re)
      0.5)
     re)
    (fma im im 2.0))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= 5e-34) {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * re) * fma(fma((im * im), 0.08333333333333333, 1.0), (im * im), 2.0);
	} else {
		tmp = (fma(((0.004166666666666667 * (re * re)) - 0.08333333333333333), (re * re), 0.5) * re) * fma(im, im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= 5e-34)
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re) * fma(fma(Float64(im * im), 0.08333333333333333, 1.0), Float64(im * im), 2.0));
	else
		tmp = Float64(Float64(fma(Float64(Float64(0.004166666666666667 * Float64(re * re)) - 0.08333333333333333), Float64(re * re), 0.5) * re) * fma(im, im, 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], 5e-34], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * 0.08333333333333333 + 1.0), $MachinePrecision] * N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.004166666666666667 * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.08333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < 5.0000000000000003e-34

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(1 + \frac{1}{12} \cdot {im}^{2}\right) \cdot {im}^{2} + 2\right) \]

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 87.8

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot \color{blue}{im}, 2\right) \]

   4. Applied rewrites 87.8%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right)} \]

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 65.4

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right) \]

   7. Applied rewrites 65.4%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right) \]

   if 5.0000000000000003e-34 < (*.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. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(1 + \frac{1}{12} \cdot {im}^{2}\right) \cdot {im}^{2} + 2\right) \]

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 88.1

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot \color{blue}{im}, 2\right) \]

   4. Applied rewrites 88.1%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right)} \]

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 31.1

          \[\leadsto \left(\mathsf{fma}\left(0.004166666666666667 \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right) \]

   7. Applied rewrites 31.1%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.004166666666666667 \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right) \]

   8. Taylor expanded in im around 0

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

      1. pow2 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 29.6

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

   10. Applied rewrites 29.6%

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

Alternative 16: 55.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq 5 \cdot 10^{-34}:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot 0.08333333333333333, im \cdot im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.004166666666666667 \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) 5e-34)
   (*
    (* (fma (* re re) -0.08333333333333333 0.5) re)
    (fma (* (* im im) 0.08333333333333333) (* im im) 2.0))
   (*
    (*
     (fma
      (- (* 0.004166666666666667 (* re re)) 0.08333333333333333)
      (* re re)
      0.5)
     re)
    (fma im im 2.0))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= 5e-34) {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * re) * fma(((im * im) * 0.08333333333333333), (im * im), 2.0);
	} else {
		tmp = (fma(((0.004166666666666667 * (re * re)) - 0.08333333333333333), (re * re), 0.5) * re) * fma(im, im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= 5e-34)
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re) * fma(Float64(Float64(im * im) * 0.08333333333333333), Float64(im * im), 2.0));
	else
		tmp = Float64(Float64(fma(Float64(Float64(0.004166666666666667 * Float64(re * re)) - 0.08333333333333333), Float64(re * re), 0.5) * re) * fma(im, im, 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], 5e-34], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * 0.08333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.004166666666666667 * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.08333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < 5.0000000000000003e-34

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(1 + \frac{1}{12} \cdot {im}^{2}\right) \cdot {im}^{2} + 2\right) \]

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 87.8

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot \color{blue}{im}, 2\right) \]

   4. Applied rewrites 87.8%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right)} \]

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 65.4

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right) \]

   7. Applied rewrites 65.4%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right) \]

   8. Taylor expanded in im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 65.1

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

   10. Applied rewrites 65.1%

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

   if 5.0000000000000003e-34 < (*.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. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(1 + \frac{1}{12} \cdot {im}^{2}\right) \cdot {im}^{2} + 2\right) \]

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 88.1

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot \color{blue}{im}, 2\right) \]

   4. Applied rewrites 88.1%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right)} \]

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 31.1

          \[\leadsto \left(\mathsf{fma}\left(0.004166666666666667 \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right) \]

   7. Applied rewrites 31.1%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.004166666666666667 \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right) \]

   8. Taylor expanded in im around 0

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

      1. pow2 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 29.6

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

   10. Applied rewrites 29.6%

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

Alternative 17: 55.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.01:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot im, im, 0.5\right) \cdot im, im, 1\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.01)
   (* (* (fma (* re re) -0.08333333333333333 0.5) re) (fma im im 2.0))
   (* (fma (* (fma (* 0.041666666666666664 im) im 0.5) im) im 1.0) re)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.01) {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * re) * fma(im, im, 2.0);
	} else {
		tmp = fma((fma((0.041666666666666664 * im), im, 0.5) * im), im, 1.0) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.01)
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re) * fma(im, im, 2.0));
	else
		tmp = Float64(fma(Float64(fma(Float64(0.041666666666666664 * im), im, 0.5) * im), im, 1.0) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.01], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.041666666666666664 * im), $MachinePrecision] * im + 0.5), $MachinePrecision] * im), $MachinePrecision] * im + 1.0), $MachinePrecision] * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(1 + \frac{1}{12} \cdot {im}^{2}\right) \cdot {im}^{2} + 2\right) \]

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 89.4

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot \color{blue}{im}, 2\right) \]

   4. Applied rewrites 89.4%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right)} \]

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 25.7

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right) \]

   7. Applied rewrites 25.7%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right) \]

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. lower-fma.f64 24.4

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

   10. Applied rewrites 24.4%

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

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

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-sin.f64 83.6

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

   4. Applied rewrites 83.6%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 65.2

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

   7. Applied rewrites 65.2%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 65.2

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

   9. Applied rewrites 65.2%

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

       1. lift-*.f64 N/A

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

       2. lift-fma.f64 N/A

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

       3. pow2 N/A

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

       4. *-commutative N/A

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

       5. pow2 N/A

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

       6. associate-*r* N/A

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

       7. lower-fma.f64 N/A

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

       8. lower-*.f64 65.2

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

   11. Applied rewrites 65.2%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot im, im, 0.5\right) \cdot im, im, 1\right) \cdot re \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 34.2% accurate, 18.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot re \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (fma -0.16666666666666666 (* re re) 1.0) re))

C

double code(double re, double im) {
	return fma(-0.16666666666666666, (re * re), 1.0) * re;
}

Julia

function code(re, im)
	return Float64(fma(-0.16666666666666666, Float64(re * re), 1.0) * re)
end

Wolfram

code[re_, im_] := N[(N[(-0.16666666666666666 * N[(re * re), $MachinePrecision] + 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. Taylor expanded in im around 0

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

   1. lift-sin.f64 51.0

      \[\leadsto \sin re \]

4. Applied rewrites 51.0%

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

5. Taylor expanded in re around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 34.2

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

7. Applied rewrites 34.2%

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

Alternative 19: 26.3% accurate, 317.0× speedup

Math

\[\begin{array}{l} \\ re \end{array} \]

FPCore

(FPCore (re im) :precision binary64 re)

C

double code(double re, double im) {
	return re;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(re, im):
	return re

Julia

function code(re, im)
	return re
end

MATLAB

function tmp = code(re, im)
	tmp = re;
end

Wolfram

code[re_, im_] := re

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in im around 0

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

   1. lift-sin.f64 51.0

      \[\leadsto \sin re \]

4. Applied rewrites 51.0%

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

5. Taylor expanded in re around 0

   \[\leadsto re \]
6. Step-by-step derivation

7. Applied rewrites 26.3%

   \[\leadsto re \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025095 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))